Self-similar processes with independent increments

Summary A stochastic process {X _t t 0} onR ^d is called wide-sense self-similar if, for eachc>0, there are a positive numbera and a functionb(t) such that {X _ct } and {aX _t +b(t)} have common finite-dimensional distributions. If {X _t } is widesense self-similar with independent increments, stochastically continuous, andX ₀=const, then, for everyt, the distribution ofX _t is of classL. Conversely, if is a distribution of classL, then, for everyH>0, there is a unique process {X _(H) ^t } selfsimilar with exponentH with independent increments such thatX ₁ has distribution . Consequences of this characterization are discussed. The properties (finitedimensional distributions, behaviors for small time, etc.) of the process {X _(H) ^t } (called the process of classL with exponentH induced by ) are compared with those of the Lévy process {Y _t } such thatY ₁ has distribution . Results are generalized to operator-self-similar processes and distributions of classOL. A process {X _t } onR ^d is called wide-sense operator-self-similar if, for eachc>0, there are a linear operatorA _c and a functionb _c (t) such that {X _ct } and {A _c X _t +b _c (t)} have common finite-dimensional distributions. It is proved that, if {X _t } is wide-sense operator-self-similar and stochastically continuous, then theA _c can be chosen asA _c =c ^Q with a linear operatorQ with some special spectral properties. This is an extension of a theorem of Hudson and Mason 4].