I struggle with the following problem. Let $s_t \in [0,1]$ be a system state at time $t$, whereby $s_0 = 0$ and $s_t = 1$, $\forall t < 0$. Let further $p_i \in [0,1]$, $\forall i \in \{1,\dots,n\}$, such that $\sum_{i=1}^n p_i \leq 1$. Note that $n$ need not be very large in my application, e.g., $n=7$ suffices. The system state at time $t$ is defined as \begin{align} s_t = p_1 (1-s_{t-1}) + p_2 (1-s_{t-2}) + \dots + p_n (1-s_{t-n}) \end{align} I am now interested in the value of $s_\infty = \lim_{t\to\infty} s_t$. I have implemented the above algorithm and it seems to converge to \begin{align} s_\infty = 1- \frac{1}{1 + \sum_{i=1}^n p_i } \end{align} but unfortunately I fail to prove (or falsify) it. I can only show it for the case $n=1$, i.e., the above algorithm reduces to $s_t = p_1 (1-s_{t-1})$. In this case: \begin{align} s_0 &= 0 \\ s_1 &= p_1 \\ s_2 &= p_1 -p_1^2 \\ s_3 &= p_1 -p_1^2 + p_1^3 \\ \dots \\ s_T &= 1- \sum_{t=0}^{T} (-1)^t p_1^t \\ s_\infty &= 1-\frac{1}{1+p_1} \end{align}
However, I cannot translate this result for $n>1$. Thank you very much in advance for hints or ideally solutions.
I think I have a proof.
See page 1 page 2
Hope this helps.