Strong law of large numbers for $\mathrm{Bin}(n, p_n)$ variables

Question

Massive edit to simplify the central question

Suppose $X_n\sim \mathrm{Bin}(n, p_n)$ be a collection of independent random variables such that $np_n\to \infty$. Can we say that $Y_n:=X_n/np_n\to 1$ almost surely?

It is easy to show using Chebyshev, for example, that indeed $Y_n\stackrel{p}{\to}1$.

I tried using the fourth moment bound which is one of the elementary methods to prove almost sure convergence, but I need summability of $\sum_n 1/(np_n)^2$ which is certainly not true for all choices of $p_n$, for example if $p_n=\log(n)/n$.

What would be a general approach to solve such kinds of questions? Thanks in advance!

Question: Is it possible to somehow use Kronecker's lemma adapted to triangular sequences?

Answer 1

It's not always true that $Y_n\to 1$ a.s.

Your example with $p_n=\log(n)/n$ is a good one.

Then $$ P(X_n=0) =\left(1-\frac{\log(n)}{n}\right)^n =\exp\left[n\log\left(1-\frac{\log(n)}{n}\right)\right] \sim\frac1n $$ as $n\to\infty$.

So $\sum_{n=1}^\infty P(X_n=0)$ is infinite. Since the events are independent, the second Borel-Cantelli Lemma tells you that with probability $1$, there are infinitely many values of $n$ such that $X_n=0$ (and so also $Y_n=0$).

Answer 2

By the Borel-Cantelli lemma and Hoeffding's inequality, the sequence $(X_n/(np_n))_{n\geqslant 1}$ converges almost surely to $1$ provided that for each positive $\varepsilon$, $$ \sum_{n=1}^\infty \exp\left(-\varepsilon^2np_n^2\right)<\infty. $$

Following the approach proposed by James Marting, we can see that a necessary condition for the almost sure convergence of $(X_n/(np_n))_{n\geqslant 1}$ to $1$ is that $\sum_{n=1}^\infty (1-p_n)^n<\infty$.