The expected value of a bunch of dependent events

Suppose we have pairs of number randomly selected in $(0,1)$.

$$(a_1, b_1) \rightarrow (a_2, b_2) \rightarrow ...$$

We start at $t = 1$ and continue to $t = 2$, and so on. At each step $t$ we randomly choose one of $a_t$ or $b_t$, and set the result to be $d_t$: $$d_t = \begin{cases} a_t & \text{ with probability } p_t \\ b_t & \text{ with probability } 1-p_t \end{cases}$$ where $p_t = \frac{ \sum_{i=1}^{t-1} d_{i} }{ \sum_{i=1}^{t-1} a_{i} + \sum_{i=1}^{t-1} b_{i} }$

Suppose $p_1 = 0.5$.

Any ideas if I can find a closed form expression for $\mathbb{E}\left[ \frac{p_k}{p_{l}} \right]$, for some $l < k$?

• - Is the expression for $p_m$ that you have provided an estimate of $p_m$ given data $\{d_m\}_{m=1}^n$? Or is it a definition? - Also, you have used $m'$ in two different ways. Unless I have misunderstood, perhaps change the last statement to "$\mathbb{E}[p_m/p_l]$, for some $l < m$", to keep sum indices separate from regular ones. – Salmonstrikes Mar 10 '16 at 4:54
• - Yes. we start with p = 1, and continue the sequential game until time $n$. - Fix the issue with $m'$. – Daniel Mar 10 '16 at 19:41
• @Salmonstrikes simplified the problem statement a bit. – Daniel Mar 25 '16 at 19:27
• It is not a nice problem even for k=2 and l=1. I suppose that there is no nice closed form solution. May be some asymptotics. – Alexey Zaytsev Mar 25 '16 at 20:02