Celebratio Mathematica — Chung

[1]K. L. Chung and B. Liu: “Elementary proof of a theorem in probability theory,” Mathematical Magazine 1 : 4 (1936). In Chinese. article

[2]K. L. Chung: “Two remarks on Viggo Brun’s method,” Sci. Rep. Nat. Tsing Hua Univ. (A) 4 (1940), pp. 249–255. MR 0004838 Zbl 0024.24901 article

[3]K. L. Chung: “Sur un théorème de M. Gumbel,” C. R. Acad. Sci. Paris 210 (1940), pp. 620–621. MR 0002727 Zbl 0023.33805 article

[4]K.-L. Chung: “Note on a theorem on quadratic residues,” Bull. Amer. Math. Soc. 47 (1941), pp. 514–516. MR 0004823 Zbl 0027.15605 article

[5]K. L. Chung: “On the probability of the occurrence of at least \( m \) events among \( n \) arbitrary events,” Ann. Math. Statistics 12 : 3 (September 1941), pp. 328–338. MR 0005538 Zbl 0026.32902 article

[6]K.-L. Chung: “A generalization of an inequality in the elementary theory of numbers,” J. Reine Angew. Math. 183 (1941), pp. 193–196. MR 0005511 Zbl 0025.30503 article

[7]K.-L. Chung: “On mutually favorable events,” Ann. Math. Statistics 13 : 3 (September 1942), pp. 338–349. MR 0007205 Zbl 0060.28313 article

[8]K. L. Chung: “Further results on probabilities of a finite number of events,” Ann. Math. Statistics 14 : 3 (September 1943), pp. 234–237. MR 0008652 Zbl 0060.28312 article

[9]K.-L. Chung: “On fundamental systems of probabilities of a finite number of events,” Ann. Math. Statistics 14 : 2 (June 1943), pp. 123–133. MR 0008651 Zbl 0060.28311 article

[10]K. L. Chung: “Generalization of Poincaré’s formula in the theory of probability,” Ann. Math. Statistics 14 (1943), pp. 63–65. MR 0008124 Zbl 0060.28309 article

[11]K.-L. Chung and L. C. Hsu: “A combinatorial formula and its application to the theory of probability of arbitrary events,” Ann. Math. Statistics 16 : 1 (March 1945), pp. 91–95. MR 0011895 Zbl 0060.28310 article

[12]P. L. Hsu and K. L. Chung: “Sur un théorème de probabilités dénombrables,” C. R. Acad. Sci. Paris 223 (1946), pp. 467–469. In French. MR 0016563 Zbl 0060.29308 article

[13]K.-L. Chung: “The approximate distribution of Student’s statistic,” Ann. Math. Statistics 17 : 4 (December 1946), pp. 447–465. MR 0018390 Zbl 0063.00894 article

It is well known that various statistics of a large sample (of size \( n \)) are approximately distributed according to the normal law. The asymptotic expansion of the distribution of the statistic in a series of powers of \( n^{-1/2} \) with a remainder term gives the accuracy of the approximation. H. Cramér [1937] first obtained the asymptotic expansion of the mean, and recently P. L. Hsu [1945] has obtained that of the variance of a sample. In the present paper we extend the Cramér–Hsu method to Student’s statistic. The theorem proved states essentially that if the population distribution is non-singular and if the existence of a sufficient number of moments is assumed, then an asymptotic expansion can be obtained with the appropriate remainder. The first four terms of the expansion are exhibited in formula (35).

[14]K. L. Chung: “On the maximum partial sum of independent random variables,” Proc. Nat. Acad. Sci. U. S. A. 33 : 5 (May 1947), pp. 132–136. MR 0021671 Zbl 0029.15203 article

[15]K.-l. Chung: “Note on some strong laws of large numbers,” Amer. J. Math. 69 : 1 (January 1947), pp. 189–192. MR 0019853 Zbl 0034.07103 article

[16]K.-L. Chung and P. Erdős: “On the lower limit of sums of independent random variables,” Ann. of Math. (2) 48 : 4 (October 1947), pp. 1003–1013. MR 0023010 Zbl 0029.15202 article

Let \( X_1, X_2, \dots, X_n, \dots \) be independent random variables and let \( S_n = \sum_{\nu=1}^n X_{\nu} \). In the so-called law of the iterated logarithm, completely solved by Feller recently, the upper limit of \( S_n \) as \( n\to\infty \) is considered and its true order of magnitude is found with probability one. A counterpart to that problem is to consider the lower limit of \( S_n \) as \( n\to\infty \) and to make a statement about its order of magnitude with probability one.

Let \( X_1,\dots,X_n,\dots \) be independent random variables with the common distribution: \( \mathrm{Pr}(X_n=1) = p \), \( \mathrm{Pr}(X_n=0)=1-p=q \). Let \( \psi(n)\downarrow\infty \) and \[ \sum_{n=1}^{\infty} \frac{1}{n\psi(n)}=\infty .\] Then we have \[ \mathrm{Pr}\bigl(\varliminf_{n\to\infty}n^{1/2}\psi(n)\,|S_n-np|=0\bigr)=1 .\]

The theorem is best possible.

[17]K.-L. Chung: On the maximum partial sum of independent random variables. Ph.D. thesis, Princeton University, 1947. Advised by H. Cramér. MR 2937763 phdthesis

[18]K.-L. Chung: “On a lemma by Kolmogoroff,” Ann. Math. Statistics 19 : 1 (March 1948), pp. 88–91. MR 0023472 Zbl 0041.24805 article

[19]K. L. Chung: “Asymptotic distribution of the maximum cumulative sum of independent random variables,” Bull. Amer. Math. Soc. 54 : 12 (1948), pp. 1162–1170. MR 0027972 Zbl 0035.08601 article

[20]K. L. Chung: “On the maximum partial sums of sequences of independent random variables,” Trans. Amer. Math. Soc. 64 (1948), pp. 205–233. MR 0026274 Zbl 0032.17102 article

[21]K. L. Chung and G. A. Hunt: “On the zeros of \( \sum^n_ 1\pm 1 \),” Ann. of Math. (2) 50 : 2 (April 1949), pp. 385–400. MR 0029488 Zbl 0032.41701 article

[22]K. L. Chung and W. Feller: “On fluctuations in coin-tossing,” Proc. Nat. Acad. Sci. U. S. A. 35 : 10 (1949), pp. 605–608. MR 0033459 article

[23]K.-L. Chung: “An estimate concerning the Kolmogoroff limit distribution,” Trans. Amer. Math. Soc. 67 : 1 (September 1949), pp. 36–50. MR 0034552 Zbl 0034.22602 article

We consider a sequence of independent random variables having the common distribution function \( F(x) \) which is assumed to be continuous. Let \( nF_n(x) \) denote the number of random variables among the first \( n \) of the sequence whose values do not exceed \( x \). Write \[ d_n = \sup_{-\infty < x < \infty}|n(F_n(x)-F(x))| .\] Kolmogoroff [1933] proved that the probability \begin{equation*}\tag{1} P(d_n\leq\lambda n^{1/2}), \end{equation*} where \( \lambda \) is a positive constant, tends as \( n\to\infty \) uniformly in \( \lambda \) to the limiting distribution \begin{equation*}\tag{2} \Phi(\lambda)=\sum_{-\infty}^{\infty}(-1)^j e^{-2j^2\lambda^2}. \end{equation*} Smirnoff [1939] extended this result and recently [Feller 1948] has given new proofs of these theorems. In this paper we shall obtain an estimate of the difference between (1) and (2) as a function of \( n \), valid not only for \( \lambda \) equal to a constant but also for \( \lambda \) equal to a function \( \lambda(n) \) of \( n \) which does not grow too fast.

[24]K. L. Chung: “Fluctuations of sums of independent random variables,” Ann. of Math. (2) 51 : 3 (May 1950), pp. 697–706. MR 0035410 Zbl 0037.08301 article

One aspect of the theory of addition of independent random variables is the frequency with which the partial sums change sign. Investigations of this nature were originated by Paul Lévy, in a paper [1939] which contains a wealth of ideas. This problem as such was mentioned by Feller in his 1945 address. In the case where the partial sums can actually vanish the problem falls under the head of “recurrent events,” a general theory of which was recently developed in a paper by Feller [1949]. A very special case had been studied in detail by Hunt and myself [Chung and Hunt 1949]. Generalizing the problem in a natural way we shall consider the number of times \( T_n \) with which the sequence of reduced partial sums \( S_k-E(S_k) \), \( k=1,2,\dots, n \) crosses a given value \( c \). We shall establish the limiting distribution of \( T_n \) in the case where the random variables have a common distributino with a finite third absolute moment.

