Probability distribution of $\sum_i^n X_i - T$ where $\sum_i^nX_i <T<\sum_i^{n+1} X_i $

Question

Let $X_{1}, X_{2}, \ldots, X_{n}$ be IID random variables with mean $\mu$ and variance $\sigma^2$. Let $S_n=\sum_i^{n}X_i.$

Let $T\gg1$ and define $\tau=T-S_n$ where $n$ satisfies the following relation: $S_n<T<S_{n+1}$.

What will be the distribution of $\tau$?

I suspect that there exist a simple answer if we assume $T\gg1$ and we know the distribution of $X_i$, however I am not sure of the best approach to find it. I am only interested in the large $T$ regime, the idea would be that at time $T$ my system is in some thermal state.

The following algorithm will generate such number $\tau$:

• Initialise $t=0$.
• While $t<T$: $t\to t+X_i$
• return $\tau = T-t$

Could I use large deviation principle to derive the distribution of $\tau$? Any suggestions are appreciated.

Here is a sketch that illustrates the question:

Answer 1

What you describe is well known as a renewal process and the random variable $\tau$ is known as deficit (or age or backward recurremce time) at time $T$. The distribution of $\tau$ is well known. You find it f.i. in the easily accessible book Heyman/Sobel (1982), Stochastic Models in Operations Research, Vol. I p.130f (Distribution of the Deficit).