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

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 …
Dieter Kadelka's user avatar