Convergence of trigonometric series in norms of Orlicz spaces (Q918029)

The author considers stochastic series of the form \[ R(t)=\sum^{\infty}_{k=0}T_ k(t)\xi_ k,\quad 0\leq t\leq 1, \] where each \(T_ k(t)\) is a trigonometric polynomial on [0,1] of degree k, and where \(\{\xi_ k\}^{\infty}_{k=0}\) is a sequence of random variables defined on some probability space (\(\Omega\),\({\mathcal B},P)\). The author connects two (related) Orlicz N-functions \(\phi\) and u to such a series. The first is used to define a notion of a ``sub-Gaussian'' random variable, and it seems to be assumed that all random variables in question (including \(\xi_ k\) for each \(k\geq 0\) and R(t) for each \(t\in [0,1])\) are sub-Gaussian with respect to this fixed function \(\phi\). The second is used to define the standard Orlicz space \(L_ u\) consisting of (equivalence classes of) Lebesgue measurable functions f(t) on [0,1], for which the norm \[ \| f\|_ u=\| f(t)\|_ u=\inf \{r\geq 0:\;\int^{1}_{0}u(r^{-1}f(t))dt<2\} \] is finite. The goal of the investigation is to find conditions under which the trajectories \(t\to R(t)(\omega)\) belong to \(L_ u\) for almost all \(\omega\in \Omega\), and under which the ``tail sequence'' \[ X_ n=\| R(t)- \sum^{n}_{k=0}T_ k(t)\xi_ k\|_ u \] converges to zero in probability. Some estimates of the rate of convergence are also provided, and the details are horrendously technical.