A Gaussian maximum without independence

Given: AC, $\sigma >0$, $m\ge 1$, and a standard normal random variable $G$.

Assume AC as required by the standard-normal law construction (The Axiom of Choice).

The law $N(0,1)$ has density $\varphi (x)=e^{-x^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }$, and $N(0,{\sigma }^2)$ is the law of $x\mapsto \sigma x$ applied to $N(0,1)$ (Standard normal and normal laws).

The standard normal density is positive and integrates to one (The standard normal density has total mass one).

For a nonnegative measurable $g$, or an integrable real $g$, expectation of $g(G)$ is its integral against the law of $G$ (Change of variables for expectation). Integration against the density law satisfies $\int gdN(0,1)=\int g(x)\varphi (x)dx$ for nonnegative measurable $g$ (Integrating against a density agrees with integrating the product); applying this to the positive and negative parts gives the signed formula once absolute integrability is established.

One-dimensional substitution satisfies ${\int }_{\phi (c)}^{\phi (d)}h={\int }_c^d(h\circ \phi ){\phi }^{\mathrm{\prime }}$ when $h$ is continuous and the stated differentiability and integrability hypotheses hold (Substitution: if $\phi$ is differentiable on $[c,d]$ with ${\phi }^{\mathrm{\prime }}$ integrable and $f$ is continuous on an interval containing $\phi ([c,d])$, then ${\int }_{\phi (c)}^{\phi (d)}f={\int }_c^d(f\circ \phi ){\phi }^{\mathrm{\prime }}$).

Increasing nonnegative measurable functions pass to their limit under the integral (Monotone convergence for the integral).

$e^{x+y}=e^x{e}^y$ (The exponential addition formula $\exp (x+y)=\exp (x)\exp (y)$).

The exponential is continuously differentiable with derivative $e^x$ (The real exponential function and the number $e$ by a power series, The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$); the chain rule differentiates compositions (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$); and $(x^2\mathrm{/}2{)}^{\mathrm{\prime }}=x$ (For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term).

Continuous functions on compact intervals are integrable (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion), and if $H^{\mathrm{\prime }}=g$ with integrable $g$, then $\int g=H(d)-H(c)$ (The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$).

A continuous real function is Borel measurable under AC (Continuous functions on Euclidean spaces are Borel measurable).

A centered variable is sub-Gaussian with proxy ${\sigma }^2$ when its moment generating function is bounded by $e^{{\lambda }^2{\sigma }^2\mathrm{/}2}$ for every real $\lambda$ (Sub-Gaussian variables and vectors, and sub-exponential variables).

For $m$ possibly dependent centered sub-Gaussian variables with common proxy ${\sigma }^2$, $\mathbb{P}({\max }_j\mathrm{\mid }{Y}_j\mathrm{\mid }\ge t)\le 2m{e}^{-t^2\mathrm{/}(2{\sigma }^2)}$ (Finite sub-Gaussian maxima and epsilon-net reduction).

Expectation is linear on integrable real random variables (Linearity, monotonicity, and the modulus bound for expectation).

Under Countable Choice, every bounded Riemann-integrable function on a nondegenerate compact interval has the same Lebesgue and Riemann integrals (A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral). AC in [A1] implies Countable Choice, so this applies to the bounded continuous integrands supplied by [F8].