Celebratio Mathematica

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

[3]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

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

[5]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

[6]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

[7]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

[8]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

[9]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

[10]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).

[11]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

[12]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.

[13]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].

[14]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

[15]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

[16]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.

[17]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.

[18]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

[19]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

[20]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.

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

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

[23]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

[24]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

[25]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.

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

[27]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).

[28]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

[29]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

[30]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

[31]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

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

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

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

Kai Lai Chung

Probability theory