Operator-norm bound for bounded complex iid matrices

If $K=0$, every entry is zero almost surely and the conclusion is immediate. Assume $K>0$. Regard the complex unit sphere in ${\mathbb{C}}^d$ as the real unit sphere in ${\mathbb{R}}^{2d}$. For $0<\epsilon <1\mathrm{/}2$, a maximal $\epsilon$-separated subset is an $\epsilon$-net; the disjoint radius-$\epsilon \mathrm{/}2$ balls about its points lie in the radius-$(1+\epsilon \mathrm{/}2)$ ball, so volume comparison bounds its cardinality by $(1+2\mathrm{/}\epsilon {)}^{2d}$. If ${\mathcal{N}}_1,{\mathcal{N}}_2$ are such nets for the two unit spheres and $A$ is an $n\times N$ complex matrix, approximating unit vectors $u,v$ attaining $\mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}=\mathrm{\mid }{u}^{\ast }Av\mathrm{\mid }$ gives $\mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le (1-2\epsilon {)}^{-1}{\max }_{u_0\in {\mathcal{N}}_1,v_0\in {\mathcal{N}}_2}\mathrm{\mid }{u}_0^{\ast }A{v}_0\mathrm{\mid }\mathrm{.}$ Indeed, the approximation error is at most $\mathrm{\parallel }u-u_0\mathrm{\parallel }\mathrm{\parallel }A\mathrm{\parallel }+\mathrm{\parallel }v-v_0\mathrm{\parallel }\mathrm{\parallel }A\mathrm{\parallel }\le 2\epsilon \mathrm{\parallel }A\mathrm{\parallel }$.

Fix deterministic unit vectors $u\in {\mathbb{C}}^n$ and $v\in {\mathbb{C}}^N$. Then $u^{\ast }{X}_{n}v={\sum }_{i,j}{\bar{u}}_i{v}_j{\xi }_{ij}$ is a sum of independent centered complex variables. Each real or imaginary summand is bounded in absolute value by $K\mathrm{\mid }{u}_i\mathrm{\mid }\mathrm{\mid }{v}_j\mathrm{\mid }$. If a centered real variable $Y$ is bounded by $b$, convexity of the exponential on $[-b,b]$ and $\mathbb{E}Y=0$ give $\mathbb{E}{e}^{\lambda Y}\le \cosh (\lambda b)\le e^{{\lambda }^2{b}^2\mathrm{/}2}$. Independence, exponential Markov inequality [F3], and optimization in $\lambda$ therefore give $\mathbb{P}\{\mathrm{\mid }{u}^{\ast }{X}_{n}v\mathrm{\mid }>t\}\le 4\exp \text{\hspace{0.17em}⁣}(-\frac{t^2}{4K^2}),$ because ${\sum }_{i,j}{K}^2\mathrm{\mid }{u}_i{\mathrm{\mid }}^2\mathrm{\mid }{v}_j{\mathrm{\mid }}^2=K^2$.

Choose $\epsilon =1\mathrm{/}8$ in step 1.1. The product of the two net cardinalities is at most ${17}^{2(n+N_n)}$. Taking $t=L\sqrt{n+N_n}$ with $L$ large enough that $L^2\mathrm{/}(4K^2)>2\log 17+1$, the union bound and step 2.1 make the probability that the net maximum exceeds $t$ at most $4e^{-(n+N_n)}$. These probabilities are summable in $n$ because $N_n\mathrm{/}n\to 1\mathrm{/}c>0$. Since all $X_n$ are defined on the common probability space in the Statement, [F2] applies to this sequence of events and the net maximum is at most $L\sqrt{n+N_n}$ eventually almost surely. The net inequality in step 1.1 and boundedness of $n\mathrm{/}{N}_n$ then yield $\mathrm{\parallel }{X}_n{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le C\sqrt{N_n}$ eventually, with deterministic $C=C(K,c)$.