Celebratio Mathematica

[1]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

[2]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

[3]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.

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

[5]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

[6]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

[7]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

[8]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

[9]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.

[10]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.

[11]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

[12]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

[13]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

[14]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

[15]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

[16]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

Kai Lai Chung

Sums of random variables