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

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 $$\varphi_{i}, i=1,\ldots,n.$$

### MSC:

 60F99 Limit theorems in probability theory