Proof of The Central Limit Theorem for Independent Identically Distributed Square-Integrable Random Variables

Step 1 (standardization). By definition $\sigma\ge0$ and $\sigma^{2}=\operatorname{Var}X_{1}>0$ (The k-th Root and the Square Root of a Nonnegative Real Number (i)), so $\sigma\neq0$, hence $\sigma>0$. Since the sequence is iid, every $X_{k}$ has the law of $X_{1}$ (Independence of Events and of Random Elements (v)); as expectations of functions of a random variable are computed from its law (Expectations Computed from the Law (ii), with the functions $x\mapsto x$ and $x\mapsto x^{2}$), $\mathbb{E}X_{k}=\mathbb{E}X_{1}=\mu$ and $\mathbb{E}X_{k}^{2}=\mathbb{E}X_{1}^{2}$, so $\operatorname{Var}X_{k}=\operatorname{Var}X_{1}=\sigma^{2}$ (Rules for Expectations, Variances and Covariances (vii)). Let $h:\mathbb{R}\to\mathbb{R}$, $h(x)=(x-\mu)/\sigma$, a continuous and hence Borel function (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv)), and $Y_{k}=h(X_{k})$. Then $\mathbb{E}Y_{k}=0$, $\operatorname{Var}Y_{k}=\operatorname{Var}X_{k}/\sigma^{2}=1$ and $\mathbb{E}Y_{k}^{2}=\operatorname{Var}Y_{k}+(\mathbb{E}Y_{k})^{2}=1$ (Rules for Expectations, Variances and Covariances (iii), Rules for Expectations, Variances and Covariances (ix), Rules for Expectations, Variances and Covariances (vii)). The $Y_{k}$ are identically distributed, the law of $Y_{k}$ being the image of the law of $X_{k}$ under $h$ (Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (ii)), and the sequence $(Y_{k})$ is independent (Independent Random Elements: Preimages, Functions and Blocks (iii)). Pointwise on $\Omega$,

$$\sum_{k=1}^{n}\frac{X_{k}-\mu}{\sigma}=\frac{1}{\sigma}\Bigl(\sum_{k=1}^{n}X_{k}-\sum_{k=1}^{n}\mu\Bigr)=\frac{\sum_{k=1}^{n}X_{k}-n\mu}{\sigma}$$

(Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality (i), Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality (ii), Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality (iv)), and $1/(\sigma\sqrt{n})=(1/\sqrt{n})(1/\sigma)$, so

$$T_{n}=\frac{1}{\sqrt{n}}\sum_{k=1}^{n}Y_{k},$$

a random variable (Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), Arithmetic of Measurable Real- and [0,∞]-Valued Maps (ii)). In particular $Y_{1},\dots,Y_{n}$ are independent for every $n$, the defining product formula of Independent Random Elements: Preimages, Functions and Blocks (i) for a subfamily being a case of that for the whole sequence.

Step 2 (the product space). Fix $n$. By Independent Random Variables and Independent Standard Normal Coordinates on a Product Space (i),

$$(\tilde\Omega,\tilde{\mathcal{F}},\tilde P)=(\Omega\times\mathbb{R}^{n},\ \mathcal{F}\otimes\mathcal{B}(\mathbb{R}^{n}),\ P\otimes\gamma_{n})$$

is a probability space; for $k\in[n]$ let $\tilde Y_{k}(\omega,z)=Y_{k}(\omega)$ and $\tilde Z_{k}(\omega,z)=z_{k}$. Applied to $Y_{1},\dots,Y_{n}$, Independent Random Variables and Independent Standard Normal Coordinates on a Product Space (ii) shows that the family $(W_{i})_{i\in[2n]}$ with $W_{k}=\tilde Y_{k}$ and $W_{n+k}=\tilde Z_{k}$ for $k\in[n]$ is independent (this defines $W$ on all of $[2n]$, since an index $i\in[2n]$ with $i>n$ is $n+k$ for a unique $k\in\mathbb{N}$, and then $k\le n$: Arithmetic and Order of the Natural Numbers (viii), Arithmetic and Order of the Natural Numbers (v), Arithmetic and Order of the Natural Numbers (ix)), and Independent Random Variables and Independent Standard Normal Coordinates on a Product Space (iii) that each $\tilde Y_{k}$ has the law of $Y_{k}$, which is the law of $Y_{1}$ by Step 1, and each $\tilde Z_{k}$ is standard normal. The function $\tilde Z=(\tilde Z_{1},\dots,\tilde Z_{n})$, that is $\tilde Z(\omega,z)=z$, is a random vector, its coordinates being random variables (Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (i)). For a Borel set $B\subseteq\mathbb{R}^{n}$ the event $\{\tilde Z\in B\}$ is the rectangle $\Omega\times B$, so by Product Measure (i)

