Abstract: | Two processes of importance in statistics and probability are the empirical and partial-sum processes. Based on d-dimensional data X1, … Xa the empirical measure is defined for any A ⊂ Rd by the sample proportion of observations in A. When normalized, Fn yields the empirical process Wn: = n1/2 (Fn - F), where F denotes the “true” probability measure. To define partial-sum processes, one needs data that are assigned to specified locations (in contrast to the above, where specified unit masses are assigned to random locations). A suitable context for many applications is that of data attached to points of a lattice, say {Xj:j ϵ Jd} where J = {1, 2,…}, for which the partial sums are defined for any A ⊂ Rd by Thus S(A) is the sum of the data contained in A. When normalized, S yields the partial-sum process. This paper provides an overview of asymptotic results for empirical and partial-sum processes, including strong laws and central limit theorems, together with some indications of their inferential implications. |