On the spectral representation of symmetric stable processes

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Abstract

The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into Lp (0 < p ≤ 2). We use the theory of Lp isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex Lq into real or complex Lp for 0 < p < q ≤ 2.

MSC

60G99
60G10
60E07
46E30

Keywords

Stable processes
spectral representation
stationary processes
complex processes
Lp spaces
imbedding theorems

Cited by (0)

This work is based on a portion of the author's Ph.D. thesis written at the University of Virginia under the supervision of Professor Loren Pitt. The author gratefully acknowledges Professor Pitt's advice, interest, and influence.