Self-similar processes with independent increments

Summary A stochastic process {X_tt 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, andX₀=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 thatX₁ 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 thatY₁ 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_cX_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].