A decomposition theorem for biadditive processes

Abstract

This paper treats a class of stochastic processes called biadditive processes and gives a proof of a decomposition of their sample functions. Informally, a biadditive proces X(s, t) is a process indexed by two time parameters whose increments over disjoint rectangles are independent. The increments of such a process are the second differences X(s2, U) - X(8U U) - X(s2, U) + X(8U ti) where s± < s2 and U < t2. The decomposition theorem states that every centered biadditive process is the sum of four independent biadditive processes: one with jumps in both variables, two with jumps in one variable and continuous in probability in the other, and a fourth process which is jointly continuous in probability. This decomposition is similar to one for processes with independent increments and in the proofs of both results a major role is played by the theory of centralized sums of independent random variables. More formally, let P1 = {slf s2, , sn} and P2 = {tl9 t2, , tm) be two partitions of [0, sn] and [0, tm] respectively. Define Px x P2 to be the corresponding partition of [0, sn] x [0, tm] into rectangles whose vertices are the (si9 ί, )'s. Let ΔiS denote the increment