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- W4302201583 abstract "In this paper, we establish an extension of a noncommutative Bennett inequality with a parameter $1leq rleq2$ and use it together with some noncommutative techniques to establish a Rosenthal inequality. We also present a noncommutative Hoeffding inequality as follows: Let $(mathfrak{M}, tau)$ be a noncommutative probability space, $mathfrak{N}$ be a von Neumann subalgebra of $mathfrak{M}$ with the corresponding conditional expectation $mathcal{E}_{mathfrak{N}}$ and let subalgebras $mathfrak{N}subseteqmathfrak{A}_jsubseteqmathfrak{M},,(j=1, cdots, n)$ be successively independent over $mathfrak{N}$. Let $x_jinmathfrak{A}_j$ be self-adjoint such that $a_jleq x_jleq b_j$ for some real numbers $a_j<b_j$ and $mathcal{E}_{mathfrak{N}}(x_j)=mu$ for some $mugeq 0$ and all $1leq jleq n$. Then for any $t>o$ it holds that begin{eqnarray*} {rm Prob}left(left|sum_{j=1}^n x_j-nmuright|geq tright)leq 2 expleft{frac{-2t^2}{sum_{j=1}^n(b_j-a_j)^2}right}. end{eqnarray*}" @default.
- W4302201583 created "2022-10-06" @default.
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- W4302201583 date "2014-06-12" @default.
- W4302201583 modified "2023-09-26" @default.
- W4302201583 title "Inequalities for sums of random variables in noncommutative probability spaces" @default.
- W4302201583 doi "https://doi.org/10.48550/arxiv.1406.3220" @default.
- W4302201583 hasPublicationYear "2014" @default.
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