On dimension of support for stochastic processes with independent increments

Abstract

Introduction.Herman Rubin [5] showed that if v(t) is a real valued function which is monotone nondecreasing on the interval (0, a), where a is possibly +00, and if 0^i;(i)^l, then there exists a "pure" stochastic process {Xt : t e [0, 00)} (i.e. for t e (0, 00) the distribution function of Xt, denoted by Ft, is either absolutely continuous or continuous singular) with stationary independent increments such that if t<a, then Ft is continuous singular with the dimension of its support, denoted by dim supp Ft, being v(t); and if (5«, then Ft is absolutely continuous.This can be interpreted as saying that for a "pure" process with stationary independent increments, the only general properties the dimension of the support of Ft has to have as a function of /, are the obvious ones of being monotone nondecreasing and being bounded between 0 and 1.One is now led to remove the restriction that the process be "pure," to isolate one's attention to the continuous singular component of Ft, denoted by (Ft)c.s., and to ask if a similar result to that of Rubin's holds for dim supp (Ft)c.s.The answer is yes.In §l of this paper, it is shown that if {Xt: t e [0,00)} is a stochastic process with independent increments then dim supp (Ft)c.s.= lim inf dim supp (Fz)c.s. for all t g (0, 00) except at possibly a countable number of points.Conversely, in §2 it is shown that if/maps (0, 00) into [0, 1] and is a lower semicontinuous function at all but possibly a countable number of points, then there exists a stochastic process {Xt : t e [0, 00)} with stationary independent increments such that dim supp (Ft)c.s.=/(/).These two results are obtained while investigating the relationship between the total variation of (Ft)c.s., denoted by T.y.(Ft)c.s., and dim supp (F()c.s. for a process with independent increment.

Locations

• Transactions of the American Mathematical Society - View - PDF