Background:

The strong law of large numbers (SLLN) is a powerful result in probability, and there has been extensive literature on when the SLLN holds. However, constructing nontrivial examples for which the SLLN seems to be (very) hard.

K.L. Chung's famous paper "The strong law of large numbers" talks about the necessary and sufficient conditions for the SLLN to hold. However, it is for independent random variabes. On the other hand, R. Lyons' paper "Strong laws of large numbers for weakly correlated random variables" is probably the best on when the SLLN holds under dependence. One of the theorems of Lyon's is quoted below:

Let $\left\{ X_{n}\right\} _{n=1}^{\infty}$ be a sequence of real-valued, zero mean random variables such that $\mathbb{E}\left[ \left\vert X_{n}\right\vert ^{2}\right] \leq1$. If $\left\vert X_{n}\right\vert \leq1$ a.s. and $$\sum\nolimits_{N=1}^{\infty}N^{-1}\mathbb{E}\left[ \left\vert N^{-1} \sum\nolimits_{n=1}^{N}X_{n}\right\vert ^{2}\right] <\infty\text{,}$$ then $\lim_{m\rightarrow\infty}N^{-1}\sum_{n=1}^{N}X_{n}=0$ a.s.

Question:

However, could anyone please give an example for which the SLLN fails to hold in the following setting? Let $\left\{ X_{n}\right\} _{n\geq1}$ be a sequence of Bernoulli random variables such that

1. $\mathbb{P}\left(X_{n}=1\right) =p_{n}$ for which $p_{i}\neq p_{j}$ whenever $i\neq j$, and $\mathbb{P}\left(X_{n}=0\right) =1-p_{n}$
2. $\lim\inf_{n\rightarrow\infty}p_{n}>0$ and $\lim\sup_{n\rightarrow\infty }p_{n}<1$
3. $X_{n}$'s are not wekly dependent in Lyon's sense (see the displayed equation above)