[25]K. L. Chung and W. H. J. Fuchs: “On the distribution of values of sums of random variables,” Mem. Amer. Math. Soc. 1951 : 6 (1951). 12 pages. MR 0040610 Zbl 0042.37502 article

[26]K. L. Chung and P. Erdős: “Probability limit theorems assuming only the first moment, I,” Mem. Amer. Math. Soc., 1951 : 6 (1951), pp. 19. MR 0040612 Zbl 0042.37601 article

[27]K. L. Chung: “The strong law of large numbers,” pp. 341–352 in Proceedings of the second Berkeley symposium on mathematical statistics and probability, 1950 (Berkeley, CA, July 31–August 12, 1950). Edited by J. Neyman. University of California Press (Berkeley, CA and Los Angeles, CA), 1951. MR 0045328 Zbl 0044.13701 inproceedings

[28]K. L. Chung and M. Kac: “Remarks on fluctuations of sums of independent random variables,” Mem. Amer. Math. Soc. 1951 : 6 (1951), pp. 11. MR 0040611 Zbl 0042.37503 article

[29]K. L. Chung and P. Erdős: “On the application of the Borel–Cantelli lemma,” Trans. Amer. Math. Soc. 72 (1952), pp. 179–186. MR 0045327 Zbl 0046.35203 article

[30]K. L. Chung and J. Wolfowitz: “On a limit theorem in renewal theory,” Ann. of Math. (2) 55 : 1 (January 1952), pp. 1–6. MR 0044769 Zbl 0047.12402 article

Let \( X \) be a random variable which assumes only integral values and define for all \( n \), \( p_n=P\{X=n\} \), where \( P\{\ \} \) is the probability of the event in braces. Let \( \{X_i\} \), \( i=1,2,\dots \) ad inf. be an infinite sequence of independent random variables with the same distribution as \( X \). Define \[ S_j=\sum_{i=1}^j X_i \quad\text{and}\quad u_n=\sum_{j=1}^{\infty}P\{S_j=n\} .\] Let \( m=E(X) \) be the expectation of \( X \). Let \( t \) be the absolute value of the greatest common divisor of all indices \( n \) such that \( p_n > 0 \). We shall prove the follow theorem: If \( 0 < m\leq\infty \), then \[ \lim_{n\to\infty}u_{nt}=t/m \quad\text{and}\quad \lim_{n\to -\infty} u_{nt} = 0 .\] If \( -\infty\leq m < 0 \) then the limits given above should be interchanged.

[31]K. L. Chung: “On the renewal theorem in higher dimensions,” Skand. Aktuarietidskr. 35 (1952), pp. 188–194. MR 0054879 Zbl 0048.11103 article

Renewal theory has been treated by many pure and applied mathematicians. Among the former we may mention Feller, Täcklind and Doob. The principal limit theorem (for one-dimensional, positive, lattice random variables) was however proved earlier by Kolmogorov in 1936 as the ergodic theorem for denumerable Markov chains. A partial result for the non-lattice case was first proved by Doob using the theory of Markov processes, and the complete result by Blackwell. The extension of the renewal theorem to random variables taking both positive and negative values was first given by Wolfowitz and the author [Chung and Wolfowitz 1952], for the lattice case. A partial result for the non-lattice case, using a purely analytical approach, was obtained by Pollard and the author [Chung and Pollard 1952]. For the literature see [Chung and Wolfowitz 1952].

[32]K. L. Chung: “An ergodic theorem for stationary Markov chains with a countable number of states,” pp. 568 in Proceedings of International Congress of Mathematicians (Cambridge, MA, August 30–September 6, 1950). Edited by L. M. Graves, E. Hille, P. A. Smith, and O. Zariski. American Mathematical Society (Providence, RI), 1952. incollection

Paul Althaus Smith	Related
Oscar Zariski	Related
Lawrence Murray Graves	Related
Carl Einar Hille	Related

[33]K. L. Chung and H. Pollard: “An extension of renewal theory,” Proc. Amer. Math. Soc. 3 (1952), pp. 303–309. MR 0048734 Zbl 0047.12403 article

[34]K. L. Chung: “Corrections to my paper ‘Fluctuations of sums of independent random variables’,” Ann. of Math. (2) 57 (1953), pp. 604–605. MR 0053424 Zbl 0051.35601 article

[35]K. L. Chung: “Contributions to the theory of Markov chains,” J. Research Nat. Bur. Standards 50 : 4 (April 1953), pp. 203–208. Research Paper 2411. MR 0055608 Zbl 0053.27201 article

The fundamentals of the theory of denumerable Markov chains with stationary transition probabilities were laid down by Kolmogorov and further work was done by Doblin. The theory of recurrent events of Feller is closely related, if not coextensive. Some new results obtained by T. E. Harris turn out to tie up very nicely with some amplifications of Doblin’s work. Harris was led to consider the probabilities of hitting one state before another, starting from a third one. This idea of considering three states, one initial, one “taboo”, and one final, is more fully developed in the present work. The notion of first passage time of the “union” or “intersection” of two states is also introduced here. The interplay between these notions is illustrated.

[36]K. L. Chung and M. Kac: “Corrections to the paper ‘Remarks on fluctuations of sums of independent random variables’,” Proc. Amer. Math. Soc. 4 : 4 (August 1953), pp. 560–563. MR 0056228 Zbl 0050.35304 article

[37]K. L. Chung: “Sur les lois de probabilité unimodales,” C. R. Acad. Sci. Paris 236 (1953), pp. 583–584. MR 0053419 Zbl 0050.13702 article

[38]K. L. Chung: “On a stochastic approximation method,” Ann. Math. Statistics 25 : 3 (1954), pp. 463–483. MR 0064365 Zbl 0059.13203 article

Asymptotic properties are established for the Robbins–Monro [1951] procedure of stochastically solving the equation \( M(x) = \alpha \). Two disjoint cases are treated in detail. The first may be called the “bounded” case, in which the assumptions we make are similar to those in the second case of Robbins and Monro. The second may be called the “quasi-linear” case which restricts \( M(x) \) to lie between two straight lines with finite and nonvanishing slopes but postulates only the boundedness of the moments of \( Y(x) - M(x) \). In both cases it is shown how to choose the sequence \( \{a_n\} \) in order to establish the correct order of magnitude of the moments of \( x_n - \theta \). Asymptotic normality of \( a^{1/2}_n(x_n - \theta) \) is proved in both cases under a further assumption. The case of a linear \( M(x) \) is discussed to point up other possibilities. The statistical significance of our results is sketched.

[39]B. V. Gnedenko and A. N. Kolmogorov: Limit distributions for sums of independent random variables. Addison-Wesley Mathematics Series. Addison-Wesley (Cambridge, MA), 1954. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. MR 0062975 Zbl 0056.36001 book

Joseph L. Doob	Related	Volume
Andrey Nikolaevich Kolmogorov	Related
B. V. Gnedenko	Related

[40]K. L. Chung: “Contributions to the theory of Markov chains, II,” Trans. Amer. Math. Soc. 76 (1954), pp. 397–419. MR 0063603 Zbl 0058.34602 article

[41]K. L. Chung: “Foundations of the theory of continuous parameter Markov chains,” pp. 29–40 in Proceedings of the third Berkeley symposium on mathematical statistics and probability, 1954–1955, vol. II: Contributions to probability theory. Edited by J. Neyman. University of California Press (Berkeley, CA and Los Angeles, CA), 1956. MR 0084884 Zbl 0071.35001 inproceedings

This paper is exclusively concerned with continuous parameter Markov processes with a denumerably infinite number of states and stationary transition matrix function. The foundations of the proper theory of such processes, as distinguished from that of the discrete parameter version, or of Markov processes which are either more special (for example, general state space; nonstationary transition matrix function), were laid by Doob [1942; 1945], and Lévy [1951; 1952; 1953]. Roughly speaking it was Lévy in 1952 who drew, in his inimitable way, the comprehensive picture while Doob, ten years earlier, had supplied the essential ingredients. The present effor aims at a synthesis of the most fundamental parts of the theory, made possible by the contributions of these two authors. While the results given here generally extend and clarify those in the cited literature, immense credit must go to Professors Lévy and Doob for the inspiration of their pioneer work. To them I am also indebted for much valuable discussion through correspondence and conversation. An attempt is made in the presentation to be quite formal and rigorous, in the spirit of Doob’s already classic treatise [1953]. Further developments of the theory will be published elsewhere.

