Stopping time for sum of iid random variables.

Question

Suppose we have $m$-sided biased die. Let $X_i$ be the outcome of the $i$'th roll with the die. Furthermore let $\mathbb{P}[X_i=k]=p_k$ with $k \in \{1,...,m\}$. We define $T=\min\{n\text{ :}\sum_{j=1}^n X_i\geq N\}$. What is there to say about the distribution of $T$?

Answer 1

What is there to say about the distribution of $T$?

About the distribution of $T$ alone, not much I am afraid. But consider the family $(T_n)_{n\geqslant0}$ defined by $T_0=0$ and by $T_n=\min\{k\geqslant1\mid X_1+\cdots+X_k\geqslant n\}$, for every $n\geqslant1$.

Then $T=T_N$ and, for every $|s|\leqslant1$, the generating functions $g_n(s)=E[s^{T_n}]$ are characterized by the relations $g_0(s)=1$ and $g_n(s)=s\sum\limits_{k\geqslant1}p_kg_{n-k}(s)$ for every $n\geqslant1$, with the convention that $g_n(s)=1$ for every $n\leqslant-1$.

For every $|t|\leqslant1$, let $G(t,s)=\sum\limits_{n\geqslant0}g_n(s)t^n$, then $G(t,s)=1+s\sum\limits_{k\geqslant1}p_kt^k\left(G(t,s)+\sum\limits_{i=1}^{k-1}t^{-i}\right)$, from which one gets $$ G(t,s)=\frac1{1-t}-\frac{t}{1-t}\,\frac{1-s}{1-sx(t)},\qquad x(t)=\sum\limits_{k\geqslant1}p_kt^k=E[t^X]. $$ This formula defines uniquely each $g_n(s)$ as the coefficient of $t^n$ in $G(t,s)$, but to get farther one should probably specify the distribution $(p_k)_{k\geqslant1}$ of the increments $(X_n)_{n\geqslant1}$ or, equivalently, their generating function $x:t\mapsto E[t^X]$.