Poisson Approximations for r-Scan Processes

Let Xi be positive i.i.d. random variables (or more generally a uniformly mixing positive-valued ergodic stationary process). The r-scan process induced by {Xi} is Ri=.i+r.1k=iXk,i=1,2,.,n.r+1. Limiting distributions for the extremal order statistics among {Ri} suitably normalized (and appropriate threshold values a=an and b=bn) are derived as a consequence of Poisson approximations to the Bernoulli sums N.(a)=.n+r.1i=1W.i(a) and N+(b)=.n.r+1i=1W+i(b), where W.i(a)[W+i(b)]=1 or 0 according as Ri.a(Ri>b) occurs or not. Applications include limit theorems for r-spacings based on i.i.d. uniform [0,1] r.v.'s, for extremal r-spacings based on i.i.d. samples from a general density and for the r-scan process with a variable time horizon.