Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type
A random variable X is said to have an operator-selfdecomposable or Lévy distribution if there are sequences \(\{X_ n\}\), \(\{A_ n\}\) and \(\{a_ n\}\) of respectively independent random variables, linear operators, and vectors such that the random variables \(A_ n(X_ 1+...+X_ n)-a_ n\) converge completely to X and in addition an infinitesimal condition is satisfied. Processes of Ornstein-Uhlenbeck type on Euclidean spaces are analogues of the Ornstein-Uhlenbeck process with the Brownian motion part replaced by general processes with homogeneous independent increments. The class of operator- selfdecomposable distributions is characterized as the class of limit distributions of processes of Ornstein-Uhlenbeck type. Integro- differential equations for operator-selfdecomposable distributions are established.
scientific article; zbMATH DE number 3846550
scientific article