**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)
4. no significant proportion (compared to $m^2$ when $m$ is large) of the sequence $X\_n$ for $1 \leq n \leq m$ should contain elements that are a scalar multiple of one another (this is to avoid types of examples given by AnthonyQuas)


**Follow-up**


Anthony and Fedor each provided an example of dependent Bernoulli's for which the SLLN fails. Their examples have special structures, i.e., $X\_n = f(Y\_1, \cdots, Y\_{n\_0}, Y\_n)$ for a fixed $n\_0$ and independent Bernoulli's $Y\_n$, where $f$ is a linear mapping whose functional norm is bounded from above and away from $0$. It seems that the key is to introduce infinite-range dependence (which is in sharp constrast to finite-range dependence or exchangeability; see a related post [Is there a McDiarmid-type inequality for sequences with a finite range of dependence?](https://mathoverflow.net/questions/151657/is-there-a-mcdiarmid-type-inequality-for-sequences-with-a-finite-range-of-depend?rq=1) )


Is this a general technique to construct examples for the failure of SLLN for dependent Bernoulli's?



Dependent Bernoulli sequence for which the strong law fails to hold
Let $(S\_n, n\geq 0)$ be the one-dimensional simple random walk started at the origin. Set $X\_n=\mathbf{1}\{S\_n\geq 0\}$. Then $n^{-1}(X\_1+\cdots+ X\_n)$ is the proportion of time that the SRW is non-negative; it doesn't converge a.s. anywhere (although converges in distribution to a non-trivial r.v., cf. the arcsine law).


Is this sort of example what you're looking for?


