Proof of Smooth Test Functions Characterize Convergence in Distribution of Real Random Variables

Call a function $f:\mathbb{R}\to\mathbb{R}$ admissible if it satisfies the conditions of (i): $f$, $f'$ and $f''$ are differentiable, $f'''$ is continuous, and $f,f',f'',f'''$ are bounded.

Step 1 (only if). An admissible $f$ is continuous, being differentiable at every point (The Increment Form of the Derivative, and Differentiability Implies Continuity (ii)), and bounded, so $f\in C_{b}(\mathbb{R})$ (Bounded Continuous Functions on a Metric Space (i)). Hence, if $X_{n}\xrightarrow{d}X$, then $\mathbb{E}_{n}f(X_{n})\to\mathbb{E}f(X)$ by Convergence in Distribution through Expectations of Bounded Continuous and of Bounded Lipschitz Functions (i).

Step 2 (a ramp $h_{\alpha,\delta}$ built from the clamp $c$ and $r=1-s$). Let $s(y)=35y^{4}-84y^{5}+70y^{6}-20y^{7}$ and $r=1-s$. Expanding,

$$r(y)=(1-y)^{4}(1+4y+10y^{2}+20y^{3}),\qquad s(y)=y^{4}\bigl(1+4(1-y)+10(1-y)^{2}+20(1-y)^{3}\bigr),$$

so $0\le r\le1$ on $[0,1]$, $r(0)=1$ and $r(1)=0$; moreover

$$r'(y)=-140\,y^{3}(1-y)^{3},\qquad r''(y)=-420\,y^{2}(1-y)^{2}(1-2y),\qquad r'''(y)=-840\,y(1-y)(1-5y+5y^{2}),$$

so $r'$, $r''$ and $r'''$ vanish at $0$ and at $1$.

Fix $\alpha\in\mathbb{R}$ and $\delta>0$, let $k=1/\delta$, and let $c:\mathbb{R}\to\mathbb{R}$ be the clamp to $[\alpha,\alpha+\delta]$,

$$c(x)=\tfrac{1}{2}\bigl(|x-\alpha|-|x-\alpha-\delta|+2\alpha+\delta\bigr),$$

so that $c(x)=\alpha$ for $x\le\alpha$, $c(x)=x$ for $\alpha\le x\le\alpha+\delta$ and $c(x)=\alpha+\delta$ for $x\ge\alpha+\delta$. By the reverse triangle inequality $\bigl||a|-|b|\bigr|\le|a-b|$, applied to each of the two absolute-value terms,

$$|c(x)-c(x')|\le\tfrac{1}{2}\Bigl(\bigl||x-\alpha|-|x'-\alpha|\bigr|+\bigl||x-\alpha-\delta|-|x'-\alpha-\delta|\bigr|\Bigr)\le|x-x'|,$$

so $c$ is Lipschitz with constant $1$. For $j=0,1,2,3,4$ put

$$p_{j}(x)=k^{j}\,r^{(j)}\bigl(k(x-\alpha)\bigr),\qquad h_{\alpha,\delta}=p_{0}\circ c .$$

Each $p_{j}$ is a polynomial function of $x$, the powers of $k$ sitting in its coefficients, so it is differentiable at every point by Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (i), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (ii), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (iv), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (v) and Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (vi), and these rules give $p_{j}'=p_{j+1}$ for $j\le3$. Indeed, by induction on $i$ the derivative of $x\mapsto(k(x-\alpha))^{i}$ is $x\mapsto i\,k\,(k(x-\alpha))^{i-1}$ (product, scalar, identity and constant rules), so if $r^{(j)}(y)=\sum_{i}a_{i}y^{i}$, then

p_{j}'(x)=k^{j+1}\sum_{i}i\,a_{i}\bigl(k(x-\alpha)\bigr)^{i-1}=k^{j+1}r^{(j+1)}\bigl(k(x-\alpha)\bigr)=p_{j+1}(x). $$ By the previous paragraph, $p_{1}$, $p_{2}$ and $p_{3}$ vanish at $\alpha$ and at $\alpha+\delta$. We claim $(p_{j}\circ c)'=p_{j+1}\circ c$ for $j\le2$, which we check from [Derivative of a Real Function on an Interval (i)](/theorems/6bd0350c-6df6-4bef-a99d-f6020d157b59?v=3fb2f993-5529-4cdc-ae16-abc03dcc05c1#clause-derivative-at). If $x<\alpha$ or $x>\alpha+\delta$, then $c$ is constant near $x$, so the difference quotients of $p_{j}\circ c$ at $x$ vanish near $x$, and its derivative at $x$ is $0=p_{j+1}(c(x))$. If $\alpha<x<\alpha+\delta$, then $c$ is the identity near $x$, so near $x$ the difference quotients of $p_{j}\circ c$ at $x$ are those of $p_{j}$, and the derivative is $p_{j}'(x)=p_{j+1}(c(x))$. At $x=\alpha$ the difference quotient $\bigl(p_{j}(c(t))-p_{j}(\alpha)\bigr)/(t-\alpha)$ is $0$ for $t<\alpha$ and is that of $p_{j}$ for $\alpha<t<\alpha+\delta$; both one-sided quotients tend to $p_{j}'(\alpha)=p_{j+1}(\alpha)=0$, so the derivative is $0=p_{j+1}(c(\alpha))$. The point $\alpha+\delta$ is treated in the same way. Hence $h_{\alpha,\delta}$, $h_{\alpha,\delta}'$ and $h_{\alpha,\delta}''$ are differentiable and

