# Upper bound for randomly weighted sum of independent random variables

I have a sequence of independent random variables {$\epsilon_j$} with mean 0. I also have another sequence of Bernoulli random variables $\delta_1, \delta_2,\dots$ which are dependent on the previous values of $\delta_t$ for $t \leq j-1$ and such that $\delta_j$ is independent of {$\epsilon_t:t \geq j$} for all $j \geq 1$. I am interested in the quantity $$P\left(\dfrac{\sum_{j=1}^n \delta_j \epsilon_j}{\sum_{j=1}^n \delta_j} > a \right)$$ where $a$ is some constant and I want to come up with a bound for this, which hopefully is decaying at some rate as $n \rightarrow \infty$ to converge to zero eventually.

I have found a paper which proves almost sure convergence for such randomly weighted sums of random variables but doesn't provide a bound. Any insights on this are much appreciated.

• I don't see why it must decay, which makes me wonder whether you have stated all your assumptions as intended. For instance, what is to prevent $\delta_1=1$ (which is needed for the fraction to be defined at step $1$), $\delta_2\sim\text{Bernoulli(1-p)}$, and $\delta_3=\cdots=\delta_j=\cdots=\delta_2$? In that case the probability at step $n\gt 1$ is bounded below by $p\Pr(\epsilon_1\gt a)$, which needn't be zero. – whuber Jul 7 '17 at 19:54
• Thanks for pointing that out. There are other assumptions that I didn't state and I should've. We are assuming that the denominator $\sum_{j=1}^n \delta_j$ is non-zero for each n and also the probability of success each Bernoulli random variable changes with each time step based on previous information. So $\delta_j \sim Bernoulli(p_j)$. – user111092 Jul 7 '17 at 20:35
• To be honest I am also trying to explore what are the assumptions I would need to make for this probability to converge to 0. – user111092 Jul 7 '17 at 20:37
• My example can easily be modified to conform with your additional requirements. I constructed it so that each partial sum of the $\delta_j$ is nonzero. Although all the $\delta_j,j\ge 2$ have the same $p_j$, you could change them by vanishingly small amounts without changing the limiting behavior of the probability. The crux of the matter concerns exactly what kinds and amounts of dependence you are asssuming among your random variables: this example shows you can't hope to achieve anything interesting unless you are more specific about those assumptions. – whuber Jul 7 '17 at 21:24