[42]K. L. Chung: “Some new developments in Markov chains,” Trans. Amer. Math. Soc. 81 (1956), pp. 195–210. MR 0075481 Zbl 0075.14001 article

The purpose of this paper is to point out some new connections between the sample function behavior and the analytical properties of the transition probability functions of a continuous parameter Markov chain with stationary transition probabilities. The main idea is that of a post-exit process derived from the original process by considering its evolution after the exit from a stable state. This leads to various relations between conditional probabilities all of which are “intuitively obvious” but require sometimes painstaking proofs if probability theory like any other branch of mathematics is to be treated as a discipline in logic. These relations imply certain analytical properties of the transition functions, obtained recently by D. G. Austin by purely analytical means, and exhibit them in connection with other quantities introduced in this paper. They also complete some results due to Doob and Lévy. The well-known differential equations of Kolmogorov are seen to be limiting cases of certain more generally valid differential equations involving a continuous parameter. One of these expresses the fundamental transition property of the post-exit process, and the other a similar property of the renewal density of an imbedded renewal process.

[43]K. L. Chung and C. Derman: “Non-recurrent random walks,” Pacific J. Math. 6 : 3 (1956), pp. 441–447. MR 0081587 Zbl 0072.35301 article

Let \( \{X_i\},\ i=1,2,\dots \) be a sequence of independent and identically distributed integral valued random variables such that 1 is the absolute value of the greatest common divisor of all values of \( x \) for which \( P(X_i=x) > 0 \). Define \( S_n=\sum_{i=1}^n X_i \). Chung and Fuchs [1951] showed that if \( x \) is any integer, \( S_n=x \) infinitely often with probability 1 according as \( EX_i=0 \) or \( \neq 0 \), provided that \( E|X_i| < \infty \). Let \( 0 < EX_i < \infty \), and \( A \) denote a set of integers containing an infinite number of positive integers. It will be shown that any such set \( A \) will be visited infinitely often with probability 1 by the sequence \( \{S_n\}\ n=1,2,\dots \). Conditions are given so that similar results hold for the case where \( X_i \) has a continuous distribution and the set \( A \) is a Lebesgue measurable set whose intersection with the positivfe real numbers has infinite Lebesgue measure.

[44]K. L. Chung: “Some aspects of continuous parameter Markov chains,” Publ. Inst. Statist. Univ. Paris 6 (1957), pp. 271–287. MR 0102128 Zbl 0082.34502 article

[45]K. L. Chung: “On a basic property of Markov chains,” Ann. of Math. (2) 68 : 1 (July 1958), pp. 126–149. MR 0100924 Zbl 0228.60023 article

[46]K. L. Chung, P. Erdős, and T. Sirao: “On the Lipschitz’s condition for Brownian motion,” J. Math. Soc. Japan 11 : 4 (1959), pp. 263–274. MR 0121873 Zbl 0091.13301 article

T. Sirao	Related
Paul Erdős	Related

[47]K. L. Chung: Markov chains with stationary transition probabilities. Die Grundlehren der mathematischen Wissenschaften 104. Springer (Berlin), 1960. MR 0116388 Zbl 0092.34304 book

[48]K.-L. Chung: “Continuous parameter Markov chains,” pp. 510–517 in Proceedings of the International Congress of Mathematicians 1958 (Edinburgh, 14–21 August, 1958). Edited by J. A. Todd. Cambridge University Press (New York), 1960. MR 0121839 Zbl 0122.36601 incollection

[49]K. L. Chung: “On last exit times,” Illinois J. Math. 4 : 4 (1960), pp. 629–639. MR 0130715 Zbl 0094.32102 article

[50]K. L. Chung: “A note on the ergodic theorem of information theory,” Ann. Math. Statist. 32 : 2 (1961), pp. 612–614. MR 0131782 Zbl 0115.35503 article

[51]K. L. Chung: “Some remarks on taboo probabilities,” Illinois J. Math. 5 : 3 (1961), pp. 431–435. MR 0131902 Zbl 0101.11601 article

[52]K. L. Chung: “Probabilistic methods in Markov chains,” pp. 35–56 in Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, vol. II. Edited by J. Neyman. University of California Press (Berkeley, CA), 1961. MR 0131901 Zbl 0106.33303 incollection

[53]K. L. Chung and D. Ornstein: “On the recurrence of sums of random variables,” Bull. Amer. Math. Soc. 68 (1962), pp. 30–32. MR 0133148 Zbl 0104.11904 article

[54]K. L. Chung: “On the exponential formulas of semi-group theory,” Math. Scand 10 (1962), pp. 153–162. MR 0145355 Zbl 0106.31201 article

The purpose of this paper is to present a simple unified approach to a group of theorems in semi-group theory called the “exponential formulas”, due to Hille, Phillips, Widder and D. G. Kendall [Hille and Phillips 1957, p. 354]. A more general and apparently new formula is arrivefd at, which includes some known cases. It turns out that these formulas are in essence summability methods which are best comprehended from the point of view of elementary probability theory. They are all in the spirit of S. Bernstein’s proof of Weierstrass’s approximation theorem, the same idea being present in M. Riesz’s proof of Hille’s first exponential formula (see [Hille et al. 1957, p. 314]). Whereas the details here are just a little simpler than in [Hille and Phillips 1957], it seems of some interest to exhibit the general pattern and to reduce the proofs to routine verifications. The reader who is not acquainted with the language of probability should have no difficulty in everything into the language of classical analysis. But the probability way of thinking is really germane to the subject.

[55]K. L. Chung: “On the Martin boundary for Markov chains,” Proc. Nat. Acad. Sci. U.S.A. 48 : 6 (June 1962), pp. 963–968. MR 0145584 Zbl 0228.60030 article

[56]K. L. Chung: “On the boundary theory for Markov chains,” Acta Math. 110 (1963), pp. 19–77. MR 0150829 Zbl 0292.60121 article

[57]K. L. Chung: Odnorodnye cepi Markova. Mir (Moscow), 1964. In Russian.

[58]K. L. Chung: “The general theory of Markov processes according to Doeblin,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 : 3 (1964), pp. 230–254. MR 0166835 Zbl 0119.34604 article

[59]K. L. Chung: Odnorodnye cepi Markova. Mir (Moscow), 1964. In Russian. Zbl 0121.12901 book

[60]K. L. Chung: “Sur une équation de convolution,” C. R. Acad. Sci. Paris 260 (1965), pp. 4665–4667. MR 0187283 Zbl 0145.15401 article

[61]K. L. Chung and J. L. Doob: “Fields, optionality and measurability,” Amer. J. Math. 87 : 2 (April 1965), pp. 397–424. MR 0214121 Zbl 0192.24604 article

[62]K. L. Chung: “Sur une équation de convolution,” C. R. Acad. Sci. Paris 260 (1965), pp. 6794–6796. MR 0187284 Zbl 0145.15402 article

[63]K. L. Chung: “On the boundary theory for Markov chains, II,” Acta Math. 115 : 1 (1966), pp. 111–163. MR 0189117 Zbl 0315.60041 article

[64]K. L. Chung: “Markov processes with infinities,” pp. 23–32 in Markov processes and potential theory (Mathematical Research Center (U.S. Army) at University of Wisconsin, Madison, WI, May 1–3, 1967). Edited by J. Chover. Wiley (New York), 1967. MR 0231447 Zbl 0189.17703 incollection

[65]K. L. Chung: Markov chains with stationary transition probabilities, 2nd edition. Die Grundlehren der mathematischen Wissenschaften 104. Springer (New York, NY), 1967. MR 0217872 Zbl 0146.38401 book

[66]K. L. Chung: A course in probability theory. Harcourt, Brace & World (New York), 1968. MR 0229268 Zbl 0159.45701 book

[67]B. V. Gnedenko and A. N. Kolmogorov: Limit distributions for sums of independent random variables, revised edition. Addison-Wesley, 1968. Translated from the Russian, annotated, and revised by K. L. Chung. With appendices by J. L. Doob and P. L. Hsu. MR 0233400 book

Joseph L. Doob	Related	Volume
B. V. Gnedenko	Related
Pao Lu Hsü	Related
Andrey Nikolaevich Kolmogorov	Related

[68]K. L. Chung and J. B. Walsh: “To reverse a Markov process,” Acta Math. 123 : 1 (1969), pp. 225–251. MR 0258114 Zbl 0187.41302 article

