

% STA2112f99 Convergence Handout


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\begin{center}   
{\Large Convergence of Sequences of Random Variables}\footnote{This material can be found in many texts. I took it from T. S. Ferguson's \emph{A course in large sample theory:} Chapman and Hall, 1996}\\
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\begin{itemize}
   \item Definitions (All quantities in boldface are vectors in $\mathbb{R}^m$ unless otherwise stated )
   \begin{itemize}
      \item[$\star$] $ \mathbf{X}_n \stackrel{a.s.}{\rightarrow} \mathbf{X}$ means
            $P\{s:\, \lim_{n \rightarrow \infty} \mathbf{X}(s) = \mathbf{X}(s)\}=1$.
      \item[$\star$] $ \mathbf{X}_n \stackrel{P}{\rightarrow} \mathbf{X}$ means
            $\forall \epsilon>0,\,\lim_{n \rightarrow \infty}
            P\{|\mathbf{X}_n-\mathbf{X}|<\epsilon  \}=1$.
      \item[$\star$] $ \mathbf{X}_n \stackrel{d}{\rightarrow} \mathbf{X}$ means
            for every continuity point $\mathbf{x}$ of $F_\mathbf{X}$, 
            $\lim_{n \rightarrow \infty}F_{\mathbf{X}_n}(\mathbf{x}) = F_\mathbf{X}(\mathbf{x})$.
   \end{itemize}
   \item $ \mathbf{X}_n \stackrel{a.s.}{\rightarrow} \mathbf{X} \Rightarrow
           \mathbf{X}_n \stackrel{P}{\rightarrow} \mathbf{X} \Rightarrow
           \mathbf{X}_n \stackrel{d}{\rightarrow} \mathbf{X} $.
   \item If $\mathbf{a}$ is a vector of constants, $ \mathbf{X}_n \stackrel{d}{\rightarrow} \mathbf{a} 
                     \Rightarrow \mathbf{X}_n \stackrel{P}{\rightarrow} \mathbf{a}$.
   \item Strong Law of Large Numbers (SLLN): Let $\mathbf{X}_1, \ldots \mathbf{X}_n$ be i.i.d. 
         random vectors with finite first moment. Then 
         $ \overline{\mathbf{X}}_n \stackrel{a.s.}{\rightarrow} E(\mathbf{X}_1)$.

  \item Central Limit Theorem: Let $\mathbf{X}_1, \ldots, \mathbf{X}_n$ be i.i.d. random vectors 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 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}

   \item Delta Method (Theorem of Cram\'{e}r, Ferguson p. 45): Let $g: \mathbb{R}^d \rightarrow \mathbb{R}^k$ be such that the elements of \.{g}$(\mathbf{x}) = 
\left[ \frac{\partial g_i}{\partial x_j} \right]_{k \times d}$ are continuous in a neighborhood of $\boldsymbol{\theta} \in \mathbb{R}^d$. If $\mathbf{X}_n$ is a sequence of $d$-dimensional random vectors such that $\sqrt{n}(\mathbf{X}_n-\boldsymbol{\theta}) \stackrel{d}{\rightarrow} \mathbf{X}$, then 
$\sqrt{n}(g(\mathbf{X}_n)-g(\boldsymbol{\theta})) \stackrel{d}{\rightarrow} 
\mbox{\.{g}} (\boldsymbol{\theta}) \mathbf{X}$. In particular, if
$\sqrt{n}(\mathbf{X}_n-\boldsymbol{\theta}) \stackrel{d}{\rightarrow} \mathbf{X} 
\sim N(\mathbf{0},\mathbf{\Sigma})$, then $\sqrt{n}(g(\mathbf{X}_n)-g(\boldsymbol{\theta})) \stackrel{d}{\rightarrow} 
\mathbf{Y} \sim N(\mathbf{0},
\mbox{\.{g}}(\boldsymbol{\theta})\mathbf{\Sigma}\mbox{\.{g}}(\boldsymbol{\theta}) ^\prime)$.


\end{itemize}


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