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Consider the metric space $X = \mathbb{R}$, $\mathcal{B}$ the Borel $\sigma$-algebra on $\mathbb{R}$ and $\mu$ a probability measure on $X$. Let $A \in \mathcal{B}$ and $\tau_n \nearrow \infty$ a sequence of positive numbers.

I can't demonstrate that the following sequence of measures is or is not convergent in the weak topology: $$\mu_n\left(A\right)=\frac{1}{\tau_n} \int_0^{\tau_n} \mu(A-t) dt.$$

Can someone help me?

Thank you!

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Observe that for any $x\in\mathbb R$ and $\delta\gt 0$, the inclusion $$\left(-\infty,x-t\right]\subset\left(-\infty,x-\delta\tau_n\right] $$ holds for $\tau_n\delta\leqslant t\leqslant \tau_n$. Therefore, $$\mu_n\left(-\infty,x\right]\leqslant \delta +\mu\left(-\infty,x-\delta\tau_n\right]$$ and it follows that $\mu_n\left(-\infty,x\right]\to 0$ as $n$ goes to infinity.

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    The characteristic functions of these measures tend to 0 at all points except 0 and they tend to 1 at 0. Hence these measures don't converge weakly. – 2017-02-17