Convergence in Probability

Definition

Let

• $\{X_n\}$ : Sequence of Random variables
• $X$ : Random variable

If ${\lim }_{n\to \infty }P[|X_n-X|\ge ϵ]=0,\forall ϵ>0$

• Or equivalently ${\lim }_{n\to \infty }P[|X_n-X|<ϵ]=1,\forall ϵ>0$

Then

• We say $\{X_n\}$ converges in probability to $X$
• We write $X_n\stackrel{P}{\to }X$

Remark

If $X_n\stackrel{P}{\to }X$, we often say that the mass of $X_n-X$ is converging to $0$.

In statistics, often the limiting random variable $X$ is a constant; i.e., $X$ has a degenerate distribution with all its mass at some constant $a$. In this case, we write $X_n\stackrel{P}{\to }a$.

Related theorems

• Weak law of large numbers : ${\bar{X}}_n\stackrel{P}{\to }\mu$
• Theorem 5.1.2 : $X_n+Y_n\stackrel{P}{\to }X+Y$
• Theorem 5.1.3 : $X_n\stackrel{P}{\to }X⟹a{X}_n\stackrel{P}{\to }aX$
• Theorem 5.1.4 : $g(X_n)\stackrel{P}{\to }g(a)$
• Theorem 5.1.5 : $X_n{Y}_n\stackrel{P}{\to }XY$
• Weak law of large numbers for sample variance : $S_n^2\stackrel{P}{\to }{\sigma }^2$
• Theorem 5.2.2 : $X_n\stackrel{D}{\to }b⟺X_n\stackrel{P}{\to }b$
• Theorem 1 : $U_n\stackrel{P}{\to }c⟹\frac{U_n}{c}\stackrel{P}{\to }1$
• Theorem 2 : $\sqrt{U_n}\stackrel{P}{\to }\sqrt{c}$
• Theorem 2.5
  • $U_n{V}_n\stackrel{P}{\to }cd$
  • $\frac{U_n}{V_n}\stackrel{P}{\to }\frac{c}{d},d\ne 0$