Celebratio Mathematica — Chung

Sums of random variables

Works connected to Paul Erdős

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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.

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