Let $\{X_n\}_{n\ge1}$ a sequence of independent random variables with same distribution function $F$.

Question

Let $\{X_n\}_{n\ge1}$ a sequence of independent random variables with same distribution function $F$. Let $x\in\mathbb{R}$ a fixed real number and $Z_i=1_{\{X_i\le x\}}$.

1. Show that, random variables $Z_i$ are independent and identically distributed and therefore $$F_n(x):=\dfrac{1}{n}\sum_{i=1}^nZ_i\to F(x)$$ almost surely.

2. Show that, for each $a<b$ $$\dfrac{1}{n}\big|\{i\in\{1,2,\ldots,n\}\mid a< X_i\le b\}\big|\to F(b)-F(a)$$ almost surely.

I have this problem, but I don't have a very clear idea of how to solve it.

Answer 1

Hint for 1) we know that the $Z_i$ have finite moments (actually they are Bernoulli(p) with $p=P(Z_i\leq x)$) so appeal to SLLN.

$$E[Z_i] = p = P(Z_i\leq x)= F(x)$$

Since $Z_i$ are a sequence of iid random variables their sample mean will converge almost surely to their expected value.

Hint for 2) You can recast it in terms of $Z_i(a),Z_i(b)$ and use SLLN