h_{\alpha,\delta}^{(j)}=p_{j}\circ c=k^{j},r^{(j)}\bigl(k(c-\alpha)\bigr)\qquad(j=0,1,2,3).

The function $h_{\alpha,\delta}$ is admissible. Indeed, $p_{3}$ is differentiable, hence continuous ([The Increment Form of the Derivative, and Differentiability Implies Continuity (ii)](/theorems/e3e1e5f6-670a-4725-8db4-1552433126ec?v=c3920d80-4da9-4005-b3cb-de29173e477e#clause-continuous)), and $c$ is continuous ([A Lipschitz Map is Uniformly Continuous and Continuous (ii)](/theorems/4f436ed2-8bd5-4842-b79b-98e96ca0e394?v=3828147a-2f34-4c42-ae67-70b89d2b1cab#clause-continuous)) with values in $[\alpha,\alpha+\delta]$, so $c\in C_{b}(\mathbb{R})$ and $h_{\alpha,\delta}'''=p_{3}\circ c\in C_{b}(\mathbb{R})$ by [Bounded Continuous Functions on a Metric Space: Measurability, Integrals, Operations, Lipschitz Functions and Approximation of Closed Sets (iv)](/theorems/96dc7fad-e062-4c39-ab47-f32c191356f2?v=fdc49679-b18d-446b-a32a-cc8ccddccf06#clause-composition). For the bounds, $k(c(x)-\alpha)\in[0,1]$ for every $x$, and on $[0,1]$ the absolute value of a polynomial is at most the sum of the absolute values of its coefficients; so $|h_{\alpha,\delta}^{(j)}|\le k^{j}S_{j}$ on $\mathbb{R}$, with $S_{j}$ that sum for $r^{(j)}$, so $M=\max_{j\le3}k^{j}S_{j}$ bounds $|h_{\alpha,\delta}^{(j)}|$ for every $j\le3$. Finally $h_{\alpha,\delta}=r(0)=1$ on $(-\infty,\alpha]$, $h_{\alpha,\delta}=r(1)=0$ on $[\alpha+\delta,\infty)$ and $0\le h_{\alpha,\delta}\le1$ everywhere, so

\mathbf{1}{(-\infty,\alpha]}\le h{\alpha,\delta}\le\mathbf{1}_{(-\infty,\alpha+\delta]}.

**Step 3 (squeezing an indicator).** Let $t$ be a continuity point of $F_{X}$ and $\delta>0$, and let $h=h_{t,\delta}$ and $g=h_{t-\delta,\delta}$. By Step 2, $h$ and $g$ are admissible and

\mathbf{1}{(-\infty,t]}\le h\le\mathbf{1}{(-\infty,t+\delta]},\qquad\mathbf{1}{(-\infty,t-\delta]}\le g\le\mathbf{1}{(-\infty,t]}.

