Extension of Kolmogorov's martingale inequality

Question

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 $M$?

Answer 1

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 $P(S_{1,T}=T/2)=2^{-T}>0$. $\quad\Box$