Remarks on the Selfdecomposability and New Examples

Abstract

Abstract. The analytic property of the seljdecomposability of characteristic functions is presented from stochastic processes point of view. This provides new examples or proofs, as well as a link between the stochastic analysis and the theory of characteristic functions. A new interpretation of the famous Levy's stochastic area formula is given. Introduction and notationsThe class of selfdecomposable probability distributions, denoted as SD, (known also as the class L or Levy class L distributions), appears in the theory of limiting distributions as laws of normalized partial sums of independent random variables but not necessarily identically distributed. However, the additional assumption of the infinitesimality of the summands guarantees their infinite divisibility; cf. Jurek & Mason (1993), Section 3.3.9.All our random variables or stochastic processes are defined on a fixed probability space (fI, T, V). For a given random variable X (for short: rv) or its probability distribution /x = C(X) or its probability density /, provided it exits (i.e.,d/j,(x) = f(x)dx), we define its characteristic function (in short: char.f.) 4>x{t) = 4>(t) as follows dV(cj) = J e itx dfi(x), t € R.n R Research supported in part by Grant no. 2 P03 A02914 from KBN, Warsaw, Poland.

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