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Convergence of quadratic forms of random elements in a Banach algebra. (Russian) Zbl 0585.60010

Teor. Veroyatn. Mat. Stat. 31, 31-39 (1984).
Let B (resp. H) be a separable Banach (resp. Hilbert) *-algebra with the norm \(\| \|\), \(\{X_ i\}\) and \(\{Y_ i\}\) sequences of independent random elements with values in B or H and \(\{a_{ij}\}^{\infty}_{i,j=1}\) a family of reals. Let \(\theta\) denote the zero element of B(H). For \(n\geq 1\) consider the quadratic forms \(S_ n=\sum^{n}_{i,j=1}a_{ij}X_ iX_ j\); \(T_ n=\sum^{n}_{i,j=1}a_{ij}X_ iY_ j.\)
In the present paper the author gives a number of theorems devoted to necessary or sufficient conditions for mean and pointwise convergence of the sequences \(\{S_ n\}\) and \(\{T_ n\}\).
Reviewer: Sh.Ayupov

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60B11 Probability theory on linear topological spaces