Strong Mixing and Operator-Selfdecomposability

Abstract

For nonstationary, strongly mixing sequences of random variables taking their values in a finite-dimensional Euclidean space, with the partial sums being normalized via matrix multiplication, with certain standard conditions being met (an “infinitesimality” assumption, and a restriction to “full” distributions), the possible limit distributions are precisely the operator-self-decomposable laws.

Appendix

For ease of reference let us quote here the following algebraic facts.

Theorem 2

Each locally compact subsemigroup S of a compact group G is a compact subgroup.

Cf. [15], Theorem 1.1.12.

Theorem 3

If $T:(0,\infty )\rightarrow \mathfrak {B}$ (a real or complex Banach algebra) satisfies

$$\begin{aligned} T (t+s)=T(t)T(s)\ \ \text{ for } \text{ all } \ \ 0< t, s < \infty \ \ \ \text{ and } \ \ \ \lim _{t\rightarrow 0} T(t)=J \ \text{(an } \text{ idempotent) }, \end{aligned}$$

then there exists an element $A \in \mathfrak {B}$ such that

$$\begin{aligned} T(t)=J+\sum _{n=1}^{\infty }\frac{t^n}{n!}\, A^n \ \ \ \text{(absolutely } \text{ convergent } \text{ series) }. \end{aligned}$$

(37)

Cf. [7], Theorem 8.4.2 or [8], Theorem 9.4.2.

Theorem 4

(Pontriagin Theorem) Suppose a topological group $G^{\prime }$, generated by a compact set, contains a compact subgroup $H^{\prime }$ such that $G^{\prime }/H^{\prime }$ is isomorphic with an $r$-dimensional real vector group $V_r$. Then $G^{\prime }$ has a vector subgroup $E_r$ such that $G^{\prime }= H^{\prime }\oplus E_r$

Cf. [13], p. 187.