[69]K. L. Chung: “On diverse questions of time reversing in Markov chains (and processes),” pp. 165–175 in Proceedings of the twelfth biennial seminar of the Canadian Mathematical Congress on time series and stochastic processes, convexity and combinatorics (University of British Columbia, Vancouver, BC, August 11–27, 1969). Edited by R. Pyke. Canadian Mathematical Congress (Montreal), 1970. MR 0310976 Zbl 0235.60061 inproceedings

[70]K. L. Chung: Lectures on boundary theory for Markov chains. Annals of Mathematics Studies 65. Princeton University Press, 1970. With the cooperation of Paul-André Meyer. MR 0267644 Zbl 0204.51003 book

[71]K. L. Chung: “A simple proof of Doob’s convergence theorem,” pp. 76 in Séminaire de probabilités V (Université de Strasbourg, année universitaire 1969–1970). Edited by A. Dold and B. Eckmann. Lecture Notes in Mathematics 191. Springer (Berlin), 1971. MR 0383551 incollection

Albrecht Dold	Related
B. Eckmann	Related

[72]K. L. Chung: “Boundary behavior of Markov chains and its contributions to general processes,” pp. 499–505 in Actes du Congrès International des Mathématiciens (Nice, France, September 1–10, 1970), Tome 2. Gauthier-Villars (Paris), 1971. MR 0420882 Zbl 0298.60047 incollection

[73]K. L. Chung: “The Poisson process as renewal process,” pp. 41–48 in Collection of articles dedicated to the memory of Alfréd Rényi, I, published as Period. Math. Hungar. 2 : 1–4 (1972). MR 0345232 Zbl 0278.60061 incollection

[74]K. L. Chung: “On the fundamental hypotheses of Hunt processes,” pp. 43–52 in Convegno di calcolo delle probabilità (INDAM, Rome, March–April, 1971). Edited by F. Severi. Symposia Mathematica IX. Academic Press (London), 1972. MR 0359019 Zbl 0242.60031 incollection

[75]K. L. Chung: “Crudely stationary counting processes,” Amer. Math. Monthly 79 : 8 (October 1972), pp. 867–877. MR 0305466 Zbl 0253.60043 article

The theorems by Khintchine, Korolyuk, and Dobrushin in the theory of stationary point processes are basic and simple theorems. Korolyuk’s theorem was originally derived from the Palm–Khintchine formulas; a direct proof was given in Cramér–Leadbetter [1967]. Its real simplicity seems to be obscured by the slightly complicated presentation of the proof. The same may be said of the proof of Dobrushin’s theorem involving an unnecessary contraposition as well as some epsilonics. Both results become quiet transparent when dealt with by standard methods of measure and integration in sample space. After all, these are problems of probability theory and nowadays students spend a lot of time learning this kind of “abstract” set-up. It would be a pity not to use the knowledge so acquired in straightforward situations such as these theorems. In doing so we arrive at certain natural extensions which seem to put the results in proper perspective. The results in \( R^d \), obtained by the same method, seem to be new.

[76]K. L. Chung: “Some universal field equations,” pp. 90–97 in Séminaire de Probabilités VI (Université de Strasbourg, année universitaire 1970–1971). Edited by A. Dold and B. Eckmann. Lecture Notes in Mathematics 258. Springer (Berlin), 1972. MR 0372973 Zbl 0238.60034 incollection

Albrecht Dold	Related
B. Eckmann	Related

[77]K. L. Chung: “Probabilistic approach in potential theory to the equilibrium problem,” Ann. Inst. Fourier (Grenoble) 23 : 3 (1973), pp. 313–322. MR 0391277 Zbl 0258.31012 article

[78]K. L. Chung: “Note on last exit decomposition,” pp. 20–21 in Séminaire de Probabilités VIII (Université de Strasbourg, année universitaire 1972–1973). Edited by C. Dellarcherie, P. Meyer, and M. Weil. Lecture Notes in Mathematics 381. Springer (Berlin), 1974. MR 0378117 Zbl 0295.60056 incollection

M. Weil	Related
Paul-André Meyer	Related
Claude Dellacherie	Related

[79]K. L. Chung: Elementary probability theory with stochastic processes. Undergraduate Texts in Mathematics. Springer (New York), 1974. MR 0362408 Zbl 0293.60001 book

[80]K. L. Chung and J. B. Walsh: “Meyer’s theorem on predictability,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 29 : 3 (1974), pp. 253–256. MR 0370787 Zbl 0284.60069 article

[81]K. L. Chung: “An expression for canonical entrance laws,” pp. 311–314 in Stochastic analysis: A tribute to the memory of Rollo Davidson. Edited by D. G. Kendall and E. F. Harding. Wiley Series in Probability and Statistics. Wiley (London), 1974. MR 0368189 incollection

Rollo Davidson	Related
David George Kendall	Related
E. F. Harding	Related

[82]K. L. Chung: A course in probability theory, 2nd edition. Probability and Mathematical Statistics 21. Academic Press (New York–London), 1974. MR 0346858 Zbl 0345.60003 book

[83]K. L. Chung: “Remarks on equilibrium potential and energy,” pp. xiv, 131–138 in Collection of articles dedicated to Marcel Brelot on the occasion of his 70th birthday, published as Ann. Inst. Fourier (Grenoble) 25 : 3–4 (1975). MR 0405601 Zbl 0338.31001 incollection

[84]K. L. Chung: “A bivariate distribution in regeneration,” J. Appl. Probability 12 : 4 (December 1975), pp. 837–839. MR 0386023 Zbl 0326.60034 article

[85]K. L. Chung: Elementary probability theory with stochastic processes, 2nd edition. Undergraduate Texts in Mathematics. Springer (New York), 1975. MR 0378007 Zbl 0328.60001 book

[86]K. L. Chung: “Maxima in Brownian excursions,” Bull. Amer. Math. Soc. 81 : 4 (July 1975), pp. 742–745. MR 0373035 Zbl 0325.60077 article

[87]P. J. Brockwell and K. L. Chung: “Emptiness times of a dam with stable input and general release function,” J. Appl. Probability 12 : 1 (March 1975), pp. 212–217. MR 0368231 Zbl 0316.60066 article

[88]K. L. Chung: “Approximation of a continuous parameter Markov chain by discrete skeletons,” Bull. London Math. Soc. 7 : 1 (1975), pp. 71–76. MR 0388540 Zbl 0301.60046 article

The approximation of a given continuous parameter Markov chain by its discrete skeletons raises many interesting questions. Some beginnings of this study were given in §11.13 of [Chung 1967], but little progress seems to have been made since. It is possible that the apparent impenetrability of the “embedding problem” (when is a discrete parameter chain a skeleton?) may have unwarrantedly hampered reasonable and useful investigations. The present note deals with a relatively easy aspect of the approximation theory: the global faithfulness of the skeletal approach as reflected in the moments of the entrance time distributions. Even so our method leans heavily on recurrence. When this hypothesis is dropped these distributions may be improper, but moments can still be defined by excluding the mass at infinity as done in §1.11 of [Chung 1967] in the discrete parameter case. Solidarity results such as Theorem 1 there can be generalized to the continuous parameter case without the intervention of skeletons (see the remarks concerning the last part of the theorem below); but approximation seems an open problem.

[89]K. L. Čžun: “A proof of a lemma of A. V. Skorohod,” Teor. Verojatnost. i Mat. Statist. 15 (1976), pp. 151–152, 157. In Russian. MR 0415718 Zbl 0398.60012 article

[90]K. L. Chung: “Excursions in Brownian motion,” Ark. Mat. 14 : 1–2 (1976), pp. 155–177. Dedicated to the Memory of Paul Lévy. MR 0467948 Zbl 0356.60033 article

[91]K. L. Chung and R. Durrett: “Downcrossings and local time,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 35 : 2 (1976), pp. 147–149. MR 0405605 Zbl 0348.60111 article

Let \( \{W(t):t\geq 0\} \) be the standard Brownian motion with all paths continuous. Let \( M(t)= \max_{0\leq s\leq t}W(s) \) be the maximum process and \( Y(t)=M(t)-W(t) \) be reflecting Brownian motion. If \( d_{\varepsilon}(t) \) is the number of times \( Y \) crosses down from \( \varepsilon \) to 0 before time \( t \), then it was Paul Lévy’s idea that \begin{equation*}\tag{1} P\bigl\{\lim_{\varepsilon\to 0} \varepsilon d_{\varepsilon}(t) = M(t),\ \forall t\geq 0\bigr\} = 1. \end{equation*} In [1974] Itô and McKean demonstrated the almost sure convergence of \( \varepsilon d_{\varepsilon}(t) \) using martingale methods. To identify the limit they used the hard fact, due to Lévy, that \begin{equation*}\tag{2} P\Bigl\{\lim_{\varepsilon\to 0} \frac{ \text{measure} \{s: Y(s) < \varepsilon,\ s\leq t\} }{2\varepsilon}=M(t), \ \forall t\geq 0\Bigr\} = 1 \end{equation*} and computed the second moment of the difference of the expressions in (1) and (2). In this paper, by examining the excursions in Brownian motion and using a new formula for the distribution of their maxima, we obtain a direct identification of the limit in (1) without using (2).

