Law of large numbers for non-identically distributed Bernoulli random variables

Let $(X_n)$ be a succession of independent r.v., such that $X_n$ ~ $Bern(p_n)$. I know then that $\lim_{n \to \infty}p_n=p$ and $p_n>p>0$ for each $n \in \mathbb{N}$. I have to prove that

$\dfrac{X_1 + \dots + X_n}{n} \rightarrow p$

almost surely.

Intuitively, by the strong law of large numbers, I would say it is true. The problem is that the succession $p_n$ is not constant, so I do not know how to conclude in a formal way.

Thanks to who will solve my doubt.

Solution 1:

This answer is based on a coupling argument, which makes rigorous the comment by LJG in the answer by Davide Giraudo. We will assume the (usual) strong law of large numbers for i.i.d. sequences, but nothing else.

Namely, let $X_n$ be independent random variables with $X_n \sim Bern(p_n)$ where $p_n \downarrow p$. Define a sequence $Y_n$ of i.i.d. Bernoulli($p$) random variables as follows. If $X_n=0$, let $Y_n=0$. If $X_n=1$, flip an (independent, unfair) coin, whose probability of heads is $1-p/p_n$. If heads, let $Y_n=0$. If tails let $Y_n=1$. Clearly $Y_n \leq X_n$ for all $n$, and the $Y_n$ are i.i.d. Bern($p$). By the strong law of large numbers $$\liminf_{n \to \infty}\frac{1}{n} \sum_1^n X_k \geq \liminf_{n \to \infty}\frac{1}{n} \sum_1^n Y_k =p, \;\;\;\;\;\;\;a.s.$$

Now let $q>p$. For large enough $n$ we have that $p_n<q$. Since the limit of Cesaro means does not depend on the omission of some finite number of terms, we may assume wlog that $p_n<q$ for all $n$. We now define a sequence $Z_n$ of i.i.d. Bernoulli($q$) random variables as follows. If $X_n=1$, let $Z_n=1$. If $X_n=0$, flip an (independent, unfair) coin, whose probability of heads is $(q-p_n)/(1-p_n)$. If heads, let $Z_n=1$. If tails let $Z_n=0$. Clearly $Z_n \geq X_n$ for all $n$, and the $Z_n$ are i.i.d. Bern($q$). By the strong law of large numbers, $$\limsup_{n \to \infty}\frac{1}{n} \sum_1^n X_k \leq \limsup_{n \to \infty}\frac{1}{n} \sum_1^n Z_k =q, \;\;\;\;\;\;\;a.s.$$ But since $q>p$ was arbitrary, we conclude (after taking the intersection over countably many $q$'s converging down to $p$) that $$\limsup_{n \to \infty}\frac{1}{n} \sum_1^n X_k \leq p, \;\;\;\;a.s.$$ which gives the result.

Solution 2:

This answer uses the following statement

Let $(Y_i)_{i \in \mathbb{N}}$ be a sequence of independent random variables. If $\sum_{i=0}^n Y_i$ converges in probability, then $\sum_{i=0}^n Y_i$ converges almost surely.

Hints:

1. Set $T_n := \sum_{i=1}^n (X_i-p_i)/i$. Use the independence of the random variables $X_i$ to show that $$\mathbb{E}((T_n-T_m)^2) \leq 4 \sum_{i=m+1}^n \frac{1}{i^2}$$ for all $n \geq m$. Conclude that the limit $$\sum_{i=1}^{\infty} \frac{X_i-p_i}{i} := \lim_{n \to \infty} T_n$$ exists in $L^2$ and, hence, in probability.
2. Deduce from Step 1 that $\sum_{i=1}^{n} (X_i-p_i)/i$ converges almost surely as $n \to \infty$.
3. Apply the Kronecker lemma to conclude that $$\lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^n (X_i-p_i)=0$$ almost surely.

Remark: Using the above reasoning it is actually possible to show the following more general result which is known as "Kolmogorov's strong law"

Let $(X_i)_{i \in \mathbb{N}} \subseteq L^2(\mathbb{P})$ be a sequence of independent random variables. If $$\sum_{i \geq 1} \frac{\text{var} \, (X_i)}{i^2} < \infty$$ then $$\lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^n (X_i-\mathbb{E}(X_i))=0 \quad \text{almost surely.}$$