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STRONG LAWS OF LARGE NUMBERS FOR LINEAR PROCESSES GENERATED BY ASSOCIATED RANDOM VARIABLES IN A HILBERT SPACE

  • Ko, Mi-Hwa
  • Received : 2008.10.13
  • Accepted : 2008.11.27
  • Published : 2008.12.25

Abstract

Let ${{\xi}_k,k{\in}{\mathbb{Z}}}$ be an associated H-valued random variables with $E{\xi}_k$ = 0, $E{\parallel}{\xi}_k{\parallel}$ < ${\infty}$ and $E{\parallel}{\xi}_k{\parallel}^2$ < ${\infty}$ and {$a_k,k{\in}{\mathbb{Z}}$} a sequence of bounded linear operators such that ${\sum}^{\infty}_{j=0}j{\parallel}a_j{\parallel}_{L(H)}$ < ${\infty}$. We define the sationary Hilbert space process $X_k={\sum}^{\infty}_{j=0}a_j{\xi}_{k-j}$ and prove that $n^{-1}{\sum}^n_{k=1}X_k$ converges to zero.

Keywords

Strong laws of large numbers;linear process;associated;linear operator;H-valued random variable

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