Jtdylktuy

If $X_n xrightarrow{text{d}} X$, then for any equicontinuous family ${f_theta:thetain Theta}$, satisfying sup${|f_theta(x)|:xin mathbf{R},thetain Theta}}<infty$, prove that $E[f_theta(X_n)]rightarrow E[f_theta(X)]$ uniformly in $thetain Theta$.

Now I know if $X_n xrightarrow{text{d}} X$, $E[f_theta(X_n)]rightarrow E[f_theta(X)]hspace{1ex}forall thetainTheta$. And I sense I have to use Arzela-Ascoli Theorem, but I don't have compact set as domain. Any help?

Arzelà-Ascoli is a good idea; in order to apply it we need to truncate the functions $f_{theta}$ in a suitable way.

Step 1: Find suitable level of truncation Fix $epsilon>0$. By the monotone convergence theorem, there exists $R>0$ such that

$$mathbb{P}(|X| geq R) leq epsilon; tag{1}$$

without loss of generality, we may assume that $mathbb{P}(|X|=R)=0$. Since $X_n to X$ weakly we have

$$mathbb{P}(|X_n| geq R) to mathbb{P}(|X| geq R) leq epsilon,$$

and therefore we can choose $N in mathbb{N}$ such that

$$mathbb{P}(|X_n| geq R) leq 2 epsilon quad text{for all $n geq N$.}$$

We can enlarge $R>0$ in such a way that

$$mathbb{P}(|X_j| geq R) leq 2 epsilon quad text{or all $j=1,ldots,N-1$}$$

(note that we only have finitely many $j$'s here!), and therefore we get

$$mathbb{P}(|X_n| geq R) leq 2 epsilon quad text{for all $n geq 1$.} tag{2}$$

Step 2: Apply Arzelà-Ascoli For fixed $epsilon>0$ let $R>0$ be as in Step 1 (i.e. such that $(1)$ and $(2)$ hold). Choose a function $chi in C_b(mathbb{R})$, $0 leq chi leq 1$ such that $chi(x)=1$ for $|x| leq R$ and $chi(x)=$ for $|x| geq R+1$. The family $${chi cdot f_{theta}; theta in Theta}$$ is uniformly bounded and equicontinuous. Moreover, the support of each function $g_{theta} := chi cdot f_{theta}$ is contained in the closed ball of radius $R+1$. By the Arzelà-Ascoli theorem, this implies that the above family is compact in $C_b(mathbb{R})$. In particular, we can choose $theta_1,ldots,theta_m in Theta$ such that $${g_{theta}; theta in Theta} subseteq bigcup_{j=1}^m B(g_{theta_j},epsilon). tag{3}$$ For each $j=1,ldots,m$ there exists $N_j$ such that $$|mathbb{E}g_{theta_j}(X_n)-mathbb{E}g_{theta_j}(X)| leq epsilon quad text{for all $n geq N_j$};$$ set $N := max{N_1,ldots,N_m}$.

Step 3: Prove uniform convergence Fix $theta in Theta$. Clearly, we have

$$|mathbb{E}f_{theta}(X_n)-mathbb{E}f_{theta}(X)| leq I_1+I_2$$

where

$$begin{align*} I_1 &:= |mathbb{E}(f_{theta}(X_n) chi(X_n))- mathbb{E}(f_{theta}(X) chi(X))| \ I_2 &:=|mathbb{E}(f_{theta}(X_n) (1-chi(X_n)))- mathbb{E}(f_{theta}(X) (1-chi(X)))| end{align*}$$

We estimate the terms separately.

Estimate of $I_1$: According to $(3)$ there exists $j in {1,ldots,m}$ such that $|g_{theta}-g_{theta_j}|_{infty} leq epsilon$. Hence, $$I_1 = |mathbb{E}g_{theta}(X_n)-mathbb{E}g_{theta}(X)| leq 2 |g_{theta}-g_{theta_j}|_{infty} + |mathbb{E}g_{theta_j}(X_n)-mathbb{E}g_{theta_j}(X)| leq 3 epsilon tag{4}$$ for all $n geq N$.

Estimate of $I_2$: Since $chi(x)=0$ for all $|x| leq R$ and $0 leq chi leq 1$, we clearly have

$$I_2 leq |f_{theta}|_{infty} left( mathbb{P}(|X_n| geq R) + mathbb{P}(|X| geq R) right).$$

It follows from $(1)$ and $(2)$ that

$$I_2 leq3 |f_{theta}|_{infty} epsilon.$$
If we set $M:= sup_{theta in Theta} |f_{theta}|_{infty}$ (which is finite by assumption) then $$I_2 leq 3 M epsilon.$$

Combining our estimates, we find that

$$|mathbb{E}f_{theta}(X_n)-mathbb{E}f_{theta}(X)| leq 3 (M+1) epsilon$$

for all $n geq N$. Since $epsilon>0$ is arbitrary and $N$ does not depend on $theta$, this proves the uniform convergence with respect to $theta in Theta$: $$sup_{theta in Theta} |mathbb{E}f_{theta}(X_n)-mathbb{E}f_{theta}(X)| xrightarrow{n to infty} 0.$$

Remark: Note that the result implies in particular the following statement.

Theorem If $X_n to X$ in distribution then $$sup_{xi in K} |mathbb{E}exp(i xi X_n)-mathbb{E}exp(i xi X)| xrightarrow{n to infty} 0$$ for any compact set $K$, i.e. the characteristic functions converge uniformly on compact sets.