$n^{-1}\sum_{i=1}^{n}(1-R_i)\overset{p}{\to}P(R_i=0)$, where $R_i$ is a binary variable

Question

Say, $R_i$ $(i=1,2,\ldots,n)$ is a binary variable.

I have an estimator $n^{-1}\sum_{i=1}^{n}(1-R_i)$.

How can I show $n^{-1}\sum_{i=1}^{n}(1-R_i)$ converges in probability to $P(R_i=0)$? That is,

$\lim_{n\to \infty}n^{-1}\sum_{i=1}^{n}(1-R_i)=?$

I know $\sum_{i=1}^{n}(1-R_i)P(1-R_i)=\mathbb E(1-R_i)$.

But is $\lim_{n\to \infty}n^{-1}\sum_{i=1}^{n}(1-R_i)=\mathbb E(1-R_i)$?

Answer 1

Assume $R_1, R_2, \ldots, R_n$ are IID Bernoulli random variables. By the Weak Law of Large Numbers, $n^{-1}\sum_{i=1}^n (1-R_i)$ converges in probability to $\mathbb{E}[1-R_i] = 1 - \mathbb{E}[R_i] = 1 - Pr(R_i = 1) = Pr(R_i = 0)$.