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Let $S_t$ a random walk with rate $a>0$ on $Z$ that such has a drift in direction of $0$, this is, if $(0,0)\in Z\times[0,\infty)$, and defined a sequence $(S_n, T_n)_{n\in N}$ such that $(S_0, T_0)=(0,0)$ and, for $n\geq1$

$T_n-T_{n-1}=\tau_n\sim Exp(a), \ \ \ \ \ \ S_n-S_{n-1}= \left\{ \begin{array}{lcc} 1, & if & S_n \leq 0 \\ -1, & if & S_n <0 \\ \end{array} \right.$

What is $E(TR_o)$ where $TR_0=\inf\{t>0: S_t=0\}$?

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  • $\begingroup$ I don't get what your $T_n$ does in here. Can you be more precise on this point ? $\endgroup$ – Joseph Budin Oct 1 '19 at 13:28
  • $\begingroup$ $T_n$ are the renewal times of random walk $\endgroup$ – Joan Jesus Amaya Triana Oct 1 '19 at 22:04
  • $\begingroup$ And where is the stochasticity of $S_n$ ? As far as I understand, its behaviour is deterministic. $\endgroup$ – Joseph Budin Oct 2 '19 at 7:23

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