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Let $B_t$ be a standard Brownian motion on $\mathbb R$ started at $0$. For $A\subset\mathbb R$ Lebesgue measurable, let $\mu_T(A) = \frac{1}{T} m(t \leq T: B_t \in A)$, where $m$ is Lebesgue measure. Then $\mu_T$ is a random measure on $\mathbb R$. Do the random measures $\mu_T$ converge, in any reasonable sense, as $T\rightarrow \infty$? If so, what is the limit measure?

I know that the event that $\mu_T \rightarrow \mu$ for some fixed deterministic measure $\mu$ is a tail event (it is in the intersection of the $\sigma$-algebras $\sigma(B_t : t \geq s)$). Therefore its probability is either 0 or 1. Beyond that, however, I do not have any real ideas as to how to approach this.

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Show what you tried and the results you know that seem related to the question. (Amazing that after 17 months and 135 questions, you so squarely neglect the procedure...) – Did Feb 11 '13 at 16:35
up vote 1 down vote accepted

Hint: $\mu_T(A)\to0$ for every bounded $A$.

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