**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 fails to hold 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)