Maximum of Random Variables

Question

Let $(X_{ni})_{i=1}^{k_n}$ be a martingale difference array. In the literature I often find the equation

$P\left(\max_{i\le k_n}|X_{ni}|>\varepsilon\right)=P\left(\sum_{i\le k_n}X_{ni}^2 1_{\{|X_{ni}|>\varepsilon\}}>\varepsilon^2\right)$.

Why does this hold?

Answer 1

Let $A:=\left\{\max_{i\leqslant k_n}|X_{ni}|>\varepsilon\right\}$ and $B:=\left\{\sum_{i\leqslant k_n}X_{ni}^2 1_{\{|X_{ni}|>\varepsilon\}}>\varepsilon^2\right\}$.

• $A\subseteq B$: if $\omega$ is such that $\max_{i\leqslant k_n}\left|X_{ni} \left(\omega\right)\right|>\varepsilon$, then there exists $i_0\leqslant k_n$ such that $\left|X_{ni_0} \left(\omega\right)\right|>\varepsilon$. Consequently, $$\sum_{i\leqslant k_n}X_{ni}\left(\omega\right)^2 1_{\{|X_{ni}\left(\omega\right)|>\varepsilon\}}\geqslant X_{ni_0}\left(\omega\right)^2 1_{\{|X_{ni_0}\left(\omega\right)|>\varepsilon\}}\gt\varepsilon^2.$$
• $A^c\subseteq B^c$: if $\omega\in A^c$, then for all $i\leqslant k_n$, $\left|X_{ni} \left(\omega\right)\right|\leqslant\varepsilon$ hence $\sum_{i\leqslant k_n}X_{ni}\left(\omega\right)^2 1_{\{|X_{ni}\left(\omega\right)|>\varepsilon\}}=0$ and $\omega\in B^c$.