The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then \begin{equation*} \left( \sum_{n = 1}^N |x_n|^2 \right)^\frac{p}{2} \approx \mathbb{E} \left| \sum_{n = 1}^N \epsilon_n x_n \right|^p. \end{equation*}

I'm curious if there is any kind of "matrix" version of this where the $\epsilon_n$'s are replaced by a sequence of independent random unitary $k \times k$ matricies satisfying certain properties (what these are is part of the question) and each $x_n \in \mathbb{C} ^k$? In particular is there any kind of new, stronger result that comes from this level of generality?

I'm very, very far from an expert in these matters so perhaps I missed something in my quick search.

  • $\begingroup$ Google "non commutative Khintchine inequality". $\endgroup$ Nov 22 '13 at 23:11
  • $\begingroup$ Are you referring to the results of Lust-Piquard and Pisier? These seem vastly different than what I'm asking and the papers I've found are closely related to their results. There is an interesting new paper of Dirksen and Ricard "Some remarks on noncommutative Khintchine inequalities" that might have something of interest $\endgroup$ Nov 22 '13 at 23:50
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    $\begingroup$ Yes, the classical results are from Lust-Piquard and Pisier, but now you have found the more recent theorems. You did not say what you have looked at, so I pointed in a direction that might help. $\endgroup$ Nov 23 '13 at 4:24

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