I am looking for the applications of non-commutative Khinchine inequality (see below) in case when Rademacher random variables are tight by the condition $\sum_{i=1}^N\varepsilon_i=M, \, -N \leq M\leq N$.

Let $\varepsilon_i, i=1, \ldots, N$ be independent Rademacher random variables. Let $\Gamma_i, i=1, \ldots, N$ be real (or complex) matrices of the same dimension. \begin{align}\label{Noncommutkhin} E \left\|\sum_{i=1}^N \varepsilon_i \Gamma_i\right\|_{S_{p}}^{p}\leq C\sqrt p \max\left\{\left\|\left(\sum_{i=1}^N\Gamma_i\Gamma_i^*\right)^{1/2}\right\|_{S_{p}}^{p}, \, \left\|\left(\sum_{i=1}^N\Gamma_i^*\Gamma_i\right)^{1/2}\right\|_{S_{p}}^{p}\right\}. \end{align}


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