Convergence in Probability

Definition

$(\forall ϵ>0,\underset{n\to \infty }{\lim }P(|X_n-X|\ge ϵ)=0)\iff (X_n\stackrel{P}{\to }X)$

Let $\{X_n\}$ be a Sequence of random variables and $X$ be a Random Variable, then $X_n$ converges in probability to $X$ if $\forall ϵ>0,\underset{n\to \infty }{\lim }P(|X_n-X|\ge ϵ)=0$, and denoted by $X_n\stackrel{P}{\to }X$

Facts

If the Sequence of random variables $\{X_n\}$ converges in probability to $X$, then it also Almost Surely converge to $X$.

$(X_n\stackrel{P}{\to }X)\Rightarrow (X_n\stackrel{D}{\to }X)$

Convergence in Probability implies convergence in distribution

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Convergence in probability has Linearity

• Additivity $({\overline{X}}_n\stackrel{P}{\to }X)\wedge ({\overline{Y}}_n\stackrel{P}{\to }Y)\Rightarrow X_n+Y_n\stackrel{P}{\to }X+Y$
• Homogeneity $\forall a\in \mathbb{R},X_n\stackrel{P}{\to }X\Rightarrow a{X}_n\stackrel{P}{\to }aX$

Continuous Mapping Theorem

Definition

Continuous functions preserve convergences (in probability, Almost Surely, or in distribution) of a Sequence of random variables to limits.

Consider a Sequence $\{X_n\}$ of random variables defined on same Probability Space, and a Continuous Function $g$ on the space. Then,

• Convergence in Probability to a Constant: $X_n\stackrel{P}{\to }a\in \mathbb{R}⟹g(X_n)\stackrel{P}{\to }g(a)$
• Convergence in Probability: $X_n\stackrel{P}{\to }X⟹g(X_n)\stackrel{P}{\to }g(X)$
• Almost Sure Convergence: $X_n\stackrel{a.s.}{\to }X⟹g(X_n)\stackrel{a.s.}{\to }g(X)$
• Convergence in Distribution: $X_n\stackrel{D}{\to }X⟹g(X_n)\stackrel{D}{\to }g(X)$

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