Let X_i be positive i.i.d. random variables (or more generally a uniformly mixing positive-valued ergodic stationary process). The r-scan process induced by \{X_i\} is R_i = \sum^{i+r-1}_{k=i} X_k, i = 1, 2, \ldots, n - r + 1. Limiting distributions for the extremal order statistics among \{R_i\} suitably normalized (and appropriate threshold values a = a_n and b = b_n) are derived as a consequence of Poisson approximations to the Bernoulli sums N^-(a) = \sum^{n+r-1}_{i=1} W^-_i(a) and N^+(b) = \sum^{n-r+1}_{i=1}W^+_i(b), where W^-_i(a) \lbrack W^+_i(b) \rbrack = 1 or 0 according as R_i \leq a (R_i > b) occurs or not. Applications include limit theorems for r-spacings based on i.i.d. uniform \lbrack 0, 1 \rbrack 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.