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About the notation of the probability measures

Our textbook uses the following notation for weak convergence of probability measure:

$\mu_{n}\overset{w}{\rightarrow}\mu$

The relationship for convengence in distribution and weak convergence of ``distribution'' measures.

$X_{n}\overset{\mathcal{D}}{\rightarrow}X\Leftrightarrow P^{X_{n}}\overset{w}{\rightarrow}P^{X}$.

I wonder whether $P^{X_{n}}$ and $\mu_{n}$ are the same thing, i.e. $\mu_{n}\overset{w}{\rightarrow}\mu\Leftrightarrow P^{X_{n}}\overset{w}{\rightarrow}P^{X}$

Thank you!

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It should be since $P^{X_n}$ is a measure on the image space of $X_n$. –  echoone Apr 10 '12 at 0:11
    
So do we have "$\mu_{n}\overset{w}{\rightarrow}\mu\Leftrightarrow P^{X_{n}}\overset{w}{\rightarrow}P^{X}$"? –  user16859 Apr 10 '12 at 0:13

1 Answer 1

up vote 1 down vote accepted

Yes. The reason $P^{X_{n}}$ is used is because it indicates that these are distributions of $X_n$.

They're not (necessarily) "the same thing", but you can show easily that the two definitions are equivalent:

(i) A sequence of probability measures $\alpha_n$ is said to converge weakly to a probability measure $\alpha$ if $\alpha_n[I] \rightarrow \alpha [I] $ for all intervals of continuity I

(ii) A sequence of probability measures $\alpha_n$ with distributions $F_n$ is said to converge weakly to a probability measure $\alpha$ with distribution $F$ if $F_n(y)\rightarrow F(y)$ for all continuity points y

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Thank you. I saw this argument somewhere. So you think we have $\mu_{n}\overset{w}{\rightarrow}\mu\Leftrightarrow P^{X_{n}}\overset{w}{\rightarrow}P^{X}$ ? –  user16859 Apr 10 '12 at 1:52
    
I checked the textbook. They are the same. –  user16859 Apr 10 '12 at 17:57

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