Distribution with no fractional moments?

AI Thread Summary
The discussion explores the existence of a random variable X for which no p in (0,1) satisfies EX^p < ∞. Examples include applying rapidly growing functions to standard random variables, such as exp(N^2) with N from a geometric distribution. The Cauchy distribution is mentioned as a notable case where p=1 is excluded. The conversation shifts to defining a random variable X on [0,1] as x^{-1/n} and questions the existence of E[|X|^p] for various p values. Participants are encouraged to identify functions larger than x^{-1/n} near zero and analyze their p'th moments.
St41n
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Is it possible that for a random variable X there exist no p in (0,1) so that EX^p < ∞ ?
Is there any example?
 
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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.
 
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.
 
ssd said:
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
 
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?
 
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