A few remarks on Riesz summability of orthogonal series

We study convergence behavior of some sequences and series related to a given orthogonal series. Following the developed technique we define (in terms of fourth mixed moments only) a class of orthonormal functions ${\left\{ {{X_i}} \right\}_{i \geq 1}}$ such that the condition: $\exists k \in \mathbb{N}\sum\nolimits_{i \geq 1} {\mu _i^2} {\left( {{{\ln }^{\left( k \right)}}i} \right)^2} < \infty$ implies almost everywhere convergence of the series $\sum\nolimits_{i \geq 1} {{\mu _i}{X_i}}$, here for every $i = 1,2, \ldots ,j = 1, \ldots ,k$,

$${\ln ^{(1)}}i = {\ln _2}i,\quad {\ln ^{(j)}}i = {\ln _2}(\max (1,{\ln ^{(j - 1)}}i)).$$