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gradstudent
gradstudent
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I am looking for a sequence of probability distributions (parameterized by $h \in \{1,2,3,4,..\}$) supported on the compact interval $x \sim [a(h),b(h)]$ such that,

  • $a(h) > \frac{b(h)}{h^{\nu^2}} >0$ for some $\nu \in \mathbb{R}$ (Hence $x \geq 0$)
  • $\frac {\mathbb{E}[x^2]}{\mathbb{E}^2[x]}$ is an (preferably polynomially) increasing function of $h$.

It would be great if someone could give an example of such a sequence or let me know if there is any argument which shows that such a thing is impossible.

EDIT : Its okay if the properties are satisfied only in the large $h$ asymptotics.

I am looking for a sequence of probability distributions (parameterized by $h \in \{1,2,3,4,..\}$) supported on the compact interval $x \sim [a(h),b(h)]$ such that,

  • $a(h) > \frac{b(h)}{h^{\nu^2}} >0$ for some $\nu \in \mathbb{R}$ (Hence $x \geq 0$)
  • $\frac {\mathbb{E}[x^2]}{\mathbb{E}^2[x]}$ is an (preferably polynomially) increasing function of $h$.

It would be great if someone could give an example of such a sequence or let me know if there is any argument which shows that such a thing is impossible.

I am looking for a sequence of probability distributions (parameterized by $h \in \{1,2,3,4,..\}$) supported on the compact interval $x \sim [a(h),b(h)]$ such that,

  • $a(h) > \frac{b(h)}{h^{\nu^2}} >0$ for some $\nu \in \mathbb{R}$ (Hence $x \geq 0$)
  • $\frac {\mathbb{E}[x^2]}{\mathbb{E}^2[x]}$ is an (preferably polynomially) increasing function of $h$.

It would be great if someone could give an example of such a sequence or let me know if there is any argument which shows that such a thing is impossible.

EDIT : Its okay if the properties are satisfied only in the large $h$ asymptotics.

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gradstudent
gradstudent
  • 2.2k
  • 16
  • 28

Looking for a specific kind of a compactly supported one dimensional distribution

I am looking for a sequence of probability distributions (parameterized by $h \in \{1,2,3,4,..\}$) supported on the compact interval $x \sim [a(h),b(h)]$ such that,

  • $a(h) > \frac{b(h)}{h^{\nu^2}} >0$ for some $\nu \in \mathbb{R}$ (Hence $x \geq 0$)
  • $\frac {\mathbb{E}[x^2]}{\mathbb{E}^2[x]}$ is an (preferably polynomially) increasing function of $h$.

It would be great if someone could give an example of such a sequence or let me know if there is any argument which shows that such a thing is impossible.