Distribution of bounded summation of i.i.d random variables

Question

We have a set of positive random variables $\boldsymbol X=\{X_1, X_2,\ldots\}$, where $X_1, X_2,\ldots$, are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for $X_i$ are known in advance.

Define $S_n=\sum_{i=1}^nX_i$.

As we can see $\boldsymbol X$ can be viewed as inter-arrival times for a renewal process, and $S_n$ denotes each arrival epoch.

Next we define a variable $K$: $K=\inf\, \{n\mid S_n>T\}$ (or $K=\min\, \{n\mid S_n > T\}$), where $T$ is a constant.

Then

1. what is the distribution of $K$?
2. what is the distribution of $S_K$?

I already know that the PDF for $S_n$, denoted by $f_n$, can be computed by $f_n=f^{*n}=f*f*\cdots *f$, the $n$-fold convolution power of $f(x)$. By Laplace Transform, we can convert the convolution to multiplication.

Answer 1

The CDF of $K$ is $$ P(K \le n) = P(S_n > T) = \int_{T}^\infty dt\; f_n(t)$$ The CDF of $S_K$ (for $s > T$) is $$\eqalign{P(S_K \le s) &= \sum_{n=1}^\infty P(K = n, S_n \le s) = \sum_{n=1}^\infty P(S_{n-1} \le T, T < S_n \le s)\cr &= \sum_{n=1}^\infty \int_0^T dt\; f_{n-1}(t) \int_T^s dr\; f(r-t)}$$

EDIT: the $n=1$ term needs to be modified since $S_0 = 0$ doesn't have a density. So (assuming of course $T > 0$) it's $$P(S_K \le s) = \int_T^s dr\; f(r) + \sum_{n=2}^\infty \int_0^T dt\; f_{n-1}(t) \int_T^s dr\; f(r-t)$$