Operator self‐similar processes on Banach spaces

Abstract

Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determined multiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such a power-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval (s 0 ,∞), for some s 0 ≥ 1, have a unique scaling family of operators of the form {s H : s > 0}, if the expectations of the process span a dense linear subspace of category 2. The existence of a scaling family of the form {s H : s > 0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.

Operator self-similar processes and functional central limit theorems

“…If t > u, the proof is exactly the same as in the case of t < u. If t = u, then we just use asymptotic relations (16) and (17). The proof of Proposition 6 is complete.…”

“…Matache and Matache [16] consider and study operator self-similar processes valued in (possibly infinite-dimensional) Banach spaces. Let E denote a Banach space and let L(E) be the algebra of all bounded linear operators on E. Matache and Matache [16] give the following definition.…”

Spectral analysis of multifractional LRD functional time series

“…As commented in Introduction, the literature on LRD modeling in functional sequences has been mainly developed in the time domain, under the context of linear processes in Hilbert spaces (see, e.g., [14,26,32,33]), paying special attention to the theory of operator self-similar processes (see [10,23,24,29], among others).…”

Operator Fractional Brownian Motion as Limit of Polygonal Lines Processes in Hilbert Space

“…As mentioned in the Introduction, self-similarity of processes introduced by Lamperti [16] is one of the sources of long memory. Operator self-similar processes appeared later in the paper by Laha and Rohatgi [15] and were investigated by Matache and Matache [20]. Let us recall that a stochastic process {X(t), t > 0} with values in a Banach space E is operator self-similar if there exists a family of linear bounded operators {A(a), a > 0} on E such that for each a > 0, …”