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.

  • $\begingroup$ A heuristic argument is as follows: Consider the Levy-Ito decomposition of the process X $$ X =\text{ drift}+ (\text{ Continuous part} + \text{ Small jumps} ) + \text{ Large Jumps}. $$ The Large jump process of $X$ is given by a compound Poisson process and the jump sizes are distributed according to the law $\Pi$. $\endgroup$ – Sayantan Apr 20 '18 at 3:50
  • $\begingroup$ So, if you restrict yourself on the set $\{|x|>a\}$, the limit as $\epsilon \to 0$, $P_0(X_{\epsilon} \in dx)$ is the probability that $X$ makes a jump of size $\,dx$ (as $|x|>a>0$, this is a large jump) immediately after staring. It might then be possible to make this argument rigorous to prove the desired result. $\endgroup$ – Sayantan Apr 20 '18 at 3:54
  • $\begingroup$ Thanks to answer @Sayantan. Is useful to understand the intuition behind this. How could express this on a formal way? $\endgroup$ – Squird37 Apr 20 '18 at 3:59
  • $\begingroup$ Are you still interested in the problem or do you have already found a solution? $\endgroup$ – saz May 8 '18 at 11:57
  • $\begingroup$ I'm still interested in the solution. To this day I haven't achieved to prove this @zas. $\endgroup$ – Squird37 May 8 '18 at 15:07

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