Convergence of the Characteristic Functions

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[Theorem 1] Vague convergence implies convergence of ch.f.

Let $\{\mu_n,\,1\leq n\leq\infty\}$ be probability measures on $\mathbb{R}$ with ch.f.'s $\{f_n,\,1\leq n\leq\infty\}$.  We have $$\mu_n\overset{v}{\rightarrow}\mu_\infty\implies f_n\rightarrow f_\infty\mbox{ uniformly in every finite interval.}$$

[Theorem 2] Convergence of ch.f. implies vague convergence.

Let $\{\mu_n,\,1\leq n<\infty\}$ be probability measures on $\mathbb{R}$ with ch.f.'s $\{f_n,\,1\leq n<\infty\}$.  Suppose that 

  (a1) $f_n$ converges everywhere in $\mathbb{R}$, say $f_n\rightarrow f_\infty$.

  (a2) $f_\infty$ is continuous at $t=0$.

Then we have 

  (b1) $\mu_n\overset{v}{\rightarrow}\mu_\infty$, where $\mu_\infty$ is a probability measure. 

  (b2) $f_\infty$ is the ch.f. of $\mu_\infty$.

General Conditions for A Series Converging as An Exponential Term

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Let $\{\theta_{nj},\,1\leq j \leq k_n,\,1\leq n\}$ be a double array of complex numbers satisfying the following conditions as $n\rightarrow\infty$:

(1) $\displaystyle\underset{1\leq j \leq k_n}{\max}|\theta_{nj}|\rightarrow0;$
(2) $\displaystyle\sum_{j=1}^{k_n}|\theta_{nj}|\leq M<\infty$, where $M$ does not depend on $n$;
(3) $\displaystyle\sum_{j=1}^{k_n}\theta_{nj}\rightarrow\theta$, where $\theta$ is a (finite) complex number.

Then we have $$\prod_{j=1}^{k_n}(1+\theta_{nj})\rightarrow e^\theta.$$

$\bullet$ Proof.

Varied Type of Borel-Cantelli Lemma I

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Let $\{E_n\}$ be arbitrary events satisfying

(1) $\underset{n}{\lim}\mathscr{P}(E_n)=0$;
(2) $\underset{n}{\sum}\mathscr{P}(E_nE_{n+1}^c)<\infty$,

then $\mathscr{P}\{\limsup_n E_n\}=0$.

 $\bullet$ Proof.

Application of the Characteristic Function (2)

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Let $X_n$ have the binomial distribution with parameter $(n,p_n)$, and suppose that $n\,p_n\rightarrow\lambda\geq0$. Prove that $X_n$ converges in dist. to the Poisson d.f. with parameter $\lambda$. (In the old days this was called the law of small numbers.)

$\bullet$ Proof.

Application of The Classical Central Limit Theorem (2)

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Let $\{X_j,\,j\geq1\}$ be independent, identically distributed r.v.'s with mean $0$ and variance $1$. Prove that both $$\frac{\displaystyle\sum_{j=1}^nX_j}{\sqrt{\displaystyle\sum_{j=1}^nX^2_j}}\quad
\mbox{ and }\quad\frac{\displaystyle{\sqrt{n}\sum_{j=1}^nX_j}}{\displaystyle\sum_{j=1}^nX^2_j}$$converge in distribution to $\Phi$.

$\bullet$ Proof.

Application of The Classical Central Limit Theorem (1)

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Let $X_\lambda$ have the Poisson distribution with parameter $\lambda$.  Consider the limit distribution of $(X_\lambda-\lambda)/\lambda^{1/2}$ as $\lambda\rightarrow\infty$.  Since $X_\lambda\sim\textit{Poi}\,(\lambda)$, we have $$\mathscr{E}(X_\lambda)=\lambda\mbox{ and }\sigma^2(X_\lambda)=\lambda.$$ $X_\lambda$ is a single random variable which of course be i.i.d.  Thus by the Classical Central Limit Theorem, we have  $$\frac{X_\lambda-\mathscr{E}(X_\lambda)}{\sigma(X_\lambda)\sqrt{1}} = \frac{X_\lambda-\lambda}{\lambda^{1/2}}\overset{\mathscr{L}}{\longrightarrow}\boldsymbol{\Phi},
$$where $\boldsymbol{\Phi}$ is normal distribution with mean 0 and variance 1.

$\Box$

Linderberg-Feller's Central Limit Theorem (completed)

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Let $\{X_{nj}\}$, $n=1,2,...$, $j=1,2,...,k_n$, be a double array of random variables and for each $n$, $X_{n1},\ldots,X_{nk_n}$ are independent.  Define $S_n=\sum_{j=1}^{k_n}X_{nj}$ and
$$\begin{array}{ll}
\mathscr{E}(X_{nj})=\alpha_{nj}, & \mathscr{E}(S_n)=\sum_{j=1}^{k_n}\alpha_{nj}=\alpha_n; \\
\sigma^2(X_{nj})=\sigma^2_{nj}, & \sigma^2(S_n)=\sum_{j=1}^{k_n}\sigma^2_{nj}=s^2_n. \\
\end{array}$$Suppose $\alpha_{nj}=0$ for all $n$ and $j$, and $s^2_n=1$.  In order that as $n\rightarrow\infty$ the two conclusions below both hold:

(1) $S_n$ converges in distribution to $\Phi$.
(2) $\{X_{nj}\}$ is uniformly asymptotically negligible (UAN);

it is necessary and sufficient that for each $\eta>0$, we have $$\underset{n\rightarrow\infty}{\lim}\sum_{j=1}^{k_n}\mathscr{E}\left[X_{nj}^2\,I\left(|X_{nj}|>\eta\right)\right]=0$$

$\bullet$ Proof.