Proof of Central Limit Theorem

Throughout, $\mathbb{N}$ denotes the natural numbers and $\mathbb{R}$ the real numbers. We use the terms jointly Borel and the $r$-fold product Borel $\sigma$-algebra as defined in Joint Distribution, Expectations, and Block Independence for Independent Random Variables, and the proof is the replacement method of Lindeberg.

Step 1 (normalization). Set $Y_m=(X_m-\mu)/\sigma$ and $W_n=(Y_1+\cdots+Y_n)/\sqrt{n}$, so that $W_n=(S_n-n\mu)/(\sigma\sqrt{n})$. The map $h:\mathbb{R}\to\mathbb{R}$, $h(x)=(x-\mu)/\sigma$, is continuous, hence Borel measurable by the generator criterion there, so each $Y_m=h(X_m)$ is a random variable. The sequence $(Y_m)_{m\in\mathbb{N}}$ is independent: for Borel sets $B_1,\dots,B_r$ and indices $m_1<\dots<m_r$ we have $\{Y_{m_i}\in B_i\}=\{X_{m_i}\in h^{-1}(B_i)\}$, so the product identities required by that definition follow from independence of $(X_m)$; and the $Y_m$ are identically distributed, since $P(Y_m\in B)=P(X_m\in h^{-1}(B))$ depends only on the common distribution of the $X_m$. By linearity (Linearity and Monotonicity of the Lebesgue Integral) and the definition of variance, $Y_1$ and $Y_1^{2}$ are integrable with

Let $\mu_Y$ denote the distribution of $Y_1$ and $N$ the standard normal distribution.

By claim 2 of Smooth Test Function Criterion for Convergence in Distribution it suffices to prove that $\mathbb{E}[f(W_n)]\to\mathbb{E}[f(Z)]$ for every admissible test function $f$ in the sense of that theorem; once $W_n\to Z$ in distribution is established, the displayed convergence in the statement holds at every $t\in\mathbb{R}$ because $\Phi$ is continuous everywhere (claim 3 of The Gaussian Weight Defines a Probability Distribution). Fix such an $f$ and set $M_2=\sup|f''|$ and $M_3=\sup|f'''|$, finite by admissibility; all expectations of compositions of $f$ (or of the bounded functions $f',f''$) with random variables below are defined by claim 1 of Smooth Test Function Criterion for Convergence in Distribution and its argument.

Step 2 (auxiliary space and reduction). Apply Existence of Independent Sequences with Prescribed Distributions to the sequence of probability measures given by $\nu_{2k-1}=\mu_Y$ and $\nu_{2k}=N$ for $k\ge 1$: there are a probability space $(\Omega',\mathcal{F}',P')$ and an independent sequence $(U_m)_{m\in\mathbb{N}}$ on it with these distributions. Write $Y'_k=U_{2k-1}$ and $Z_k=U_{2k}$; every finite subfamily of an independent family is independent directly from Independence of Events and of Random Variables, so for each $n$ the variables $Y'_1,\dots,Y'_n,Z_1,\dots,Z_n$ are independent, each $Y'_k$ with distribution $\mu_Y$ and each $Z_k$ standard normal. Expectation on $(\Omega',\mathcal{F}',P')$ is written $\mathbb{E}'$. Set

By claim 4 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables, the map $w_n(x_1,\dots,x_n)=f\bigl((x_1+\cdots+x_n)/\sqrt{n}\bigr)$ is jointly Borel (the addition map is jointly Borel, and $t\mapsto f(t/\sqrt{n})$ is continuous, hence Borel), and it is bounded. By claim 2 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables, applied on $(\Omega,\mathcal{F},P)$ to the independent tuple $Y_1,\dots,Y_n$ and on $(\Omega',\mathcal{F}',P')$ to $Y'_1,\dots,Y'_n$ — all entries with distribution $\mu_Y$ — both $\mathbb{E}[f(W_n)]$ and $\mathbb{E}'[f(W'_n)]$ equal the integral of $w_n$ against the $n$-fold product of $\mu_Y$ with itself, so

By claim 3 of Moments and Stability of the Standard Normal Distribution, $G_n$ is standard normal; since $f$ is bounded and Borel, Change of Variables for Expectations applied to $G_n$ and to $Z$ (which share the distribution $N$) gives $\mathbb{E}'[f(G_n)]=\mathbb{E}[f(Z)]$. It therefore suffices to show

