Abstract.A general representation for an operator-self-similar process is obtained and its class of exponents is characterized. It is shown that such a process is the limit in a certain sense of an operator-normed process and any limit of an operator-normed process is operator-self-similar.1. Introduction. In 1962 Lamperti [8] introduced the notion of a self-similar process, {X(t): t s* 0), taking values in a real finite-dimensional inner product space T. A stochastic process {X(t)} is called self-similar (s.s.) if it is continuous in probability at each t > 0 and if for every a > 0 there exist a positive number B(a) and a vector b(a) in Tsuch that the process {X(at)} and {B(a)X(t) + b(a)} have the same finite-dimensional distributions. Lamperti actually called such processes " semi-stable" but currently this term is being used in another sense. These processes arise in diverse areas (see Taqqu [14]). Self-similar processes are also obtained by various limiting procedures (e.g., see Gorodetskii [4], Kesten and Spitzer [6], Sinai

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A general representation for an operator-self-similar process is obtained and its class of exponents is characterized. It is shown that such a process is the limit in a certain sense of an operator-normed process and any limit of an operator-normed process is operator-self-similar. 1. Introduction. In 1962 Lamperti [8] introduced the notion of a self-similar process, {X(t): t s* 0), taking values in a real finite-dimensional inner product space T. A stochastic process {X(t)} is called self-similar (s.s.) if it is continuous in probability at each t > 0 and if for every a > 0 there exist a positive number B(a) and a vector b(a) in Tsuch that the process {X(at)} and {B(a)X(t) + b(a)} have the same finite-dimensional distributions. Lamperti actually called such processes " semi-stable" but currently this term is being used in another sense. These processes arise in diverse areas (see Taqqu [14]). Self-similar processes are also obtained by various limiting procedures (e.g., see Gorodetskii [4], Kesten and Spitzer [6], Sinai