[92]K. L. Chung: “Pedagogic notes on the barrier theorem,” pp. 27–33 in Séminaire de probabilités XI (Université de Strasbourg, année universitaire 1975–1976). Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 581. Springer (Berlin), 1977. MR 0445620 Zbl 0376.60080 incollection

M. Weil	Related
Paul-André Meyer	Related
Claude Dellacherie	Related

[93]K. L. Chung and R. K. Getoor: “The condenser problem,” Ann. Probability 5 : 1 (February 1977), pp. 82–86. MR 0428480 Zbl 0381.60064 article

[94]K. L. Chung: Elementare Wahrscheinlichkeitstheorie und stochastische Prozesse, 2nd edition. Hochschultext. Springer (Berlin), 1978. MR 0517119 Zbl 0375.60002 book

[95]K. L. Chung and M. Rao: On existence of a dual process. Technical report 25, Mathematics Institute, Aarhus University, 1978. 23 pages. Zbl 0392.60052 techreport

[96]K. L. Chung: “Proof of a lemma of A. V. Skorohod,” Theory Probab. Math. Stat. 15 (1978), pp. 156. English translation of Russian original. Zbl 0449.60007 article

[97]K. L. Chung and M. Rao: Potential theory without duality. Technical report 22, Mathematics Institute, Aarhus University, 1978. 15 pages. Zbl 0369.31006 techreport

[98]K. L. Chung: “Correction to: ‘Pedagogic notes on the barrier theorem’,” pp. 739 in Séminaire de probabilités XII (Université de Strasbourg, année universitaire 1976–1977). Lecture Notes in Mathematics 649. Springer (Berlin), 1978. MR 520034 incollection

[99]T. W. Anderson, K. L. Chung, and E. L. Lehmann: “Pao Lu Hsü: 1909–1970,” Ann. Statist. 7 : 3 (1979), pp. 467–470. MR 527480 article

Erich Leo Lehmann	Related
Theodore W. Anderson	Related
Pao Lu Hsü	Related

[100]K. L. Chung: “Hsü’s work in probability,” Ann. Statist. 7 : 3 (May 1979), pp. 479–483. MR 527483 article

[101]K. L. Chung and J. Glover: “Left continuous moderate Markov processes,” Z. Wahrsch. Verw. Gebiete 49 : 3 (1979), pp. 237–248. MR 547825 Zbl 0413.60063 article

[102]K. L. Chung: Elementary probability theory with stochastic processes, 3rd edition. Undergraduate Texts in Mathematics. Springer (New York), 1979. MR 560506 Zbl 0404.60002 book

[103]K. L. Chung and S. R. S. Varadhan: “Kac functional and Schrödinger equation,” Studia Math. 68 : 3 (1980), pp. 249–260. MR 599148 Zbl 0448.60054 article

[104]K. L. Chung and K. M. Rao: “Sur la théorie du potentiel avec la fonctionnelle de Feynman–Kac,” C. R. Acad. Sci. Paris Sér. A-B 290 : 13 (1980), pp. A629–A631. MR 572652 Zbl 0439.31005 article

[105]T. W. Anderson, K. L. Chung, and E. L. Lehmann: “Pao Lu Hsü: 1910–1970,” Knowledge Practice Math. 1980 : 3 (1980), pp. 3–5. MR 615332 Zbl 0567.01028 article

Erich Leo Lehmann	Related
Theodore W. Anderson	Related
Pao Lu Hsü	Related

[106]K. L. Chung: “Hsu Pao-Lu’s work with respect to probability theory,” Knowledge Practice Math. 3 (1980), pp. 12–15. In Chinese. MR 615335 Zbl 0524.01015 article

[107]K. L. Chung and M. Rao: “Equilibrium and energy,” Probab. Math. Statist. 1 : 2 (1980), pp. 99–108. MR 626304 Zbl 0502.60060 article

In this paper it is shown that the equilibrium measure \( \nu \) for a compact \( K \) in potential theory can be related with a unique invariant measure \( \pi \) for a discrete time Markov process by the formula \[ \pi(dy) = \phi(y)\,\nu(dy) .\] The chain has the transition function \( L(x,A) \), where \( L \) is the last-exit kernel in [Chung 1973]. For a general non-symmetric potential density \( u \) the modified energy \[ I(\lambda) = \iint \lambda(dx)\,u(x,y)\,\phi(y)^{-1}(dy) \] and the Gauss quadratic \[ G(\lambda) = I(\lambda) - 2\lambda(K) \] are introduced. Then \( G \) is minimized by \( \pi \) among all signed measures \( \lambda \) on \( K \) of finite modified energy, provided \( I \) is positive. This includes the classical symmetric case of Newtonian and M. Riesz potentials as a special case. The modification corresponds to a time change for the underlying Markov process. The positivity of \( I \) is established for a class of signed measures associated with continuous additive functionals in the sense of Revuz.

[108]K. L. Chung and T. Lindvall: “On recurrence of a random walk in the plane,” Proc. Amer. Math. Soc. 78 : 2 (1980), pp. 285–287. MR 550514 Zbl 0433.60072 article

[109]R. G. Miller: “Editorial note on: ‘Pao Lu Hsü: 1909–1970’ by T. W. Anderson, K. L. Chung and E. L. Lehmann,” Ann. Statist. 8 : 2 (1980), pp. 456. MR 560742 article

[110]K. L. Chung and K. M. Rao: “A new setting for potential theory, I,” Ann. Inst. Fourier (Grenoble) 30 : 3 (1980), pp. 167–198. MR 597022 Zbl 0424.31004 article

In Hunt’s theory of Markov processes certain duality assumptions are made to generalize well known classical potential theoretic results such as F. Riesz representation theorem, uniqueness, existence of equilibrium potential, etc. A standard treatment developed by several subsequent authors can be found in [Blumenthal and Getoor 1968]. In a different direction, it was shown in [Chung 1973] under simple analytic conditions that the equilibrium measure is inherently linked to the last exit distribution of the process. It thus appears feasible that this last result, namely on “equilibrium principle” may be made the starting point to which other major results are related.

In this paper we exploit the line of thought in [Chung 1973] to derive some of these major results under the same set of conditions as in [Chung 1973]. It turns out that these sets of conditions automatically imply the existence of a dual. However, this will not be proved here.

[111]K. L. Chung: “On stopped Feynman–Kac functionals,” pp. 347–356 in Séminaire de probabilités XIV (Paris, 1978–1979). Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 784. Springer (Berlin), 1980. MR 580141 Zbl 0444.60061 incollection

Marc Yor	Related	Volume
Jacques Azéma	Related

[112]K. L. Chung and K. M. Rao: “Feynman–Kac functional and the Schrödinger equation,” pp. 1–29 in Seminar on stochastic processes, 1981 (Northwestern University, Evanston, IL, April 1981). Edited by E. Çinlar, K. L. Chung, and R. K. Getoor. Progress in Probability and Statistics 1. Birkhäuser (Boston, MA), 1981. MR 647779 Zbl 0492.60073 incollection

Erhan Çinlar	Related
K. Murali Rao	Related
Ronald K. Getoor	Related

[113]A. D. Wentzell: A course in the theory of stochastic processes. McGraw-Hill (New York), 1981. With a foreword by K. L. Chung. Translated from the Russian by S. Chomet. MR 614594 book

[114]K. L. Chung: “Brownian motion on the line, I,” J. Math. Res. Exposition 1 : 1 (1981), pp. 74–82. MR 617355 Zbl 0583.60074 article

[115]K. L. Chung and K. M. Rao: “Erratum: ‘A new setting for potential theory, I’,” Ann. Inst. Fourier (Grenoble) 31 : 2 (1981). loose errata. MR 617248 article

[116]Seminar on stochastic processes, 1981 (Northwestern University, Evanston, IL, April 1981). Edited by E. Çinlar, K. L. Chung, and R. K. Getoor. Progress in Probability and Statistics 1. Birkhäuser (Boston, MA), 1981. MR 647778 Zbl 0471.00025 book

