Exit times for integrated random walks

We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time $n$. Assuming that the moment of order $2+δ$ is finite, we show that the exact asymptotics for this probability are $n^{-1/4}$. To show these asymptotics we develop a discrete potential theory for the integrated random walk.

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