202409302056

tags : Probability

Kolmogorov’s Law of Large Numbers

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Kolmogorov's Strong law of Large Numbers

1. Let $(X_n)$ be a series of independent mean 0 random variables such the series of numbers $\sum \frac{1}{2}\text{var}(X_n)$ converges, then $\frac{1}{n}\sum X_n\to 0$
2. In particular, for a sequence of independent identically distributed random variables with finite mean and variance, their average converges to the mean almost surely.

From Kolmogorov’s theorem, we know that $\sum \frac{X_n}{n}$ converges almost surely.

To prove that

$$\sum \frac{x_i}{i}\to c⟹\frac{1}{n}\sum x_i\to 0$$

Then we are done. let $s_m={\sum }_{i=1}^n\frac{x_i}{i}$ then we can write $x_i=m(s_m-s_{m-1})$ Then using Cesaro’s theorem, we get the answer trivially.

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