Erhan Çinlar	Related
Ronald K. Getoor	Related

[117]K. L. Chung: “Review of two scientific communications,” J. Math. Res. Exposition 2 : 3 (1982), pp. 141–142. In Chinese. article

[118]K. L. Chung: “A cluster of great formulas,” Acta Math. Acad. Sci. Hungar. 39 : 1–3 (1982), pp. 65–67. MR 653674 Zbl 0495.60020 article

[119]K. L. Chung: Lectures from Markov processes to Brownian motion. Grundlehren der Mathematischen Wissenschaften 249. Springer (New York), 1982. MR 648601 Zbl 0503.60073 book

[120]K. L. Chung: “Brownian motion on the line, II,” J. Math. Res. Exposition 2 : 3 (1982), pp. 87–98. MR 672218 Zbl 0583.60075 article

[121]Seminar on stochastic processes, 1982 (Northwestern University, Evanston, IL, March 1982). Edited by E. Çinlar, K. L. Chung, and R. K. Getoor. Progress in Probability and Statistics 5. Birkhäuser (Boston, MA), 1983. MR 733662 Zbl 0516.00023 book

Erhan Çinlar	Related
Ronald K. Getoor	Related

[122]K. L. Chung, R. Durrett, and Z. X. Zhao: “Extension of domains with finite gauge,” Math. Ann. 264 : 1 (1983), pp. 73–79. MR 709862 Zbl 0519.60080 article

Zhong-xin Zhao	Related
Richard Timothy Durrett	Related

[123]P.-L. Hsü: Collected papers. Edited by K. L. Chung. Springer (New York), 1983. With the cooperation of Ch’ing Shui Chêng and Tse Pei Chiang. MR 682921 Zbl 0508.01028 book

Pao Lu Hsü	Related
Tse Pei Chiang	Related
Ch'ing Shui Chêng	Related

[124]K. L. Chung and R. J. Williams: Introduction to stochastic integration. Progress in Probability and Statistics 4. Birkhäuser (Boston, MA), 1983. MR 711774 Zbl 0527.60058 book

[125]K. L. Chung: “An inequality for boundary value problems,” pp. 111–122 in Seminar on stochastic processes, 1982 (Northwestern University, Evanston, IL, March 1982). Edited by E. Çinlar, K. L. Chung, and R. K. Getoor. Progress in Probability and Statistics 5. Birkhäuser (Boston, MA), 1983. MR 733667 Zbl 0528.60071 incollection

Erhan Çinlar	Related
Ronald K. Getoor	Related

[126]K. L. Chung: “Properties of finite gauge with an application to local time,” pp. 16–24 in Probability and mathematical statistics: essays in honour of Carl-Gustav Esseen (Uppsala, Sweden, 1983). Edited by A. Gut and L. Holst. Uppsala University, 1983. MR 727123 Zbl 0519.60081 incollection

A. Gut	Related
L. Holst	Related
Carl-Gustav Esseen	Related

[127]K. L. Chung and Z. X. Zhao: “Some probabilistic results on the Sturm–Liouville equation,” J. Systems Sci. Math. Sci. 4 : 4 (1984), pp. 320–324. MR 820144 Zbl 0591.60075 article

[128]K. L. Chung: “The lifetime of conditional Brownian motion in the plane,” Ann. Inst. H. Poincaré Probab. Statist. 20 : 4 (1984), pp. 349–351. MR 771894 Zbl 0573.60070 article

[129]K. L. Chung: “Mathematics and applications,” Math. Medley 12 : 1 (1984), pp. 1–8. MR 781647 article

[130]Seminar on stochastic processes, 1983 (University of Florida, Gainesville, FL, 1983). Edited by E. Çinlar, K. L. Chung, and R. K. Getoor. Progress in Probability and Statistics 7. Birkhäuser (Boston, MA), 1984. MR 902407 Zbl 0546.00025 book

Erhan Çinlar	Related
Ronald K. Getoor	Related

[131]K. L. Chung, M. Liao, and K. M. Rao: “Duality under a new setting,” pp. 23–38 in Seminar on stochastic processes, 1983 (University of Florida, Gainesville, FL, 1983). Edited by E. Çinlar, K. L. Chung, and R. K. Getoor. Progress in Probability and Statistics 7. Birkhäuser (Boston, MA), 1984. MR 902410 Zbl 0558.60056 incollection

Erhan Çinlar	Related
Ronald K. Getoor	Related
K. Murali Rao	Related
Ming Liao	Related

[132]K. L. Chung: “Conditional gauges,” pp. 17–22 in Seminar on stochastic processes, 1983 (University of Florida, Gainesville, FL, 1983). Edited by E. Çinlar, K. L. Chung, and R. K. Getoor. Progress in Probability and Statistics 7. Birkhäuser (Boston, MA), 1984. MR 902409 Zbl 0552.60073 incollection

Erhan Çinlar	Related
Ronald K. Getoor	Related

[133]K. L. Chung: “The gauge and conditional gauge theorem,” pp. 496–503 in Séminaire de probabilités XIX (Université de Strasbourg, année universitaire, 1983–84). Edited by J. Azéma. Lecture Notes in Mathematics 1123. Springer (Berlin), 1985. MR 889497 Zbl 0561.60084 incollection

[134]K. L. Chung and Z. X. Zhao: “A simple proof of a new theorem on conditional Brownian motion,” Kexue Tongbao (Chinese) 30 : 5 (1985), pp. 329–332. MR 803587 article

[135]K. L. Chung: “Probabilistic approach to boundary value problems for Schrödinger’s equation,” Exposition. Math. 3 : 2 (1985), pp. 175–178. MR 816403 Zbl 0577.60066 article

[136]K. L. Chung: “A probability anecdote,” J. Math. Res. Exposition 5 : 2 (1985), pp. 141–143. MR 842116 article

[137]K. L. Chung and P. Li: “Comparison of probability and eigenvalue methods for the Schrödinger equation,” pp. 25–34 in Probability, statistical mechanics, and number theory. Edited by G.-C. Rota. Advances in Mathematics Supplementary Studies 9. Academic Press (Orlando, FL), 1986. Volume dedicated to Mark Kac. MR 875443 Zbl 0608.60061 incollection

Peter Li	Related
Gian-Carlo Rota	Related
Marek Kac	Related

[138]K. L. Chung and P. Hsu: “Gauge theorem for the Neumann problem,” pp. 63–70 in Seminar on stochastic processes, 1984 (Northwestern University, Evanston, IL, 1984). Edited by E. Çinlar, K. L. Chung, and R. K. Getoor. Progress in Probability and Statistics 9. Birkhäuser (Boston, MA), 1986. MR 896722 Zbl 0585.60064 incollection

Erhan Çinlar	Related
Pei Hsu	Related
Ronald K. Getoor	Related

[139]K. L. Chung, P. Li, and R. J. Williams: “Comparison of probability and classical methods for the Schrödinger equation,” Exposition. Math. 4 : 3 (1986), pp. 271–278. MR 880127 Zbl 0623.60097 article

Peter Li	Related
Ruth J. Williams	Related

[140]K. L. Chung: “Notes on the inhomogeneous Schrödinger equation,” pp. 55–62 in Seminar on stochastic processes, 1984 (Northwestern University, Evanston, IL, 1984). Edited by E. Çinlar, K. L. Chung, and R. K. Getoor. Progress in Probability and Statistics 9. Birkhäuser (Boston, MA), 1986. MR 896721 Zbl 0585.60063 incollection

Erhan Çinlar	Related
Ronald K. Getoor	Related

[141]K. L. Chung: “Remark on the conditional gauge theorem,” pp. 423–425 in Séminaire de Probabilités XX (Université de Strasbourg, année universitaire 1984/85). Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 1204. Springer (Berlin), 1986. MR 942036 incollection

Marc Yor	Related	Volume
Jacques Azéma	Related

[142]Seminar on stochastic processes, 1984 (Northwestern University, Evanston, IL, 1984). Edited by E. Çinlar, K. L. Chung, and R. K. Getoor. Progress in Probability and Statistics 9. Birkhäuser (Boston, MA), 1986. MR 896718 Zbl 0575.00017 book

Erhan Çinlar	Related
Ronald K. Getoor	Related

[143]K. L. Chung: “‘Markov chain must have a beginning.’ In memory of Prof. Loo Keng Hua,” J. Math. Res. Expo. 1 (1986), pp. 1–3. MR 858088 Zbl 0632.60064 article

[144]K. L. Chung: “Doubly-Feller process with multiplicative functional,” pp. 63–78 in Seminar on stochastic processes, 1985 (University of Florida, Gainesville, FL, March, 1985). Edited by E. Çinlar, K. L. Chung, R. K. Getoor, and J. Glover. Progress in Probability and Statistics 12. Birkhäuser (Boston, MA), 1986. MR 896735 Zbl 0603.60066 incollection

