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I'm trying to show

$$ \pi(\theta_n|X) \overset{a.s.}{\to} \pi(\theta|X) $$

where $\theta_n \sim N(0, n)$, and $\pi(\theta) = 1$(improper), and $X$ is normal.

Is there any reference, or hint to approach this problem?

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    $\begingroup$ I think you are describing the bayesian central limit theorem which takes your idea a step further and says the posterior distribution will converge approximately to a normal distribution. Here's the first reference I could find with Google: www2.myoops.org/twocw/mit/NR/rdonlyres/Mathematics/… $\endgroup$ – Great38 Oct 5 '18 at 1:53

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