If this question has already been answered, please link me because I could not find anything online.

I have been using exponential Brownian motion in my models of stochastic population dynamics. The hitting times form a Levy distribution, which we cannot compute the expectation of. I am aware that the expectation of a Levy distribution diverges due to the heavy tails of the distribution, but how can I justify this analytically? We could set up the integral as $$\int xp(x)dx$$ but what exactly are we computing?

Thank you in advance for any insight!

  • $\begingroup$ If you integrate over the support you'll get $\infty$. This justifies it analytically. $\endgroup$ – Wintermute May 23 '18 at 22:23
  • $\begingroup$ Over the support? $\endgroup$ – liveFreeOrπHard May 24 '18 at 16:29
  • $\begingroup$ Over the region where $x$ is not zero.For a Levy distribution this is $x \in [\mu,\infty)$ where $\mu$ is the location parameter. $\endgroup$ – Wintermute May 24 '18 at 16:34
  • $\begingroup$ I read that μ is the shift parameter of a specific distribution, but how could I show this in the general case to prove our expectations will diverge? Does this have to do with the moment-generating function? $\endgroup$ – liveFreeOrπHard May 24 '18 at 16:52

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