While reading Loeve's book on Probability (page 246, 5th edition) I found the following statement:
It is of some interest to observe that Borel's law of large numbers can also be obtained by means of the Bienaymé-Tchebichev inequality (see Introductory part)
I've studied that introductory part and found nothing. So, can anybody figure out how to proof Borel's law using Bienaymé-Tchebichev inequality?
Loéve is referring to his discussion on pp. 19-20 (in the fourth edition), in section 6 of Chapter II.
Briefly, if $X_1, X_2,\ldots$ are i.i.d Bernoulli random variables, with mean $p\in(0,1)$ and if $\bar X_n:=n^{-1}\sum_{k=1}^n X_k$, then $\sum_{m=1}^\infty\Bbb P[|\bar X_{m^2}- p|>\epsilon]\le \epsilon^{-2}\sum_{m=1}^\infty p(1-p)m^{-2}<\infty$ (by Chebyshev's inequality), so $\bar X_{m^2}\to p$ a.s. (as $m\to\infty$) by Borel-Cantelli. A general (large) $n$ satisfies $m^2\le n<(m+1)^2$ for a unique (large) $m$, in which case $|\bar X_n-\bar X_{m^2}|\le 4/m$ (by an easy estimate), whence $|X_n-p|\le 4/m+|X_{m^2}-p|$, which tends a.s. to $0$ as $m$ and $n$ go to $\infty$.