# Cumulative product of independent random variables with a stopping time

Suppose $$X_1,X_2, \dots, X_N$$ are i.i.d. random variables with $$\mathbb{E}[X_1]=\mu\neq 0$$. Extending the analysis for Wald's equation to where $$\tau$$ is a stopping time gives their partial sum subject to a stopping time $$\tau$$ to be $$\mathbb{E}[\sum_{i=1}^{\tau} X_i] = \mathbb{E}[\sum_{i=1}^{\tau} \mathbb{E}[X_1]] = \mathbb{E}[\sum_{i=1}^{\tau} \mu].$$ I am interested in the partial product $$\prod_{i=1}^{\tau} X_i,$$ where $$\{\tau=i\}$$ is dependent on $$X_1,X_2,\dots,X_i$$ and the empty product equals one. Ruben states that such a partial product satisfies $$\mathbb{E}[\prod_{i=1}^{\tau} X_i\mu^{-\tau}] = 1.$$ Can we say anything more about the properties of $$\mathbb{E}[\prod_{i=1}^{\tau} X_i]$$? Alternatively, if we knew that $$\mu>1$$, can we say that $$\mathbb{E}[\prod_{i=1}^{\tau} X_i] > 1?$$

I'm not sure what an empty product is defined to be, but let's assume an empty product is just $$1$$. I'm also assuming that the $$X_i$$ are independent of $$\tau$$, which is needed for Wald's formula anyway. In that case, we have \begin{align} \mathbb{E}[\prod_{i=1}^{\tau} X_i]&=\sum_{n=0}^{\infty}\Pr[\tau = n]\mathbb{E}[\prod_{i=1}^{n} X_i\vert \tau = n ]\\ &=\sum_{n=0}^{\infty}\Pr[\tau = n] \mu^n\\ &=\Pr[\tau=0]+\sum_{n=1}^{\infty} \Pr[\tau=n]\mu^n\\ &>\Pr[\tau=0]+\sum_{n=1}^{\infty} \Pr[\tau=n]\\ &=1, \end{align} where the last inequality follows from the assumption $$\mu>1$$.
• I did mean $\tau$ is dependent on $X_i$'s. I have updated the question and corrected some of the equations to reflect that. – thisisenfield Aug 12 '19 at 22:04