Erhan Çinlar	Related
Ronald K. Getoor	Related
Joseph Glover	Related

[145]Seminar on stochastic processes, 1985 (University of Florida, Gainesville, FL, March, 1985). Edited by E. Çinlar, K. L. Chung, R. K. Getoor, and J. Glover. Progress in Probability and Statistics 12. Birkhäuser (Boston, MA), 1986. MR 896731 Zbl 0591.00011 book

Erhan Çinlar	Related
Ronald K. Getoor	Related
Joseph Glover	Related

[146]G. L. Alexanderson and L. H. Lange: “Obituary: George Pólya,” Bull. London Math. Soc. 19 : 6 (1987), pp. 559–608. K. L. Chung provided an essay on Pólya’s contributions to probability as part of this extended obituary, covering pp. 570–576. MR 915433 Zbl 0611.01012 article

L. H. Lange	Related
George Pólya	Related
Gerald Lee Alexanderson	Related

[147]K. L. Chung: “Green’s function for a ball,” pp. 1–13 in Seminar on stochastic processes, 1986 (University of Virginia, Charlottesville, VA, March, 1986). Edited by E. Çinlar, K. L. Chung, R. K. Getoor, and J. Glover. Progress in Probability and Statistics 13. Birkhäuser (Boston, MA), 1987. MR 902422 Zbl 0643.31007 incollection

Erhan Çinlar	Related
Ronald K. Getoor	Related
Joseph Glover	Related

[148]Seminar on stochastic processes, 1986 (University of Virginia, Charlottesville, VA, March, 1986). Edited by E. Çinlar, K. L. Chung, R. K. Getoor, and J. Glover. Progress in Probability and Statistics 13. Birkhäuser (Boston, MA), 1987. MR 902421 Zbl 0607.00014 book

Erhan Çinlar	Related
Ronald K. Getoor	Related
Joseph Glover	Related

[149]K. Chung and R. Williams: Vvedenie v stokhasticheskoe integrirovanie. Mir (Moscow), 1987. MR 901297 Zbl 0647.60060 book

[150]K. L. Chung: “Carleson’s proof of a remarkable equation,” J. Math. Res. Expo. 7 : 1 (1987), pp. 117–123. MR 928088 article

[151]K. L. Chung: “Reminiscences of some of Paul Lévy’s ideas in Brownian motion and in Markov chains,” pp. 29–36 in Colloque Paul Lévy sur les processus stochastiques (Palaiseau, France, 22–26 June, 1987). Edited by M. E. Métivier, J. E. Neveu, and S. R. S. Varadhan. Astérisque 157–158. 1988. MR 976211 Zbl 0665.60076 incollection

S. R. S. Varadhan	Related	Volume
Paul Lévy	Related
Michel E. Métivier	Related
Jacques E. Neveu	Related

[152]Seminar on stochastic processes, 1987 (Princeton University, Princeton, NJ, March 26–28, 1987). Edited by E. Çinlar, K. L. Chung, R. K. Getoor, and J. Glover. Progress in Probability and Statistics 15. Birkhäuser (Boston, MA), 1988. MR 1046408 Zbl 0635.00011 book

Erhan Çinlar	Related
Ronald K. Getoor	Related
Joseph Glover	Related

[153]K. L. Chung and K. M. Rao: “General gauge theorem for multiplicative functionals,” Trans. Amer. Math. Soc. 306 : 2 (April 1988), pp. 819–836. MR 933320 Zbl 0647.60083 article

We generalize our previous work on the gauge theorem and its various consequences and complements, initiated in [Chung and Rao 1981] and somewhat extended by subsequent investigations (see [Chung and Zhao]). The generalization here is two-fold. First, instead of the Brownian motion, the underlying process is now a fairly broad class of Markov processes, not necessarily having continuous paths. Second, instead of the Feynman–Kac functional, the exponential of a general class of additive functionals is treated. The case of Schrödinger operator \( \Delta/2+\nu \), where \( \nu \) is a suitable measure, is a simple special case. The most general operator, not necessarily a differential one, which may arise from our potential equations is briefly discussed toward the end of the paper. Concrete instances of applications in this case should be of great interest.

[154]K. L. Chung: “Probability methods in potential theory,” pp. 42–54 in Potential theory — surveys and problems (Prague, July 19–24, 1987). Edited by J. Král, J. Lukeš, I. Netuka, and J. Veselý. Lecture Notes in Mathematics 1344. Springer (Berlin), 1988. MR 973880 Zbl 0677.60085 incollection

J. Veselý	Related
J. Král	Related
J. Lukeš	Related
I. Netuka	Related

[155]K. L. Chung: “Gauge theorem for unbounded domains,” pp. 87–98 in Seminar on stochastic processes, 1988 (University of Florida, Gainesville, FL, March 3–5, 1988). Edited by E. Çinlar, K. L. Chung, R. K. Getoor, and J. Glover. Progress in Probability 17. Birkhäuser (Boston, MA), 1989. MR 990475 Zbl 0689.60076 incollection

Erhan Çinlar	Related
Ronald K. Getoor	Related
Joseph Glover	Related

[156]Seminar on stochastic processes, 1988 (University of Florida, Gainesville, FL, March 3–5, 1988). Edited by E. Çinlar, K. L. Chung, R. K. Getoor, and J. Glover. Progress in Probability 17. Birkhäuser (Boston, MA), 1989. MR 990471 Zbl 0659.00010 book

Erhan Çinlar	Related
Ronald K. Getoor	Related
Joseph Glover	Related

[157]K. L. Chung: “Reminiscences of some of Paul Lévy’s ideas in Brownian motion and in Markov chains,” pp. 99–107 in Seminar on stochastic processes, 1988 (University of Florida, Gainesville, FL, March 3–5, 1988). Edited by E. Çinlar, K. L. Chung, R. K. Getoor, and J. Glover. Birkhäuser (Boston, MA), 1989. MR 990476 Zbl 0677.60002 incollection

Erhan Çinlar	Related
Ronald K. Getoor	Related
Joseph Glover	Related
Paul Lévy	Related

[158]K. L. Chung and R. J. Williams: Introduction to stochastic integration, 2nd edition. Probability and its Applications. Birkhäuser (Boston, MA), 1990. MR 1102676 Zbl 0725.60050 book

[159]Seminar on stochastic processes, 1989 (University of California, San Diego, CA, March 30–April 1, 1989). Edited by E. Çinlar, K. L. Chung, R. K. Getoor, P. J. Fitzsimmons, and R. J. Williams. Progress in Probability 18. Birkhäuser (Boston, MA), 1990. MR 1042337 Zbl 0686.00017 book

Erhan Çinlar	Related
P. J. Fitzsimmons	Related
Ronald K. Getoor	Related
Ruth J. Williams	Related

[160]A. A. Balkema and K. L. Chung: “Paul Lévy’s way to his local time,” pp. viii+351 in Seminar on stochastic processes, 1990 (University of British Columbia, Vancouver, BC, May 10–12, 1990). Edited by E. Çinlar, P. J. Fitzsimmons, and R. J. Williams. Progress in Probability 24. Birkhäuser (Boston, MA), 1991. MR 1118437 Zbl 0748.60066 incollection

Erhan Çinlar	Related
A. A. Balkema	Related
P. J. Fitzsimmons	Related
Paul Lévy	Related
Ruth J. Williams	Related

[161]Seminar on stochastic processes, 1991 (University of California, Los Angeles, CA, March 23–25, 1991). Edited by E. Çinlar, K. L. Chung, M. J. Sharpe, P. Fitzsimmons, S. Port, and T. Liggett. Progress in Probability 29. Birkhäuser (Boston, MA), 1992. Dedicated to the memory of Steven Orey. MR 1172137 Zbl 0745.00038 book

Erhan Çinlar	Related
P. J. Fitzsimmons	Related
Thomas Milton Liggett	Related	Volume
M. J. Sharpe	Related
Steven Orey	Related
S. Port	Related

[162]K. L. Chung: “Greenian bounds for Markov processes,” Potential Anal. 1 : 1 (1992), pp. 83–92. MR 1245226 Zbl 1081.60543 article

In Section 1, a temporal bound is estimated by a spatial bound, for a Markov process whose transition density satisfies a simple condition. This includes the Brownian motion, for which comparison with a more special method is made. In Section 2, the result is related to the Green operator and examples are given. In Section 3, the result is applied to an old problem of eigenvalues of the Laplacian. In Section 4, recent extensions from the Laplacian to the Schrödinger circle-of-ideas are briefly described. In this case, time is measured by an exponential functional of the process, commonly known under the names Feynman–Kac.

