# How can I calculate this repeating power?

I'm considering the stochastic tree. When root node is activated, the $$n$$ child nodes get the signal. But the probability of activation is $$p$$. $$d$$ is index of layer and starts from 1 that is indicating the first child layer. The activated nodes of layer $$d$$ is called $$a_d$$. The probability that a layer $$d$$ has no activated nodes is $$P(a_d=0) = q^n(r(r(r(...r(r+1)^n...+1)^n+1)^n+1)^n+1)^n$$ where $$q=1-p$$ and $$r=pq^{n-1}$$. The $$n$$ power related to $$r$$ is repeated $$d$$ times. What I want to know is the analytic expression of $$P(a_d=0)$$ or at least proving, $$\lim_{d \rightarrow \infty} P(a_d=0)=1$$ I conducted numerical calculation and what I got is that when $$np < 1$$ the sequence is converge to 1 very fast($$d=10^1$$). When $$np = 1$$, converge speed is decreased($$d=10^5$$). When $$np > 1$$, the sequence seems not converge to 1($$d=10^6$$, the value was 0.16).

• "I'm considering the stochastic tree" : Which model do you refer to ?Don't you think that some precisions would be useful (or at least a reference to an apropriate site) ? Oct 4, 2022 at 12:22
• Any comment, some 5 hours later ? Oct 4, 2022 at 18:16