% STA2112f99 Convergence Handout


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\begin{center}   
{\Large Convergence of Sequences of Random Variables}\\
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\begin{itemize}
   \item Definitions
   \begin{itemize}
      \item[$\star$] $ X_n \stackrel{a.s.}{\rightarrow} X$ means
            $P\{s:\, \lim_{n \rightarrow \infty} X(s) = X(s)\}=1$.
      \item[$\star$] $ X_n \stackrel{P}{\rightarrow} X$ means
            $\forall \epsilon>0,\,\lim_{n \rightarrow \infty}
            P\{|X_n-X|<\epsilon  \}=1$.
      \item[$\star$] $ X_n \stackrel{d}{\rightarrow} X$ means
            for every continuity point $x$ of $F_X$, 
            $\lim_{n \rightarrow \infty}F_{X_n}(x) = F_X(x)$.
   \end{itemize}
   \item $ X_n \stackrel{a.s.}{\rightarrow} X$ if and only if 
   $\forall \epsilon>0,\,\lim_{n \rightarrow \infty}
            P( \cap_{k=n}^\infty\{|X_k-X|<\epsilon\})=1$.
   \item $ X_n \stackrel{a.s.}{\rightarrow} X \Rightarrow
           X_n \stackrel{P}{\rightarrow} X \Rightarrow
           X_n \stackrel{d}{\rightarrow} X $.
   \item If $a$ is a constant, $ X_n \stackrel{d}{\rightarrow} a 
                     \Rightarrow X_n \stackrel{P}{\rightarrow} a$.
   \item If $\lim_{n \rightarrow \infty}f_{X_n}(x) = f_X(x)$ for each $x$, 
            $X_n \stackrel{d}{\rightarrow} X$. 
   \item Let $\mathbf{X}$ and $\mathbf{X}_n$ be random vectors in $\mathbb{R}^k$.
         $\mathbf{X}_n \stackrel{d}{\rightarrow} \mathbf{X}$ if and only if
         $\lim_{n \rightarrow \infty}E[g(\mathbf{X}_n)] = E[g(\mathbf{X})]$ for 
         every bounded continuous function $g:\,\mathbb{R}^k \rightarrow
                                             \mathbb{R}$.
   \item Slutsky Theorems for Convergence in Distribution:
        \begin{enumerate}
            \item If $\mathbf{X}_n \in \mathbb{R}^m$, 
                     $\mathbf{X}_n \stackrel{d}{\rightarrow} \mathbf{X}$ and if
                     $f:\,\mathbb{R}^m \rightarrow \mathbb{R}^q$ 
                     (where $q \leq m$) is
                      continuous except possibly on a set $C$ with 
                      $P(\mathbf{X} \in C)=0$, then
                      $f(\mathbf{X}_n) \stackrel{d}{\rightarrow} f(\mathbf{X})$.
            \item If $\mathbf{X}_n \stackrel{d}{\rightarrow} \mathbf{X}$ and
                     $(\mathbf{X}_n - \mathbf{Y}_n) \stackrel{P}{\rightarrow} 0$,
                     then $\mathbf{Y}_n \stackrel{d}{\rightarrow} \mathbf{X}$.
            \item If $\mathbf{X}_n \in \mathbb{R}^d$,
                     $\mathbf{Y}_n \in \mathbb{R}^k$, 
                     $\mathbf{X}_n \stackrel{d}{\rightarrow} \mathbf{X}$ and
                     $\mathbf{Y}_n \stackrel{d}{\rightarrow} \mathbf{c}$, then
                     \begin{displaymath}
                        \left( \begin{array}{cc}  
                        \mathbf{X}_n \\ \mathbf{Y}_n
                        \end{array} \right)
                          \stackrel{d}{\rightarrow}
                        \left( \begin{array}{cc}  
                        \mathbf{X} \\ \mathbf{c}
                        \end{array} \right)
                     \end{displaymath}
        \end{enumerate}
   \item Slutsky Theorems for Convergence in Probability:
        \begin{enumerate}
            \item If $\mathbf{X}_n \in \mathbb{R}^m$, 
                     $\mathbf{X}_n \stackrel{P}{\rightarrow} \mathbf{X}$ and if