Step 3 (two Taylor bounds). For $x,u\in\mathbb{R}$ put

By Taylor Expansion with Third-Order Remainder Bound, $|e(x,u)|\le M_3|u|^{3}/6$. Moreover, carrying out Steps 1 and 2 of the proof of that lemma one order lower — Fundamental Theorem of Calculus, Part II in One Dimension applied to $f'$ gives $|f'(x+t)-f'(x)|\le M_2|t|$, and then applied to $f$ gives $|f(x+u)-f(x)-f'(x)u|\le M_2u^{2}/2$ (for negative increments exchange the endpoints, as there) — while $|\tfrac12 f''(x)u^{2}|\le M_2u^{2}/2$; adding these two bounds,

The function $g$ is continuous (a minimum of two continuous functions), hence Borel, and $g\ge 0$.

Step 4 (the replacement estimate). Fix $n\ge 1$ and $k\in\{1,\dots,n\}$, and define on $(\Omega',\mathcal{F}',P')$

Then $H_1=W'_n$, $\widetilde{H}_n=G_n$, and $\widetilde{H}_k=H_{k+1}$ for $1\le k<n$, so

Taylor expansion at $x=R_k$ (Step 3) gives the pointwise identities

and the same with $Z_k$ in place of $Y'_k$ for $\widetilde{H}_k$.

The variable $R_k$ is a jointly Borel function (claims 4 and hence 3 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables) of the block of the independent family $(U_m)$ with indices in $I_k=\{2i:i<k\}\cup\{2j-1:j>k,\ j\le n\}$, while $Y'_k$ and $Z_k$ are functions of the singleton blocks $\{2k-1\}$ and $\{2k\}$, each disjoint from $I_k$. Composing with the Borel maps $f'$, $f''$, and $t\mapsto t^{2}$ (claim 4), claim 3 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables shows that each of the pairs $(f'(R_k),\,Y'_k)$, $(f''(R_k),\,(Y'_k)^{2})$, $(f'(R_k),\,Z_k)$, $(f''(R_k),\,Z_k^{2})$ consists of independent random variables. The first factors are bounded, hence integrable on a probability space, and the second factors are integrable: $Y'_k$ and $(Y'_k)^{2}$ share the integrability and moments of $Y_1$ by Change of Variables for Expectations (equal distributions), and $Z_k$, $Z_k^{2}$ are integrable with $\mathbb{E}'[Z_k]=0$, $\mathbb{E}'[Z_k^{2}]=1$ by claim 1 of Moments and Stability of the Standard Normal Distribution. Hence Expectation of a Product of Independent Random Variables gives

Taking expectations in the two Taylor identities (each term integrable as just noted, the remainder terms being differences of integrable terms) and subtracting, the zeroth-, first-, and second-order terms cancel, so by linearity and monotonicity (Linearity and Monotonicity of the Lebesgue Integral) together with the bound $|e(R_k,u)|\le g(u)$ of Step 3,

For the Gaussian term, $g(u)\le M_3|u|^{3}/6$ and claim 1 of Moments and Stability of the Standard Normal Distribution give

the value $\gamma$ being shared by all the $Z_k$ (Change of Variables for Expectations, equal distributions). For the sample term, the $Y'_k$ share the distribution $\mu_Y$ and $u\mapsto g(u/\sqrt{n})$ is nonnegative Borel, so $\mathbb{E}'[g(Y'_k/\sqrt{n})]=\mathbb{E}'[g(Y'_1/\sqrt{n})]$ for every $k$ (Change of Variables for Expectations). Summing the $n$ replacement estimates,

Step 5 (vanishing of the error). For $y\in\mathbb{R}$,

Each $g_n$ is continuous, hence Borel; $0\le g_n(y)\le M_2y^{2}$; and $g_n(y)\to 0$ as $n\to\infty$ for every fixed $y$, since the second entry of the minimum tends to $0$. Hence $g_n(Y'_1)\to 0$ pointwise on $\Omega'$, dominated by the integrable function $M_2\,(Y'_1)^{2}$ (Step 1 and equality of distributions). By Dominated Convergence Theorem,

and also $M_3\gamma/(6\sqrt{n})\to 0$. Combining with Step 2,

Since $f$ was an arbitrary admissible test function, claim 2 of Smooth Test Function Criterion for Convergence in Distribution yields $W_n\to Z$ in distribution in the sense of Almost Sure Convergence, Convergence in Probability, and Convergence in Distribution, and as noted in Step 1 the convergence $P(W_n\le t)\to\Phi(t)$ holds at every $t\in\mathbb{R}$ because $\Phi$ is continuous everywhere. $\blacksquare$