Kolmogorov's 0-1 law with indicator function.

Asked Feb 23 '19 at 16:24

Active Feb 23 '19 at 16:24

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Let $A_1, A_2, \ldots$ be any independent sequence of events and let $S_x := \{ \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} 1_{A_n} \leq x \}$. Prove that for each $x \in \mathbb{R}$ we have $\mathbb{P}(S_x) \in \{0, 1 \}.$

I know we have to make use of Kolmogorov's 0-1 law which is defined as:
Define $\tau = \cap_{n=0}^{\infty} \sigma(A_n, A_{n+1}, \ldots)$
Given a sequence of independent events $\{A_n; n \geq 1\}$. If $A \in \tau$ then $\mathbb{P}(A) \in \{0,1\}$.

So I have to prove that $S_x \in \tau$. However I don't really understand what $S_x$ is.

asked Feb 23 '19 at 16:24

Jasper

I worked on exactly this exercise yesterday, you might find this post useful. – Jelle Dijkstra Feb 23 '19 at 16:40
$S_x$ is the set of sample points in which the limit of teh sequence $a_n = 1/n \sum_1^n A_i$ is at most $x.$ Notice $\tau = \cap_{n = m}^\infty \sigma(A_k; k \geq n)$ for every $m$ and use that $\lim a_n \leq x$ if and only if $\lim a_{n,m} \leq x,$ where $a_{n,m} = 1/n \sum_m^n A_i.$ – William M. Feb 23 '19 at 19:01

Kolmogorov's 0-1 law with indicator function.

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