$$\tilde P(\tilde Z\in B)=P(\Omega)\,\gamma_{n}(B)=\gamma_{n}(B).$$

Thus the law of $\tilde Z$ is $\gamma_{n}$, and $\tilde Z$ is a standard normal vector (Standard Normal Vectors: Density, Moments, Orthogonal Invariance and Concatenation (iii)). Finally $\tilde T_{n}=\frac{1}{\sqrt{n}}\sum_{k}\tilde Y_{k}$ satisfies $\tilde T_{n}(\omega,z)=T_{n}(\omega)$, so for a Borel set $A\subseteq\mathbb{R}$ the event $\{\tilde T_{n}\in A\}$ is the rectangle $\{T_{n}\in A\}\times\mathbb{R}^{n}$, and by Product Measure (i) (with $\gamma_{n}(\mathbb{R}^{n})=1$ by Standard Normal Vectors: Density, Moments, Orthogonal Invariance and Concatenation (i))

$$\tilde P(\tilde T_{n}\in A)=P(T_{n}\in A)\,\gamma_{n}(\mathbb{R}^{n})=P(T_{n}\in A);$$

that is, $\tilde T_{n}$ has the law of $T_{n}$.

Step 3 (the two ends). Let $a=1/\sqrt{n}$; since $\sqrt{n}\ge0$ and $(\sqrt{n})^{2}=n$ (The k-th Root and the Square Root of a Nonnegative Real Number (i)), $a>0$ and $a^{2}=1/n$. With $c=(a,\dots,a)\in\mathbb{R}^{1\times n}$, $cc^{\top}=na^{2}=1$, so $c\tilde Z$ is a standard normal vector in $\mathbb{R}^{1}$ (Standard Normal Vectors: Density, Moments, Orthogonal Invariance and Concatenation (vi)), and its first coordinate $N_{n}=\frac{1}{\sqrt{n}}\sum_{k}\tilde Z_{k}$ is a standard normal random variable (Standard Normal Vectors and the Standard Normal Distribution on ℝᵏ (iii)). So, by Step 2, for every bounded Borel function $f:\mathbb{R}\to\mathbb{R}$, expectations being computed from laws (Expectations Computed from the Law (ii)),

$$\mathbb{E}f(T_{n})=\tilde{\mathbb{E}}f(\tilde T_{n}),\qquad\tilde{\mathbb{E}}f(N_{n})=\int f\,d\gamma=\mathbb{E}'f(Z),$$

with $\mathbb{E}'$ the expectation on $(\Omega',\mathcal{F}',P')$.

Step 4 (the replacement bound). Let $f$ be a function of the class used in Smooth Test Functions Characterize Convergence in Distribution of Real Random Variables (i), with a real $M$ such that $|f|,|f'|,|f''|,|f'''|\le M$ on $\mathbb{R}$, so that $M\ge|f''(0)|\ge0$; take $M_{2}=M_{3}=M$. Being differentiable, $f$ is continuous (The Increment Form of the Derivative, and Differentiability Implies Continuity (ii), $\mathbb{R}$ being an interval with two distinct points by Basic Facts about Intervals of the Real Line (i)) and hence Borel (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv)), so $f$ is a bounded Borel function and Step 3 applies to it. For $n\in\mathbb{N}$ let $g_{n}$ be the function of The Integrals of min(My², M|y|³/(2√n)) Tend to Zero under a Finite Second Moment for this $M\ge0$,

$$g_{n}(y)=\min\Bigl(My^{2},\frac{M|y|^{3}}{2\sqrt{n}}\Bigr),$$

a Borel function by The Integrals of min(My², M|y|³/(2√n)) Tend to Zero under a Finite Second Moment (i). Apply Lindeberg's Replacement Bound on $(\tilde\Omega,\tilde{\mathcal{F}},\tilde P)$ to $\tilde Y_{1},\dots,\tilde Y_{n},\tilde Z_{1},\dots,\tilde Z_{n}$ with $a=1/\sqrt{n}$ and $M_{2}=M_{3}=M$, so that its function $g$ is $g_{n}$. Its hypotheses hold. Its family $(W_{i})_{i\in[2n]}$ is the family of Step 2, which is independent. Each $\tilde Y_{k}$ has the law of $Y_{1}$. The function $x\mapsto|x|^{2}$ on $\mathbb{R}$ is continuous, hence Borel (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv)), and nonnegative, hence measurable into $[0,\infty]$ (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii)); so, computing both expectations from the common law (Expectations Computed from the Law (i)),

$$\int|\tilde Y_{k}|^{2}\,d\tilde P=\tilde{\mathbb{E}}|\tilde Y_{k}|^{2}=\int|x|^{2}\,dP_{Y_{1}}(x)=\mathbb{E}|Y_{1}|^{2}=\int|Y_{1}|^{2}\,dP<\infty,$$

