Distribution with no fractional moments?

[St41n]

Is it possible that for a random variable X there exist no p in (0,1) so that EX^p < ∞ ?
Is there any example?

 

[bpet]

Yes, many examples can be constructed by applying a sufficiently rapidly growing function to one of the standard random variables, e.g. exp(N^2) where N has the geometric distribution.

 

[ssd]

Well, the nearest I readily remember is Cauchy distbn, the case of p=1 (which of course is excluded by your condition). The condition 0<p<1 restricts us to only non negative variables.

 

[St41n]

Well, the nearest I readily remember is Cauchy distbn, the case of p=1 (which of course is excluded by your condition). The condition 0<p<1 restricts us to only non negative variables.

I actually meant E|X|^p < ∞ to include RV's on R

 

[micromass]

Try to find one yourself.

Let your phase space be [0,1] with the usual Borel structure and the usual measure.

Define the random variable $X:[0,1]\rightarrow \mathbb{R}:x\rightarrow x^{-1/n}$. What can you say about $E[|X|^p]$? For which p does this exist?

Can you find a function on $[0,1]$ which is larger then any $x^{-1/n}$ as soon as x is close to 0?? What about the p'th moments of those functions?