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Quadratic variation and the convergence of random sequences. (English. Russian original) Zbl 0623.60060

Sib. Math. J. 27, 876-879 (1986); translation from Sib. Mat. Zh. 27, No. 6(160), 111-115 (1986).
Let (\(\Omega\),\({\mathcal F},{\mathbb{F}},P)\) be a filtered probability space and \(X=\{X_ n\), \(n=0,1,...\}\) be an \({\mathbb{F}}\)-adapted sequence, \(\Delta X_ 0=0\), \(\Delta X_ k=X_ k-X_{k-1}\), \(k\geq 1\), \(\epsilon_ k=E(\Delta X_{k+1}| {\mathcal F}_ k)\), \(k\geq 0\). Two distributions of the subset \(\{S(X)<\infty \}\cap \{-\infty <\lim X_ n<\infty \}\), where \(S^ 2(X)=\sum^{\infty}_{k=0}(\Delta X_ k)^ 2\), are presented up to equivalence with respect to the measure P.
If \(X_ n\geq 1\), \(EX_ n<\infty\) for all \(n\geq 0\) and \(E\sup_{n\geq 0}(\Delta X_ n)^ 2<\infty\), then \[ \{S(X)<\infty \}\quad \cap \quad \{\lim X_ n<\infty \} = \{\sum^{\infty}_{k=0}\epsilon_ k/X_ k<\infty \}. \] If \(E| X_ n| <\infty\) for all \(n\geq 0\) and \(E\sup_{n\geq 0}(\Delta X_ n)^ 2<\infty\), then \[ \{S(X)<\infty \}\quad \cap \quad \{-\infty <\lim X_ n<\infty \} = \{\inf X_ n>- \infty \}\quad \cap \quad \{-\infty <\sum^{\infty}_{k=0}\epsilon_ n<\infty \}\quad \cap \quad \{\sum^{\infty}_{n=0}\epsilon^ 2_ n<\infty \}. \] Some extensions and applications of these results are also discussed.
Reviewer: B.Grigelionis

MSC:

60G42 Martingales with discrete parameter
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