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$fin C^{1}$. If $X, f(X)$ are heavy-tailed, so is $left| f'(X) cdot f(X)right|$.

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$begingroup$

Let $X$ be a $mathbb{R}$-valued heavy-tailed and continuous random variable with density function $p_X$. Furthermore, let $f:mathbb{R}tomathbb{R}$ be continuously differentiable.

I am trying to show whether the following assertion is true:

If $X$ and $f(X)$ are heavy-tailed, so is
begin{equation}
left|f'(X) cdot f(X)right|.
end{equation}

To prove heavy-tailedness we need to show that for all $t>0$

begin{equation}
mathbb{E}left[expleft(t cdot left|f'(x) cdot f(x)right|right) right]
= int_{mathbb R} expleft(t cdot left|f'(x) cdot f(x)right|right) p_X(x) dx = infty
end{equation}

Any hints?

probability probability-theory statistics probability-distributions

share|cite|improve this question

edited Mar 13 at 16:09

Carl

asked Mar 11 at 12:59

CarlCarl

11311

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You edited your question and no longer assume $f$ to be Lipschitz - in other words you no longer assume $f'$ to be bounded. Is that really what you wanted?
  $endgroup$
  – Mars Plastic
  Mar 13 at 16:11
  
  
  

  
  
  
  

add a comment | 

1

1

$begingroup$

Let $X$ be a $mathbb{R}$-valued heavy-tailed and continuous random variable with density function $p_X$. Furthermore, let $f:mathbb{R}tomathbb{R}$ be continuously differentiable.

I am trying to show whether the following assertion is true:

If $X$ and $f(X)$ are heavy-tailed, so is
begin{equation}
left|f'(X) cdot f(X)right|.
end{equation}

To prove heavy-tailedness we need to show that for all $t>0$

begin{equation}
mathbb{E}left[expleft(t cdot left|f'(x) cdot f(x)right|right) right]
= int_{mathbb R} expleft(t cdot left|f'(x) cdot f(x)right|right) p_X(x) dx = infty
end{equation}

Any hints?

probability probability-theory statistics probability-distributions

share|cite|improve this question

edited Mar 13 at 16:09

Carl

asked Mar 11 at 12:59

CarlCarl

11311

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You edited your question and no longer assume $f$ to be Lipschitz - in other words you no longer assume $f'$ to be bounded. Is that really what you wanted?
  $endgroup$
  – Mars Plastic
  Mar 13 at 16:11
  
  
  

  
  
  
  

add a comment | 

$begingroup$

Let $X$ be a $mathbb{R}$-valued heavy-tailed and continuous random variable with density function $p_X$. Furthermore, let $f:mathbb{R}tomathbb{R}$ be continuously differentiable.

I am trying to show whether the following assertion is true:

If $X$ and $f(X)$ are heavy-tailed, so is
begin{equation}
left|f'(X) cdot f(X)right|.
end{equation}

To prove heavy-tailedness we need to show that for all $t>0$

begin{equation}
mathbb{E}left[expleft(t cdot left|f'(x) cdot f(x)right|right) right]
= int_{mathbb R} expleft(t cdot left|f'(x) cdot f(x)right|right) p_X(x) dx = infty
end{equation}

Any hints?

probability probability-theory statistics probability-distributions

share|cite|improve this question

edited Mar 13 at 16:09

Carl

asked Mar 11 at 12:59

CarlCarl

11311

$endgroup$

share|cite|improve this question

edited Mar 13 at 16:09

Carl

asked Mar 11 at 12:59

CarlCarl

11311

share|cite|improve this question

edited Mar 13 at 16:09

Carl

asked Mar 11 at 12:59

CarlCarl

11311

asked Mar 11 at 12:59

CarlCarl

11311

asked Mar 11 at 12:59

CarlCarl

11311