lln

The Law of the Large Numbers

Strong Law

$$P({\bar{X}}_n\to \mu )=1$$

Weak Law

For all $ϵ>0$, $P(|{\bar{X}}_n-\mu |>ϵ)\to 0$ as $n\to \infty$.

This is the convergence in probability.

Every time we perform simulations, we are assuming LLN implicitly.

To understand this concept, consider the proportion of heads in a series of coin tosses.

• SLLN suggests that ${\bar{X}}_1$, ${\bar{X}}_2$, … crystallizes into ==a sequence which converge to $1/2$==.
• WLLN suggests that for $P({\bar{X}}_n>1/2)$ ==can be made as close to $0$ as possible by increasing $n$==.

The distribution of ${\bar{X}}_n$ along the way of becoming a constant is given by Central Limit Theorem.

recurrence