- Define $r(t)=\frac{1+X_1+...+X_t}{1+Y_1+...+Y_t}$
- $X_t\sim Binomial(Y_t,\alpha)$
- $Y_t \sim Binomial(\lfloor 1+r(t-1)\rfloor,\delta)$
I would like to show that $E[r(t)]$ is a monotonic function of $t$.
Intuitively $E[r(t)]$ should converge to $\alpha$ and therefore $E[r(t)]$ should be a monotonic function.
Any comments or inputs would be appreciated.