The maps $h(X_{n})$, $g(X_{n})$ and $\mathbf{1}_{\{X_{n}\le t\}}$ are measurable ([Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv)](/theorems/976863ed-33e9-40f1-bfce-4f217ac8d0b5?v=e1164f23-e371-47cb-a6d8-d57ba1daddfc#clause-continuous), [Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (ii)](/theorems/976863ed-33e9-40f1-bfce-4f217ac8d0b5?v=e1164f23-e371-47cb-a6d8-d57ba1daddfc#clause-composition); [Basic Properties of the Integral of Nonnegative Measurable Maps (ii)](/theorems/5a501fa7-400d-485f-810c-3ed6c68b3cc6?v=5bfa5b66-c5c9-4e2c-808a-7776db88f5ee#clause-indicator), [Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii)](/theorems/976863ed-33e9-40f1-bfce-4f217ac8d0b5?v=e1164f23-e371-47cb-a6d8-d57ba1daddfc#clause-nonnegative-real)) and bounded, hence integrable ([Bounded Measurable Functions on a Finite Measure Space Are Integrable (i)](/theorems/510eea35-7520-4127-bb30-35225f2798ce?v=e6b7696f-3b16-40cf-8003-92a0df95b817#clause-integrable)), and likewise $h(X)$ and $g(X)$. By [Rules for Expectations, Variances and Covariances (iv)](/theorems/e9f6faad-3ccd-4ad7-850b-71cff6386622?v=13c0eee0-3d4c-41e2-bdae-1b6ac64b7f72#clause-monotone) and [Rules for Expectations, Variances and Covariances (i)](/theorems/e9f6faad-3ccd-4ad7-850b-71cff6386622?v=13c0eee0-3d4c-41e2-bdae-1b6ac64b7f72#clause-events), $\mathbb{E}_{n}g(X_{n})\le F_{X_{n}}(t)\le\mathbb{E}_{n}h(X_{n})$, $\mathbb{E}h(X)\le F_{X}(t+\delta)$ and $\mathbb{E}g(X)\ge F_{X}(t-\delta)$. **Step 4 (if).** Suppose $\mathbb{E}_{n}f(X_{n})\to\mathbb{E}f(X)$ for every admissible $f$, and let $t$ be a continuity point of $F_{X}$. Since $0\le F_{X_{n}}(t)\le1$, the sequence $(F_{X_{n}}(t))$ is bounded, and so are the convergent sequences $(\mathbb{E}_{n}h(X_{n}))$ and $(\mathbb{E}_{n}g(X_{n}))$ of Step 3 ([Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (iii)](/theorems/95a5b76f-d75c-452b-b2d8-f62f8dba6988?v=f709a9d2-9414-4142-8f79-c71a263835a0#clause-bounded)). Step 3, [Properties of the Limit Superior and the Limit Inferior: Tail Formula, Order, Comparison, Bounds, Eventual and Frequent Inequalities, Subsequences, Negation and Convergence (iii)](/theorems/4dde6a82-9796-4aea-bd47-b92934c04e77?v=8928746d-71e5-465a-9703-f3743f4e7723#clause-comparison) and [Properties of the Limit Superior and the Limit Inferior: Tail Formula, Order, Comparison, Bounds, Eventual and Frequent Inequalities, Subsequences, Negation and Convergence (xi)](/theorems/4dde6a82-9796-4aea-bd47-b92934c04e77?v=8928746d-71e5-465a-9703-f3743f4e7723#clause-convergence) give, for every $\delta>0$,

F_{X}(t-\delta)\le\liminf_{n}F_{X_{n}}(t)\le\limsup_{n}F_{X_{n}}(t)\le F_{X}(t+\delta).

Let $L=\limsup_{n}F_{X_{n}}(t)$ and suppose $L>F_{X}(t)$. With $\varepsilon=L-F_{X}(t)$, continuity of $F_{X}$ at $t$ ([Continuity of Real Functions on Subsets of ℝ in ε-δ Form (i)](/theorems/214f5394-efc9-463e-a71a-2e55b6802091?v=e558c708-6a57-455e-8097-1abf87015f8e#clause-at-point)) gives $\delta_{0}>0$ with $|F_{X}(y)-F_{X}(t)|<\varepsilon$ whenever $|y-t|<\delta_{0}$, and the display with $\delta=\delta_{0}/2$ gives

L\le F_{X}(t+\delta_{0}/2)<F_{X}(t)+\varepsilon=L,

a contradiction. So $L\le F_{X}(t)$, and symmetrically, at the point $t-\delta_{0}/2$, $\liminf_{n}F_{X_{n}}(t)\ge F_{X}(t)$. As $\liminf_{n}F_{X_{n}}(t)\le\limsup_{n}F_{X_{n}}(t)$ ([Properties of the Limit Superior and the Limit Inferior: Tail Formula, Order, Comparison, Bounds, Eventual and Frequent Inequalities, Subsequences, Negation and Convergence (ii)](/theorems/4dde6a82-9796-4aea-bd47-b92934c04e77?v=8928746d-71e5-465a-9703-f3743f4e7723#clause-order)), both equal $F_{X}(t)$, and $F_{X_{n}}(t)\to F_{X}(t)$ by [Properties of the Limit Superior and the Limit Inferior: Tail Formula, Order, Comparison, Bounds, Eventual and Frequent Inequalities, Subsequences, Negation and Convergence (xi)](/theorems/4dde6a82-9796-4aea-bd47-b92934c04e77?v=8928746d-71e5-465a-9703-f3743f4e7723#clause-convergence). As $t$ was an arbitrary continuity point, $X_{n}\xrightarrow{d}X$ by [Convergence in Distribution of Real Random Variables through Distribution Functions (ii)](/theorems/c00a8285-2517-43f9-a4c7-5765d64170df?v=53c26637-7b0f-44da-b80b-14dd43ddf35d#clause-backward).