Now I have a question, in which I need to find the probability mass function and the cumulative distribution function. But now I only have the recurrence relation. Here is the details:
Assume $p_n=\Pr\left(X=n\right)$ for $n\in\mathbb{N}$, $p_0=\frac{1}{1+ab_0}$, and for $n\geq1$, $ p_n=\sum_{i=0}^{n-1}ab_{n-i}p_i, $ where the constants $a>0$, $b_i>0$ for $i\in\mathbb{N}$ and $\sum_{i=1}^{\infty}b_i=b_0$. Also, we know that $\sum_{n=0}^{\infty}p_n=1$.
From these conditions, can we obtain the closed-form expression of $p_n$? And its CDF $F(M)=\Pr\left(X\leq M\right)$?
If the closed-form expression is complicated, could we find some approximation for it?
Thank you so much~~
FTI:
Actually, now I have derived the probability generating function: $ G\left(z\right)=\sum_{i=0}^{\infty}p_iz^i=\frac{1}{1+ab_0-a\sum_{i=1}^\infty b_iz^i}. $ But I have no idea whether it will help to derive the closed-form expression of $p_n$.