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Anand
Anand
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Dear friends,

Denote the function

$$ G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;. $$

It is well known that if $a\in ]0,2]$, $G_a(x)$ is the density of the symmetric stable lawlaws. The problem is what happens if $a>2$ ? It is not non-negative any more. But $\int_R G_a(x)d x=1$. Where can I find more properties of this function? For example, how does it decay (modulo the small oscillation)?

Thank you very much for any hints or references.

Anand

Dear friends,

Denote the function

$$ G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;. $$

It is well known that if $a\in ]0,2]$, $G_a(x)$ is the density of the symmetric stable law. The problem is what happens if $a>2$ ? It is not non-negative any more. But $\int_R G_a(x)d x=1$. Where can I find more properties of this function? For example, how does it decay (modulo the small oscillation)?

Thank you very much for any hints or references.

Anand

Dear friends,

Denote the function

$$ G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;. $$

It is well known that if $a\in ]0,2]$, $G_a(x)$ is the density of symmetric stable laws. The problem is what happens if $a>2$ ? It is not non-negative any more. But $\int_R G_a(x)d x=1$. Where can I find more properties of this function? For example, how does it decay (modulo the small oscillation)?

Thank you very much for any hints or references.

Anand

deleted 1 characters in body
Source Link
Anand
Anand
  • 1.6k
  • 2
  • 22
  • 33

Dear friends,

Denote the function

$$ G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;. $$

It is well known that if $a\in ]0,2]$, $G_a(x)$ is the density of the symmetric stable law. The problem is what happens if $a>2$ ? It is not non-negative any more. But $\int_R G_a(x)d x=1$. Where can I find more properties of this function? For example, how does it decay (modulo the small oscillation)?

Thank you very much for any hints andor references.

Anand

Dear friends,

Denote the function

$$ G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;. $$

It is well known that if $a\in ]0,2]$, $G_a(x)$ is the density of the symmetric stable law. The problem is what happens if $a>2$ ? It is not non-negative any more. But $\int_R G_a(x)d x=1$. Where can I find more properties of this function? For example, how does it decay (modulo the small oscillation)?

Thank you very much for any hints and references.

Anand

Dear friends,

Denote the function

$$ G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;. $$

It is well known that if $a\in ]0,2]$, $G_a(x)$ is the density of the symmetric stable law. The problem is what happens if $a>2$ ? It is not non-negative any more. But $\int_R G_a(x)d x=1$. Where can I find more properties of this function? For example, how does it decay (modulo the small oscillation)?

Thank you very much for any hints or references.

Anand

Source Link
Anand
Anand
  • 1.6k
  • 2
  • 22
  • 33

Asking for a Fourier inverse transform, which is related to stable laws

Dear friends,

Denote the function

$$ G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;. $$

It is well known that if $a\in ]0,2]$, $G_a(x)$ is the density of the symmetric stable law. The problem is what happens if $a>2$ ? It is not non-negative any more. But $\int_R G_a(x)d x=1$. Where can I find more properties of this function? For example, how does it decay (modulo the small oscillation)?

Thank you very much for any hints and references.

Anand