Let $X_1,X_2,\ldots$ be iid random variable with $E[X_i]=0$ and $P(X_i>0)>0$. Let $S_0=x>0$ and define $S_n=S_0+\sum_{i=1}^nX_i$. Then define $T=\inf\{n \geq 0: S_n \leq 0 \text{ or } S_n \geq b\}$. I have shown that $T$ has finite expectation, and am trying to show $X_T$ is integrable. Any hint or help would be great!

  • $\begingroup$ You probably need to assume that the $X_i$ are integrable also $\endgroup$ – Lorenzo Najt May 17 '18 at 22:25
  • $\begingroup$ Try writing $X_T = \Sigma_{i = 0}^{\infty} 1_{T = i} X_i$ and compute the L1 norm... $\endgroup$ – Lorenzo Najt May 17 '18 at 22:26
  • $\begingroup$ @AreaMan It is assumed that $\mathbb E[X_i]=0$. $\endgroup$ – Math1000 May 18 '18 at 0:37
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    $\begingroup$ Since the $X_i$ are i.i.d. with finite expectation and $\mathbb E[T]<\infty$, Wald's identity yields $$ \mathbb E[S_T] = \mathbb E[S_1]\mathbb E[T] = S_0\mathbb E[T]<\infty. $$ $\endgroup$ – Math1000 May 18 '18 at 0:39
  • $\begingroup$ @Math1000 but how does this emply $E[|X_{T}|]< \infty$? $\endgroup$ – user0617 May 18 '18 at 8:44

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