Proof of Moments and Stability of the Standard Normal Distribution

Throughout, $g(x)=\exp(-x^{2}/2)$ with the exponential function, $\lambda$ is Lebesgue measure, and $N(B)=\frac1c\int\mathbf{1}_B\,g\,d\lambda$ with $c=\int g\,d\lambda\in(0,\infty)$ is the standard normal distribution, with cumulative distribution function $\Phi$; the needed properties of $c$, $N$, and $\Phi$ are claims 1–3 of The Gaussian Weight Defines a Probability Distribution. From the series definition of $\exp$: $\exp$ is positive and nondecreasing on $\mathbb{R}$ with $\exp(s+t)=\exp(s)\exp(t)$ and $\exp'=\exp$ (claims of Basic Properties of the Exponential Function), $\exp(t)\ge t^{2}/2$ and $\exp(t)\ge t^{4}/24$ for $t\ge0$ (single terms of the defining series, all of whose terms are nonnegative for $t\ge0$), and $0<g\le1$.

Preliminary (P1): densities. For every nonnegative Borel measurable $h:\mathbb{R}\to\mathbb{R}$,

Standard machine, as in the proof of Change of Variables for Expectations: for indicators this is the definition of $N$; it extends to nonnegative simple $h$ by linearity (claim 1 of Linearity and Monotonicity of the Lebesgue Integral); for general nonnegative $h$, the dyadic staircase compositions $\varphi_L\circ h$ increase to $h$ and $(\varphi_L\circ h)g$ increases to $hg$ pointwise, so Monotone Convergence Theorem on both sides concludes.

Preliminary (P2): affine changes of variable for $\lambda$. Let $T(x)=\alpha x+\beta$ with $\alpha\neq0$. From the Lebesgue outer measure via interval covers: images of intervals under $T$ are intervals with lengths multiplied by $|\alpha|$, so covers map to covers and $\lambda^{*}(T(E))\le|\alpha|\lambda^{*}(E)$; applying the same to $T^{-1}$ gives equality, $\lambda^{*}(T(E))=|\alpha|\lambda^{*}(E)$. Caratheodory measurability is preserved: for any test set $A$,

Since Borel sets are Lebesgue measurable and $T^{-1}$ is again affine, $\lambda(T^{-1}(B))=|\alpha|^{-1}\lambda(B)$ for every Borel $B$ ($T^{-1}(B)$ is Borel because $T$ is continuous, so preimages of the generating open sets are open). By the standard machine again (indicators: $\int\mathbf{1}_B\circ T\,d\lambda=\lambda(T^{-1}(B))=|\alpha|^{-1}\int\mathbf{1}_B\,d\lambda$; simple functions; Monotone Convergence Theorem), for every nonnegative Borel $h$,

In particular ($\alpha=-1$, $\beta=0$) reflection preserves all such integrals.

Step 1 (Claim 1). First, finiteness. For $|x|\ge4$ we have $x^{2}/2\ge2|x|$, so by monotonicity and the product property of $\exp$, together with $\exp(s)\ge s^{4}/24$ (i.e. $\exp(-s)\le 24 s^{-4}$ for $s>0$),

Also $\int_{\mathbb{R}}\exp(-|x|)\,d\lambda\le4$: by (P2) with reflection, it is at most twice $\int_{[0,\infty)}\exp(-x)\,d\lambda$; by Monotone Convergence Theorem along $\mathbf{1}_{[0,L)}\exp(-x)\uparrow\mathbf{1}_{[0,\infty)}\exp(-x)$ and monotonicity on each $[k,k+1)$ (where $\exp(-x)\le\exp(-k)$, a constant on an interval of length one), $\int_{[0,\infty)}\exp(-x)\,d\lambda\le\sum_{k\ge0}\exp(-1)^{k}=\frac{1}{1-\exp(-1)}\le2$, the geometric series bound using $\exp(1)\ge2$ (series) hence $\exp(-1)\le1/2$. On $|x|<4$, $|x|^{3}g\le64$. Hence, splitting $\mathbb{R}$ into $[-4,4]$ and its complement (additivity via linearity applied to the two indicator pieces),

Since $x^{2}\le1+|x|^{3}$ and $|x|\le1+|x|^{3}$ pointwise, the functions $|x|g$, $x^{2}g$, $|x|^{3}g$ are all $\lambda$-integrable, so by (P1) the functions $|t|$, $t^{2}$, $|t|^{3}$ are $N$-integrable. By claims 1 and 2 of Change of Variables for Expectations (the maps $t$, $t^{2}$, $|t|^{3}$ are continuous, hence Borel), $Z$, $Z^{2}$, $|Z|^{3}$ are integrable with $\mathbb{E}[Z]=\int t\,dN$, $\mathbb{E}[Z^{2}]=\int t^{2}\,dN$.

$\mathbb{E}[Z]=0$: writing $t=t^{+}-t^{-}$, (P1) gives $\int t^{+}dN=\frac1c\int\mathbf{1}_{(0,\infty)}(x)\,x\,g(x)\,d\lambda$ and $\int t^{-}dN=\frac1c\int\mathbf{1}_{(-\infty,0)}(x)(-x)g(x)\,d\lambda$; the reflection $x\mapsto-x$ of (P2) carries the second integrand into the first ($g$ is even), so the two are equal and finite, and their difference is $0$.

