Vol. 42, No. 2, 1972

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A decomposition theorem for biadditive processes

W. N. Hudson

Vol. 42 (1972), No. 2, 333–341

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,t2)− X (s1,t2)− X (s2,t1) +X (s1,t1)

where s1 < s2 and t1 < 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.

Mathematical Subject Classification
Primary: 60J30
Received: 10 March 1972
Published: 1 August 1972
W. N. Hudson