I am studying the [Law of large numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers) for [independent and identically distributed (i.i.d)](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables) random variables.


Assume there are i.i.d variables $(\xi\_k)\_{k\ge 1}$ taking values in $(0,1)$. From the law of large numbers, we know $\frac{\sum\_{k\le n} \xi\_k}{n} \to \mathbb{E}(\xi\_1)$ almost surely (a.s.).


Then we know, if $\mathbb{E}(\log\xi\_1^{-1})<\infty$:


$$\sup\_{n}\sum\_{k\le n} \xi\_1\cdot \xi\_2....\xi\_k=\sup\_{n}\sum\_{k\le n} e^{-(\log\xi\_1^{-1} + \log \xi\_2^{-1}+....+\log \xi\_k^{-1})}\precsim\sup\_n\sum\_{k\le n}e^{-i\mathbb{E}(\log\xi\_1^{-1})} < \infty$$
a.s. (geometric series).


Let's consider $\sup\_{n}\sum\_{k\le n} \xi\_n\cdot \xi\_{n-1}....\xi\_k$.


Can we prove this is also finite?



LLN large number law of Probability
If the $\xi$ take values arbitrarily close to 1 (e.g. the $\xi\_i$ are Unif[0,1]), this quantity is infinite: for any $\epsilon>0$, there exist arbitrarily long strings where all of the $\xi\_j$’s are at least $1-\epsilon$, which means that the sup is larger than$1/\epsilon$.