since $Y_{1}\in\mathcal{L}^{2}(P)$. Hence $\tilde Y_{k}\in\mathcal{L}^{2}(\tilde P)$ (Power-Integrable Functions and the p-Seminorm (ii)). Applying Expectations Computed from the Law (ii) in the same way to the Borel functions $x\mapsto x$ and $x\mapsto x^{2}$, which are integrable for the law of $Y_{1}$ (by Expectations Computed from the Law (ii), $Y_{1}$ and $Y_{1}^{2}$ being integrable by Step 1), gives $\tilde{\mathbb{E}}\tilde Y_{k}=\mathbb{E}Y_{1}=0$ and $\tilde{\mathbb{E}}\tilde Y_{k}^{2}=\mathbb{E}Y_{1}^{2}=1$ by Step 1. Since $\tilde Z$ is a standard normal vector, $\tilde Z\in\mathcal{L}^{2}(\tilde P;\mathbb{R}^{n})$ with $\tilde{\mathbb{E}}\tilde Z=0$ and $\operatorname{Cov}(\tilde Z)=I_{n}$ (Standard Normal Vectors: Density, Moments, Orthogonal Invariance and Concatenation (iv)); so $\tilde Z_{k}\in\mathcal{L}^{2}(\tilde P)$ (Square-Integrable Vector-Valued Functions: Components, Dot Products and Almost-Everywhere Equality (i)), $\tilde{\mathbb{E}}\tilde Z_{k}=0$ and $\operatorname{Var}\tilde Z_{k}=1$, the $k$-th diagonal entry of $\operatorname{Cov}(\tilde Z)$ (Variance and Covariance (iv)), and

$$\tilde{\mathbb{E}}\tilde Z_{k}^{2}=\operatorname{Var}\tilde Z_{k}+(\tilde{\mathbb{E}}\tilde Z_{k})^{2}=1$$

(Rules for Expectations, Variances and Covariances (vii)). By Lindeberg's Replacement Bound (i) the random variables $g_{n}(\tilde Y_{k})$ and $g_{n}(\tilde Z_{k})$ are integrable, and since $a\sum_{k}\tilde Y_{k}=\tilde T_{n}$, $a\sum_{k}\tilde Z_{k}=N_{n}$ and $a^{2}=1/n$, Lindeberg's Replacement Bound (ii) gives

$$\bigl|\tilde{\mathbb{E}}f(\tilde T_{n})-\tilde{\mathbb{E}}f(N_{n})\bigr|\le\frac1n\sum_{k=1}^{n}\bigl(\tilde{\mathbb{E}}\,g_{n}(\tilde Y_{k})+\tilde{\mathbb{E}}\,g_{n}(\tilde Z_{k})\bigr).$$

As $\tilde Y_{k}$ has the law of $Y_{1}$ and $\tilde Z_{k}$ has law $\gamma$, $\tilde{\mathbb{E}}\,g_{n}(\tilde Y_{k})=\mathbb{E}\,g_{n}(Y_{1})$ and $\tilde{\mathbb{E}}\,g_{n}(\tilde Z_{k})=\int g_{n}\,d\gamma$ for every $k$ (Expectations Computed from the Law (ii)), so the right-hand side is an average of $n$ equal terms (Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality (iv)), and with Step 3,

$$\bigl|\mathbb{E}f(T_{n})-\mathbb{E}'f(Z)\bigr|\le\mathbb{E}\,g_{n}(Y_{1})+\int g_{n}\,d\gamma.$$

Step 5 (the limit). The function $y\mapsto y^{2}$ is integrable for the law $P_{Y_{1}}$ of $Y_{1}$, because $Y_{1}\in\mathcal{L}^{2}(P)$ (Expectations Computed from the Law (ii)), and for $\gamma$, a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Standard Normal Vectors: Density, Moments, Orthogonal Invariance and Concatenation (i)), because $\tilde Z_{1}\in\mathcal{L}^{2}(\tilde P)$ has law $\gamma$ (Steps 2 and 4, Expectations Computed from the Law (ii)). So The Integrals of min(My², M|y|³/(2√n)) Tend to Zero under a Finite Second Moment (iii), for this $M$ and the two measures $P_{Y_{1}}$ and $\gamma$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, gives

$$\mathbb{E}\,g_{n}(Y_{1})=\int g_{n}\,dP_{Y_{1}}\to0,\qquad\int g_{n}\,d\gamma\to0$$

(the equality by Expectations Computed from the Law (ii), $g_{n}(Y_{1})$ being integrable as in Step 4), so the bound of Step 4 is a null sequence (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (vi)). Hence $\mathbb{E}f(T_{n})\to\mathbb{E}'f(Z)$ (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (xii)).

Step 6 (conclusion). Since this holds for every $f$ of that class, $T_{n}\xrightarrow{d}Z$ by Smooth Test Functions Characterize Convergence in Distribution of Real Random Variables (i).