Convergence in Distribution

Definition

Let

• $\{X_n\}$ : Sequence of random variables, with
  • $F_{X_n}$ : cdf of $X_n$
• $X$ : Random variable
  • $F_X$ : cdf of $X$
• $C(F_X)$ denote the set of all points where $F_X$ is continuous

If

$$\underset{x\to \infty }{\lim }{F}_{X_n}(x)=F_X(x),\forall x\in C(F_X)$$

Then

• We say $X_n$ converges in distribution to $X$
• We write $X_n\stackrel{D}{\to }X$

Remark

1. The notation $X_n\stackrel{D}{\to }c$ means ${\lim }_{n\to \infty }{F}_{X_n}(x)=F_c(x)$ at continuity points of $F_c$
2. If $X_n\stackrel{D}{\to }X$, we often say the distribution of $X$ is the asymptotic distribution or the limiting distribution of $\{X_n\}$.
3. If $X$ is degenerate, with all its mass at some constant $a$. In that case, we may write $X_n\stackrel{P}{\to }a$.
4. If $X_n\stackrel{D}{\to }X$ where $X\sim N(0,1)$, we may write $X_n\stackrel{D}{\to }N(0,1)$

Related theorems

• Theorem 5.2.1 : $X_n\stackrel{P}{\to }X⟹X_n\stackrel{D}{\to }X$
• Theorem 5.2.2 : $X_n\stackrel{D}{\to }b⟺X_n\stackrel{P}{\to }b$
• Theorem 5.2.3 : $X_n+Y_n\stackrel{D}{\to }X$
• Theorem 5.2.4 : $g(X_n)\stackrel{D}{\to }g(X)$
• Slutsky’s theorem : $A_n+B_n{X}_n\stackrel{D}{\to }a+bX$
• Theorem: MGF Technique : ${\lim }_{n\to \infty }{M}_{X_n}(t)=M(t)⟹X_n\stackrel{D}{\to }X$
• Theorem 3 : $W_n\stackrel{D}{\to }F(w)$
• Central limit theorem : $Y_n\stackrel{D}{\to }N(0,1)$