This article was written as script for two talks, one general and one technical, given in Taiwan in January 1991. In the spirit of the occasion, the style of exposition is deliberately paced and discursive.

[163]Seminar on stochastic processes, 1992 (University of Washington, Seattle, WA, March 26–28, 1992). Edited by E. Çinlar, K. L. Chung, M. J. Sharpe, R. F. Bass, and K. Burdzy. Progress in Probability 33. Birkhäuser (Boston, MA), 1993. MR 1278074 Zbl 0780.00020 book

Erhan Çinlar	Related
R. F. Bass	Related
Krzysztof Burdzy	Related
M. J. Sharpe	Related

[164]K. L. Chung: “Reminiscences of one of Doeblin’s papers,” pp. 65–71 in Doeblin and modern probability (Blaubeuren, Germany, November 2–7, 1991). Contemporary Mathematics 149. American Mathematical Society (Providence, RI), 1993. Zbl 0788.60081 incollection

[165]K. L. Chung: “Doeblin’s big limit theorem,” J. Theoret. Probab. 6 : 3 (1993), pp. 417–426. Article contributed to further celebrate the conference “50 Years after Doeblin: Developments in the theory of Markov chains, Markov processes and sums of random variables” (Blaubeuren, Germany, November 2–7, 1991). MR 1230338 Zbl 0786.60099 article

[166]K. L. Chung: “Stochastic analysis of \( Q \)-matrix,” pp. 3–25 in Selected topics on stochastic modelling. Edited by R. Gutiérrez and M. J. Valderrama. World Scientific (River Edge, NJ), 1994. MR 1320971 incollection

M. J. Valderrama	Related
R. Gutiérrez	Related

[167]K. L. Chung and Z. X. Zhao: From Brownian motion to Schrödinger’s equation. Grundlehren der Mathematischen Wissenschaften 312. Springer (Berlin), 1995. MR 1329992 Zbl 0819.60068 book

[168]K. L. Chung: Green, Brown, and probability. World Scientific (River Edge, NJ), 1995. MR 1371379 Zbl 0871.60001 book

[169]K. L. Chung: “Multinomial ratio: Paul Erdős solves a problem,” Methods Appl. Anal. 5 : 2 (1998), pp. 143–156. MR 1636550 Zbl 0915.60037 article

[170]K. L. Chung: “Probability and Doob,” Amer. Math. Monthly 105 : 1 (January 1998), pp. 28–35. MR 1613994 Zbl 0918.60007 article

[171]K. L. Chung: “Sul problema del ritorno all’equilibrio” [On the return to equilibrium], Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 : 3 (1999), pp. 213–218. In Italian. MR 1769158 Zbl 1008.60015 article

[172]B. Bru, M.-F. Bru, and K. L. Chung: “Borel et la martingale de Saint-Pétersbourg,” Rev. Histoire Math. 5 : 2 (1999), pp. 181–247. MR 1793102 Zbl 0979.01018 article

On se propose d’examiner sur un exemple, le paradoxe de Saint-Pétersbourg, la façon dont Borel « expose » la science de son temps. La première partie indique sommairement la place singulière de la vulgarisation dans l’oeuvre de Borel. Les deux parties suivantes présentent dans l’ordre chronologique les contributions boréliennes au paradoxe de Saint-Pétersbourg qui s’échelonnent sur plus de cinquante ans; elles indiquent comment Borel aborde le problème en le replaçant dans une réflexion au long cours, scientifiquement très riche, sur le « paradoxe des martingales », ces systèmes de jeu qui prétendent faire la fortune d’un joueur au jeu de pile ou face. Borel donne de ce problème une solution originale qui anticipe une égalité fondamentale de la théorie mathématique naissante des martingales. On signale en particulier le rôle paradoxal joué par Félix Le Dantec dans le développement de la pensée borélienne sur ces thèmes. Une annexe rétablit en langue moderne les « martingales de Borel ».

Marie-France Bru	Related
Armand Borel	Related
Bernard Bru	Related

[173]K. L. Chung and J.-C. Zambrini: Introduction to random time and quantum randomness. Monographs of the Portuguese Mathematical Society 1. Amadora (McGraw-Hill) (Lisbon), 2001. MR 1937277 Zbl 1021.81500 book

[174]K. L. Chung: A course in probability theory, 3rd edition. Academic Press (San Diego, CA), 2001. MR 1796326 Zbl 0980.60001 book

[175]K. L. Chung and Z. Zhao: From Brownian motion to Schrödinger’s equation, 2nd, corrected edition. Grundlehren der Mathematischen Wissenschaften 312. Springer (Berlin), 2001. Zbl 1195.60002 book

[176]K. L. Chung: Green, Brown, and probability & Brownian motion on the line, 2nd edition. World Scientific (River Edge, NJ), 2002. Part 1 of this book constitutes the second edition of “Green, Brown and probability.” Part 2 comprises lecture notes from the short course on “Brownian motion on the line,” given at Stanford University. MR 1904742 Zbl 1014.60001 book

[177]K. L. Chung: “New solution of famous theorem,” J. Math. Res. Exposition 23 : 4 (2003). Inside front cover. MR 2083552 article

[178]K. L. Chung and F. AitSahlia: Elementary probability theory: With stochastic processes and an introduction to mathematical finance, 4th edition. Undergraduate Texts in Mathematics. Springer (New York), 2003. MR 1961879 Zbl 1019.60001 book

[179]K. L. Chung and J.-C. Zambrini: Introduction to random time and quantum randomness, new edition. Monographs of the Portuguese Mathematical Society 1. World Scientific (River Edge, NJ), 2003. MR 1999000 Zbl 1041.81074 book

[180]K. L. Chung: Chance & choice: Memorabilia. World Scientific (Hackensack, NJ), 2004. MR 2113168 Zbl 1059.01017 book

[181]K. L. Chung: “Note on open problem,” J. Math. Res. Exposition 24 : 3 (2004). Inside front cover. MR 2084466 article

[182]K. L. Chung and J. B. Walsh: Markov processes, Brownian motion, and time symmetry, 2nd edition. Grundlehren der Mathematischen Wissenschaften 249. Springer (New York), 2005. MR 2152573 Zbl 1082.60001 book

[183]K. L. Chung: “An unsolved problem by Feller,” J. Math. Res. Exposition 25 : 3 (2005), pp. 463–464. MR 2163725 article

[184]K. L. Chung: Selected works of Kai Lai Chung. Edited by F. AitSahlia, E. Hsu, and R. Williams. World Scientific (Hackensack, NJ), 2008. MR 2841270 Zbl 1165.60302 book

Farid AitSahlia	Related
Ruth J. Williams	Related
Elton Pei Hsu	Related

[185]“In memory of Kai-Lai Chung,” J. Math. Res. Exposition 29 : 6 (2009), pp. 951–952. MR 2590210 article

[186]H. W. Gould: “Kai Lai Chung (1917–2009) in memoriam,” J. Math. Res. Exposition 29 : 6 (2009), pp. 952–953. MR 2590211 article

[187]B. Bru, M.-F. Bru, and K. L. Chung: “Borel and St. Petersburg martingale,” J. Électron. Hist. Probab. Stat. 5 : 1 (2009). Article no. 3, consisting of 58pp. MR 2520662 Zbl 1170.01367 article

This paper examines — by means of the example of the St. Petersburg paradox — how Borel exposited the science of his day. The first part sketches the singular place of popularization in Borel’s work. The two parts that follow give a chronological presentation of Borel’s contributions to the St. Petersburg paradox, contributions that evolved over a period of more than fifty years. These show how Borel attacked the problem by positioning it in a long — and scientifically very rich — meditation on the paradox of martingales, those systems of play that purport to make a gambler tossing a coin rich. Borel gave an original solution to this problem, anticipating the fundamental equality of the nascent mathematical theory of martingales. The paradoxical role played by Félix Le Dantec in the development of Borel’s thought on these themes is highlighted. An appendix recasts Borel’s martingales in modern terms.

Marie-France Bru	Related
Armand Borel	Related
Bernard Bru	Related

[188]K. L. Chung: Collected articles from LNM: A special selection on the occasion of the memorial conference for Kai Lai Chung (Beijing, China, June 13–16, 2010). Lecture Notes in Mathematics. Springer-Verlag (Berlin), 2010. MR 2722042 Zbl 1194.01174 book

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