# How can I find the long-term summation of this random recursive sequence?

Suppose there is a stocastic random recursive sequence $$\{b_n\}$$, $$n=1,2,...$$, it evolves as follows: $$$$b_i=\begin{cases} b_{i-1}+1, & \text{w.p. } \lambda, \\ 0, & \text{w.p. } 1-\lambda. \end{cases}$$$$ Here $$i\geq2$$ and suppose $$b_1=0$$.

Then I construct another recursive sequence $$\{a_n\}$$, $$n=1,2,...$$, it evolves as follows: $$$$a_i=\begin{cases} a_{i-1}+1, & \text{if } a_{i-1}-b_i<\gamma, \\ (1-\epsilon)(b_i+1)+\epsilon(a_{i-1}+1), & \text{if } a_{i-1}-b_i\geq \gamma. \end{cases}$$$$ Here $$i\geq2$$ and suppose $$a_1=1$$. $$\lambda$$ and $$\epsilon$$ are both constant ranging from $$[0,1]$$. $$\gamma$$ is also a non-negative constant number, you can regard it as a "threshold" to influence the recursion.

What I want to find is the long-term summation of $$\{a_n\}$$, i.e., $$\lim_{N\rightarrow\infty}{\frac{1}{N}\sum_{i=1}^Na_i}$$, can I derive a closed form of it? Or can I find some references to solve it?

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• Have you plotted a simulation of this? It feels as though the $a$ process should fairly closely track the $b$ process. What I would expect to happen would be that the $a$ process increases until it gets $\gamma$ ahead of the $b$ process; then gets scaled back. As a rough guess, I would expect the long term average of the $a$'s to be $\frac\gamma 2$ plus the long-term average of the $b$'s. It should be fairly straightforward to compute the long-term $b$ average. Nov 24 at 19:24