The strong mixing and the selfdecomposability properties

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Abstract

It is proved that infinitesimal triangular arrays obtained from normalized partial sums of strongly mixing (but not necessarily stationary) random sequences can produce as limits only selfdecomposable distributions.

Section snippets

Notations and basic notions

Let (Ω,F,P) be a probability space. Let E be a real separable Banach space, with norm and Borel sigma-algebra E. By PP(E) we denote the set of all Borel probability measures on E, with the convolution operation denoted by “” and weak convergence denoted by “”, which make P a topological convolution semigroup.

Measurable functions ξ:ΩE are called Banach space valued random variables (in short: E-valued rv’s) and L(ξ)(A)P{ωΩ:ξ(ω)A}, for AE, is the probability distribution of ξ. Then

Strong mixing and selfdecomposability

Here is the main result of this note.

Theorem 1

Let X(X1,X2,) be a sequence of Banach space E valued random variables with partial sums SnX1+X2++Xn, and let (an) and (bn) be sequences of real numbers and elements in E respectively, and suppose the following conditions are satisfied:

  • (i)

    α(n)0 as n, i.e. the sequence X is strongly mixing;

  • (ii)

    an>0 and the triangular array (anXj,1jn,n1) is infinitesimal;

  • (iii)

    anSn+bnμ as n for some non-degenerate probability measure μP.

Then the limit distribution μ is

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There are more references available in the full text version of this article.
1

Research funded by Narodowe Centrum Nauki (NCN) grant no Dec2011/01/B/ST1/01257.

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