                      $f:\,\mathbb{R}^m \rightarrow \mathbb{R}^q$ 
                     (where $q \leq m$) is
                      continuous except possibly on a set $C$ with 
                      $P(\mathbf{X} \in C)=0$, then
                      $f(\mathbf{X}_n) \stackrel{P}{\rightarrow} f(\mathbf{X})$.
            \item If $\mathbf{X}_n \stackrel{P}{\rightarrow} \mathbf{X}$ and
                     $(\mathbf{X}_n - \mathbf{Y}_n) \stackrel{P}{\rightarrow} 0$,
                     then $\mathbf{Y}_n \stackrel{P}{\rightarrow} \mathbf{X}$.
            \item If $\mathbf{X}_n \in \mathbb{R}^d$,
                     $\mathbf{Y}_n \in \mathbb{R}^k$, 
                     $\mathbf{X}_n \stackrel{P}{\rightarrow} \mathbf{X}$ and
                     $\mathbf{Y}_n \stackrel{P}{\rightarrow} \mathbf{Y}$, then
                     \begin{displaymath}
                        \left( \begin{array}{cc}  
                        \mathbf{X}_n \\ \mathbf{Y}_n
                        \end{array} \right)
                          \stackrel{P}{\rightarrow}
                        \left( \begin{array}{cc}  
                        \mathbf{X} \\ \mathbf{Y}
                        \end{array} \right)
                     \end{displaymath}
        \end{enumerate}
\pagebreak
   \item Let $g(x)$ have a second derivative that is continuous at $x=\theta$,
         and let $\sqrt{n}(T_n-\theta) \stackrel{d}{\rightarrow} T$. Then 
         $\sqrt{n}(g(T_n)-g(\theta)) \stackrel{d}{\rightarrow} 
         g^\prime(\theta)T$. 
   \item Strong Law of Large Numbers (SLLN): Let $X_1, \ldots X_n$ be i.i.d. 
         random variables with finite first moment. Then 
         $ \overline{X}_n \stackrel{a.s.}{\rightarrow} E(X_1)$.
  \item Let $\mathbf{X}_1, \ldots, \mathbf{X}_n$ be i.i.d. random vectors in $\mathbb{R}^k$ with expected value vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$. Then $\sqrt{n}(\overline{\mathbf{X}}_n-\boldsymbol{\mu})$ converges in distribution to a multivariate normal with mean \textbf{0} and covariance matrix $\boldsymbol{\Sigma}$.
   \item Lindeberg Central Limit Theorem: Consider the triangular array of 
         random variables
                     \begin{displaymath}
                        \begin{array}{lll}
                             X_{11}                          \\
                             X_{21}, &  X_{22}               \\
                             X_{31}, &  X_{32},  &  X_{33}   \\
                             \ldots,
                        \end{array}
                     \end{displaymath}
         where the random variables in each row are assumed independent with
          $E(X_{ij})=0$ and $Var(X_{ij})=\sigma^2_{ij}$. Let 
          $S_n = \sum_{j=1}^n X_{nj}$, and let 
          $v^2_n = Var(S_n) = \sum_{j=1}^n \sigma^2_{nj}$. Then 
          $\frac{S_n}{v_n}$ converges in distribution to a standard normal
          provided, for all $\epsilon>0$,
            \begin{displaymath}
                \lim_{n \rightarrow \infty}\frac{1}{v^2_n} \sum_{j=1}^n 
                 E[X^2_{nj}I(|X_{nj}| \geq \epsilon v_n)] = 0
            \end{displaymath}

   \item $U_n = O_p(V_n)$ means $\forall \epsilon>0, \, \exists M=M(\epsilon)$ and $N=N(\epsilon)$ such that if $n>N$, $P\{\frac{U_n}{V_n} \leq M\} - P\{\frac{U_n}{V_n} \leq -M\} > 1-\epsilon$. Usually, $V_n = n^{-t}$ for some $t>0$.

\end{itemize}


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