Suppose that $X\_1,X\_2,\ldots$ are independent random variables. Denote $F\_t$ the $\sigma$-algebra generated by $X\_1,\ldots,X\_t$. Let $T, M \geq 1$, and let $f\_1,\ldots,f\_M$ be $\mathbb{R} \to \mathbb{R}$ functions such that for every $m =1,\ldots, M$, $t =1,\ldots, T$, $E[f\_m(X\_t) | F\_{t-1}]= 0$.
Denote for every $t, m$, $S\_{m,t} = \sum\_{s=1}^t f\_m(X\_t)$.


Suppose in addition that there exists $\sigma^2 < \infty$ such that for every $t, m$, $E[f\_m^2(X\_t) | F\_{t-1}] \leq \sigma^2$.


For any fixed $m$, we know from Kolmogorov's inequality for martingales that, for any $x > 0$
$$P\left[ \max\_{t=1,\ldots, T} \frac{S\_{m,t}}{\sqrt{T}} > x \right] \leq x^{-2} \sigma^2.$$


Do we have something like
$$P\left[\max\_{m=1,\ldots,M} \max\_{t=1,\ldots, T} \frac{S\_{m,t}}{\sqrt{T}} > x \right] \leq C x^{-2} \sigma^2$$ for a constant that doesn't depend on $M$?



Extension of Kolmogorov's martingale inequality
Of course not. E.g., suppose that the $X\_i$'s are independent random variables each uniformly distributed on the interval $(0,1)$. Let $f\_m(X\_i):=B\_{i,m}-1/2$, where $B\_{i,m}$ is the $m$th digit in the binary expansion of $X\_i$. Then all your conditions hold, with $\sigma^2=1/4$. Also, the $f\_m(X\_i)$'s are i.i.d.


So,
$$P\Big(\max\_{1\le m\le M} \max\_{1\le t\le T} \frac{S\_{m,t}}{\sqrt{T}}>x\Big)
=1-P\Big(\max\_{1\le t\le T}\frac{S\_{1,t}}{\sqrt{T}}\le x\Big)^M \\ 
\ge1-P\Big(\frac{S\_{1,T}}{\sqrt{T}}\le x\Big)^M
\underset{M\to\infty}\longrightarrow1
 \not\le C x^{-2} \sigma^2$$
for any real $C>0$ and any $x>\sigma\sqrt C$ such that
$$P\Big(\frac{S\_{1,T}}{\sqrt{T}}\le x\Big)<1$$
-- which will hold if $T>4x^2$, because $P(S\_{1,T}=T/2)=2^{-T}>0$. $\quad\Box$