$\mathbb{E}[Z^{2}]=1$: define $A(x)=x\,g(x)$. For differentiable real functions the product rule $(fg)'(x)=f'(x)g(x)+f(x)g'(x)$ follows from the factorization $f(x+s)g(x+s)-f(x)g(x)=(f(x+s)-f(x))g(x+s)+f(x)(g(x+s)-g(x))$ and limit arithmetic (Derivative at an Interior Point); and $g'(x)=-x\,g(x)$ by the one-dimensional chain rule applied to $\exp$ after the map $x\mapsto-x^{2}/2$, using $\exp'=\exp$. Hence $A'(x)=(1-x^{2})g(x)$, continuous, so $A$ is an antiderivative of $(1-x^{2})g$ on every $[-L,L]$, and Fundamental Theorem of Calculus, Part II in One Dimension gives the Riemann integral $\int_{-L}^{L}(1-x^{2})g(x)\,dx=A(L)-A(-L)=2Lg(L)$. By Agreement of the Riemann and Lebesgue Integrals for Continuous Functions on a Closed Interval and linearity (all pieces integrable by the bounds above),

As $L\to\infty$: $\mathbf{1}_{[-L,L]}g\uparrow g$ and $\mathbf{1}_{[-L,L]}x^{2}g\uparrow x^{2}g$ pointwise, so by Monotone Convergence Theorem the left side tends to $c-\int x^{2}g\,d\lambda$; and $0\le 2Lg(L)\le2L\cdot 8/L^{4}=16/L^{3}\to0$ using $\exp(L^{2}/2)\ge(L^{2}/2)^{2}/2=L^{4}/8$. Hence $\int x^{2}g\,d\lambda=c$ and $\mathbb{E}[Z^{2}]=\frac1c\cdot c=1$ by (P1). Finally $\operatorname{Var}(Z)=\mathbb{E}[(Z-\mathbb{E}[Z])^{2}]=\mathbb{E}[Z^{2}]=1$ by Expectation, Variance, and Moments and $\mathbb{E}[Z]=0$.

Step 2 (Claim 2). We use the vocabulary of Joint Distribution, Expectations, and Block Independence for Independent Random Variables. The map $(x,y)\mapsto ax+by$ is jointly Borel (claim 4: scalar multiples are Borel compositions of projections, sums of jointly Borel maps are jointly Borel), so $aZ_1+bZ_2$ is a random variable, and by claims 1 and 2 there, with $C_t=\{(x,y):ax+by\le t\}$,

By Tonelli and Fubini Theorems (Tonelli; $N$ is a probability measure, hence $\sigma$-finite) and then (P1) twice with (P2),

where the inner rewriting used (P2) with the map $u\mapsto(u-by)/a$ (slope $1/a>0$): $\int\mathbf{1}_{\{ax\le t-by\}}g(x)\,d\lambda(x)=\frac1a\int\mathbf{1}_{(-\infty,t]}(u)g((u-by)/a)\,d\lambda(u)$; the map $y\mapsto N((-\infty,(t-by)/a])=\Phi((t-by)/a)$ is Borel since $\Phi$ is continuous (claim 3 of The Gaussian Weight Defines a Probability Distribution) and the composition is with an affine map. Since $a^{2}+b^{2}=1$, expanding both sides of

(multiply by $a^{2}$; both sides become $u^{2}+y^{2}-2buy$) and using $\exp(s)\exp(t)=\exp(s+t)$ gives the pointwise identity $g\bigl(\tfrac{u-by}{a}\bigr)g(y)=g(u)\,g\bigl(\tfrac{y-bu}{a}\bigr)$. The function $(u,y)\mapsto\mathbf{1}_{(-\infty,t]}(u)\,g\bigl(\tfrac{u-by}{a}\bigr)g(y)$ is nonnegative and $\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})$-measurable: linear combinations of projections are jointly Borel, compositions with the continuous (hence Borel) maps $g$ and $\mathbf{1}_{(-\infty,t]}$ preserve this (claim 4 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables), and products of jointly Borel functions are jointly Borel since $uv=((u+v)^{2}-u^{2}-v^{2})/2$. So Tonelli permits interchanging the order of integration; using the pointwise identity and then (P2) with the map $y\mapsto(y-bu)/a$,

So the cumulative distribution function of $aZ_1+bZ_2$ is $\Phi$. The distributions $P_{aZ_1+bZ_2}$ and $N$ are probability measures agreeing on the closed rays $(-\infty,t]$ and on $\mathbb{R}$, which form a generating $\pi$-system of the Borel $\sigma$-algebra (every open interval with rational endpoints arises from rays by countable set operations, and every open set is a countable union of such intervals by density of the rationals); the class where two probability measures agree contains $\mathbb{R}$ and is closed under proper differences and increasing countable unions, so Dynkin's Pi-Lambda Theorem gives $P_{aZ_1+bZ_2}=N$, i.e. $aZ_1+bZ_2$ is standard normal.

Step 3 (Claim 3). Induction on $n$. For $n=1$, $\sqrt1=1$ (uniqueness in Existence and Uniqueness of the Nonnegative Square Root) and $G_1=Z_1$. Let $n\ge2$ and suppose the claim holds for $n-1$. Given independent standard normal $Z_1,\dots,Z_n$, the variable $G_{n-1}=(Z_1+\cdots+Z_{n-1})/\sqrt{n-1}$ is standard normal by the induction hypothesis (the first $n-1$ variables form an independent family, immediately from Independence of Events and of Random Variables), and it is a jointly Borel function of $(Z_1,\dots,Z_{n-1})$ (claim 4 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables), so $G_{n-1}$ and $Z_n$ are independent by claim 3 there (blocks $\{1,\dots,n-1\}$ and $\{n\}$). Put $\alpha=\sqrt{(n-1)/n}$ and $\beta=\sqrt{1/n}$, positive reals with $\alpha^{2}+\beta^{2}=1$; by Step 2, $\alpha G_{n-1}+\beta Z_n$ is standard normal. Finally, $\alpha=\sqrt{n-1}/\sqrt{n}$ and $\beta=1/\sqrt{n}$ — in each case both sides are positive with equal squares, so they coincide by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root — whence