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For each column $X_j$, the normalized score isThe errors are independent Rademacher random variables, and $\Vert X_j{\Vert }_2^2=n$. The Hoeffding lemma therefore makes $W_j$ a sub-Gaussian random variable with variance proxy $1/n$, soThe union bound with $t=\lambda /2$ givesFor $\lambda =A\sqrt{\log p/n}$ this becomesIn particular, if $A>\sqrt{8}$, the lower bound tends to one as $p\to \infty$.

Hoeffding lemma states that if $a\le X\le b$ almost surely, then for every real $\lambda$,Convexity of $e^{\lambda x}$ bounds it on $[a,b]$ by the secant joining its endpoint values. Taking expectations reduces the centered moment-generating function to that of a two-point variable on $\{a,b\}$ having the same mean. After rescaling to $[0,1]$, its logarithm iswhere $q$ is its mean. Twice differentiating in $\lambda$ shows that the second derivative is a Bernoulli variance and hence at most $1/4$. The value and first derivative vanish at zero, so Taylor's theorem gives at most ${\lambda }^2/8$. Rescaling proves the claim.