Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 1 , pp. 25--43

IDENTICALLY DISTRIBUTED STOCHASTIC INTEGRALS, STABLE PROCESSES AND SEMI-STABLE PROCESSES

By

B. RAMACHANDRAN, *Indian Statistical Institute*

SUMMARY. Let {X(t), t ³ 0} be a stochastic process, continuous in probability, homogeneous and with independent increments. In this paper we consider conditions under which the identical distribution, to within a shift, of two stochastic integrals defined (in the sense of convergence probability) with respect to the process implies that the process is semistable. Our discussions is mostly based on Riedel (1980b), which concerns itself with stable processes in the same context after identifying some important consequences of the equidistribution assumption. Our main result is Theorem I. We also examine two particular cases, each in the light of Theorem I and independently through more ‘elementary’ arguments. The first (Theorem 2) is a stronger version of a result due to Prakasa Rao and Ramachandran (1983, 1984) and incidentally shows that the technical conditions (3) imposed in (the preamble to) Theorem I, while sufficient, are not necessary. The second (Theorem 3) is a strong version of a result due to Lukacs (1969) on stable processes.

*AMS (1991) subject classification.* Primary 62E10; secondary 60E10, 60E07.

*Key words and phrases.* Stochastic integrals, stable processes, semi-stable processes, equi distribution problems.

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