LetNbe a positive integer,ca positive constant and(ξn)n≥1be a sequence of independent identically distributed pseudorandom variables. We assume that theξn’s take their values in the discrete set{-N,-N+1,…,N-1,N}and that their common pseudodistribution is characterized by the (positive or negative) real numbersℙ{ξn=k}=δk0+(-1)k-1c(2Nk+N)for anyk∈{-N,-N+1,…,N-1,N}. Let us finally introduce(Sn)n≥0the associated pseudorandom walk defined onℤbyS0=0andSn=∑j=1nξjforn≥1. In this paper, we exhibit some properties of(Sn)n≥0. In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for(Sn)n≥0as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudorandom walk to the pseudo-Brownian motion driven by the high-order heat-type equation∂/∂t=(-1)N-1c∂2N/∂x2N. We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudoprocess.
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