The strong mixing and the selfdecomposability properties
Section snippets
Notations and basic notions
Let be a probability space. Let be a real separable Banach space, with norm and Borel sigma-algebra . By we denote the set of all Borel probability measures on , with the convolution operation denoted by “” and weak convergence denoted by “”, which make a topological convolution semigroup.
Measurable functions are called Banach space valued random variables (in short: -valued rv’s) and , for , is the probability distribution of . Then
Strong mixing and selfdecomposability
Here is the main result of this note. Theorem 1 Let be a sequence of Banach space E valued random variables with partial sums , and let and be sequences of real numbers and elements in respectively, and suppose the following conditions are satisfied: as , i.e. the sequence is strongly mixing; and the triangular array is infinitesimal; as for some non-degenerate probability measure .
Then the limit distribution is
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Research funded by Narodowe Centrum Nauki (NCN) grant no Dec2011/01/B/ST1/01257.