Lower bound for the maximum of a stochastic process (Q1905341)

The authors extend a well-known result on Rademacher functions to functions bounded in \(L_3\)-norm, and show that if both \(\xi_j\) and \(\varphi_j (x)\) are independent random variables on possibly different spaces, and both of them are bounded in \(L_3\)-norm, then (on the product space) the expectation of the \(L_\infty\)-norm of the sum \(\sum^n_{j = 1} \xi_j \varphi_j (x)\) is bounded from below by a constant times \((n \log n)^{1/2}\). A major step in the proof is to establish a lower bound of \[ E [\max_{1 \leq u \leq n} |\sum^n_{j = 1} \xi_j \varphi_j (x_u) |], \] where \(x_u\) is a properly chosen sequence of points.