The strong law of large numbers states that if $X$ is a RV and $X_1, X_2 \ldots$ are independent and identically distributed copies of $X$, and $\overline{X}_n= \frac{1}{n}(X_1 + \ldots + X_n)$, and the first moment of $X$ is finite, then

$P(\lim_{n\to\infty} \overline{X}_n = \Bbb E X) =1$.

But by definition of limit this means that is almost certain that for every $\epsilon$ there exists a $n_0$ such that $n>n_0 \implies |\overline{X}_n - \Bbb E X|<\epsilon$.

What meaning can we give to "the probability that something exists" (in this case $n_0$)?

  • 2
    $\begingroup$ Don't forget that $n_0$ is a random variable itself and depends on $\omega$. $\endgroup$ – A.S. Nov 6 '15 at 21:06
  • $\begingroup$ The SLLN does not claim that all realizations of $X_1, X_2, \ldots, X_n, \ldots$ have the property that the sequence $s_n = \frac 1n\sum_{i=1}^n x_n$ converges to $\mathbb EX$. It says that the set of realizations for which $s_n$ converges to something else or does not converge at all has probability $0$. It is an event of probability $0$ but it is not the empty set. $\endgroup$ – Dilip Sarwate Nov 7 '15 at 11:59

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