An interpolation inequality for moments of sums of random vectors. (English. Ukrainian original) Zbl 0990.60034

Shklyar, S. V.

Theory Probab. Math. Stat. 63, 171-177 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 156-162 (2000).

Let us denote \(M(\nu,\sigma,\eta)=\sum_{i=1}^{n}(E\|\eta_{i}\|^{\nu+\sigma})^{\nu/(\nu+\sigma)}\), where \(\eta_{i}\in B, i=1,\ldots,n\), are random vectors, \(B\) is a separable Banach space. The author proves the following result: Let \(F=\{F_{i}\}_{i=1}^{n}\) be a collection of \(\sigma\)-fields. If for any centered and adapted to \((F,B)\) (\(E\eta_{i}=0\), \(\eta_{i}\) is \(F_{i}\)-measurable) collection of random vectors \(\eta_{i}, i=1,\ldots,n\), \(E\|\sum_{i=1}^{n}\eta_{i}\|^{\nu}\leq cQ(\nu,\sigma,\eta)\), where \(c\geq 1, \nu\geq 1, \sigma \geq 1\), then we have the inequality \[ E\Biggl\|\sum_{i=1}^{n}\varphi_{i}\Biggr\|^{t}\leq 2^{2t+4\nu-1}cQ(\nu,\sigma,\varphi), \] where \[ Q(\nu,\sigma,\eta)= \begin{cases} \max\{M(\nu,\sigma,\eta),(M(2,\sigma,\eta))^{\nu/2}\},& \text{as } \nu\geq 2, \\ M(\nu,\sigma,\eta),& \text{as } 1\leq \nu\leq 2,\end{cases} \] for all \(t\), \(1\leq t\leq \nu, t\geq \nu/(2+\sigma)\), and any centered, adapted to \((F,B)\) collection of random vectors \(\varphi_{i}, i=1,\ldots,n.\)

Reviewer: A.D.Borisenko (Kyïv)

MSC:

60F99

Limit theorems in probability theory

Keywords:

interpolation inequality; moments; sums of random vectors; Banach space