Distribution with no fractional moments?

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Discussion Overview

The discussion revolves around the existence of random variables for which no fractional moments exist, specifically questioning whether there are random variables X such that for no p in (0,1) is the expectation EX^p finite. Participants explore examples and propose functions to illustrate their points.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the existence of a random variable X for which no p in (0,1) satisfies EX^p < ∞.
  • Another participant suggests that examples can be constructed using rapidly growing functions applied to standard random variables, such as exp(N^2) where N follows a geometric distribution.
  • A participant references the Cauchy distribution as a related case, noting that the condition 0
  • Clarification is provided that the expectation E|X|^p < ∞ is intended to include random variables defined on the real line.
  • One participant challenges others to find a specific random variable defined on [0,1] and questions the existence of its p-th moments, particularly focusing on the behavior of functions like x^{-1/n} as x approaches 0.

Areas of Agreement / Disagreement

Participants have differing views on the existence and construction of random variables without finite fractional moments, with no consensus reached on specific examples or the implications of the proposed functions.

Contextual Notes

Participants express uncertainty regarding the conditions under which certain expectations exist and the implications of the definitions used, particularly concerning the behavior of functions near zero.

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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