Skip to main content
MathOverflow

Return to Question

Became Hot Network Question
deleted 1 character in body
Source Link
trisct
trisct
  • 283
  • 1
  • 9

Let $f$ be the characteristic function of a real-valued random variable $X$. It is known that if $f$ has a $k$-th order derivative (for some even $k$) then $\mu$ has a finite $k$-th order moment. Is there any reference or discussion on a weaker condition that achieves the samecase when $k$ is odd? For example, if we only know that $f$ has its first order derivative, do we have the same resultknow that $\mu$ has finite first moment?

Let $f$ be the characteristic function of a real-valued random variable $X$. It is known that if $f$ has a $k$-th order derivative (for some even $k$) then $\mu$ has a finite $k$-th order moment. Is there any reference or discussion on a weaker condition that achieves the same? For example, if we only know that $f$ has its first order derivative, do we have the same result?

Let $f$ be the characteristic function of a real-valued random variable $X$. It is known that if $f$ has a $k$-th order derivative (for some even $k$) then $\mu$ has a finite $k$-th order moment. Is there any reference or discussion the case when $k$ is odd? For example, if we only know that $f$ has its first order derivative, do we know that $\mu$ has finite first moment?

Source Link
trisct
trisct
  • 283
  • 1
  • 9

Differentiability of characteristic functions and moments of the corresponding measure

Let $f$ be the characteristic function of a real-valued random variable $X$. It is known that if $f$ has a $k$-th order derivative (for some even $k$) then $\mu$ has a finite $k$-th order moment. Is there any reference or discussion on a weaker condition that achieves the same? For example, if we only know that $f$ has its first order derivative, do we have the same result?