Boundedness of random series Let $(a_{ij})_{(i,j) \in \mathbb{N}^2}$ be a doubly-indexed sequence of real numbers such that,
for every $j \in \mathbb{N}$,
$ \sum_i a_{ij}^2 < \infty$. Set then, for each $j \in \mathbb{N}$, $Z_j = \sum_i a_{ij} X_i$, where
$(X_i)_{i \in \mathbb{N}}$ is a sequence of independent symmetric $\pm 1$ Bernoulli random variables.
Is there a sufficient (and necessary) condition in order that $\sup_j Z_j  < \infty $ almost surely?
 A: $\newcommand\N{\mathbb N}\newcommand\ga\gamma$Consider the set
$$T_a:=\{ (a_{ij})_{i\in\N}\colon j\in\N\}\subset\ell^2$$
of the columns of the matrix $a:=(a_{ij})_{(i,j)\in\N^2}$. For such a matrix $a$, let $Z^a_j:=\sum_{i\in\N} a_{i,j}X_i$.
Then, according to the Bednorz--Latała solution of the so-called Bernoulli Conjecture, there is a universal constant $c\in(1,\infty)$ such that for any matrix $a$ with $T_a\subset\ell^2$ there exist subsets $R_a$ and $S_a$ of $\ell_2$ such that $T_a\subset R_a+S_a:=\{r+s\colon r\in R_a, s\in S_a\}$ and
$$B(a)/c\le E\sup_j Z^a_j\le cB(a),$$
where
$$B(a):=\sup_{r\in R_a}\|r\|_1+g(S_a),$$
$$g(S):=E\sup_{s\in S}\sum_{i\in\N}s_iG_i,$$
and the $G_i$'s are independent standard Gaussian random variables.
In turn, estimates of the form
$$\ga(S)/C\le g(S)\le C\ga(S) $$
are available for some universal constant $C\in(1,\infty)$, where $\ga(S)$ is expressed in terms of a so-called majorizing measure or in terms of a suitable sequence of admissible partitions of the set $S$ -- see the displayed formulas on page 1168 of the cited paper by Bednorz and Latała.
