I'm reading about Lévy processes. Durig this, I've found with the next proposition without proof:
For every fixed $a>0,$ the measure $\frac{1}{\epsilon}P_{0}(X_{\epsilon}\in dx)$ converges vaguely on $\{|x|>a\}$ as $\epsilon\rightarrow 0+$ to $\Pi(dx).$
I'm stuck prove this. I've tried to use the Fourier transform, indeed I was triying to use Fourier inversion to get such convergence but I don't have any useful.
Is there a easy way to prove this?
Any kind of help is thanked in advanced.