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show that if $(\sqrt{n}(Y_n - \theta) \overset{d}{\to} N(0, 1))$ then $(Y_n \overset{P}{\to} \theta)$

Fix $\epsilon>0$. Let $K>0$. Notice that for large enough $n$ you have $\sqrt{n}\epsilon>K$. Therefore $$ \eqalign{ \limsup_nP\left(|Y_n-\theta|>\epsilon\right) &=\limsup_nP\left(\sqrt{...
John Dawkins's user avatar
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show that if $(\sqrt{n}(Y_n - \theta) \overset{d}{\to} N(0, 1))$ then $(Y_n \overset{P}{\to} \theta)$

According to Slutsky's theorem, if $Z_n \xrightarrow{d} Z$ and $a_n \xrightarrow{p} a$, then $a_n Z_n \xrightarrow{d} a Z$. Here, $Z_n=\sqrt{n}\left(Y_n-\theta\right)$ and $Z \sim N(0,1)$, and $\frac{...
bruno's user avatar
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probability convergence as compactness for a.s. convergence

First, the mentioned result actually implies that there is no "topology of almost sure convergence." Indeed, let $(X_n)$ be a sequence of random variables that converges in probability to a ...
Michael Greinecker's user avatar

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