On the concentration phenomenon for \(\varphi\)-subgaussian random elements (Q2489814)

Let \(X=(X_1,X_2,\dots)\) be a random element with values in \(l^2\) that is scalarly \(\varphi\)-subgaussian. It means that \(E\exp\{tX_k\}\leq \exp\{\varphi(at) \}\) for all \(t\) and \(a\geq\tau_\varphi(X_k)\) with \(\sum \tau_\varphi(X_k)= \tau_\varphi(X)<\infty\), where \(\varphi\) is an \(N\)-function. If for some \(p\geq 1\) the function \(\varphi(|x|^{1/p})\) is convex, then it is shown that \(P(|\,\|X \|-E\|X\|\,|>t)\leq 4\exp\{-\varphi^*(Ct/\tau_\varphi(X))\}\), where \(\varphi^*\) is the Young-Fenchel transform of \(\varphi\) and \(C=2/(\pi\max\{1,2^{(2-p)/2p}\} )\). The case of classically subgaussian random elements \((\varphi(x)=x^2/2)\) is included as a special case.