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Suppose $X_1,X_2,\dots,$ are i.i.d random variables with mean $\mu\in(0,\infty)$ and finite variance. Define the stopping time $N=\min\{ n: \sum_{i=1}^n X_i \geq B\}$. Why it is true that $E \sum_{i=1}^N X_i = B + O(1)$?

I read about this statement in an article, which doesn't seem like Wald's equation or something. I wonder how the term $O(1)$ comes from and how to evaluate it. Thanks ahead for any guidance!

Percy W
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This follows from Lorden's inequality (Wikipedia), which bounds the overshoot by $$ \mathbf{E}\Bigl[\sum_{i = 1}^N X_i \Bigr] - B \leq \frac{\mathbf{E}[X^2]}{\mathbf{E}[X]}. $$ See Chang (1994) for one proof, which relates the overshoot to the stationary excess of a renewal process with interrenewal time $X$.

Ziv
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