Characteristic Function of Gamma Distributed Random Variables
$begingroup$
I have the following characteristic function
$$sum_{m=0}^{infty} frac{(is)^m}{m!} sigma_{m,k} frac{Gamma(beta + m)}{Gamma(beta)},$$
where $i$ is the imaginary unit, $beta>0$, $Gamma(cdot)$ is the Gamma function and
$$sigma_{m,k} equiv frac{Gamma(beta+alpha)Gamma(beta+k)}{Gamma(beta + m)Gamma(beta + alpha + k)},$$
where $k$ can be any non-negative integer and $alpha > 0$ is another parameter. If there exists a $sigma_k > 0$ (a positive constant that does not depend on $m$, but can depend on $k$), such that
$$(star) qquad qquad sum_{m=0}^{infty} frac{(is)^m}{m!} sigma_{m,k} frac{Gamma(beta + m)}{Gamma(beta)} = sum_{m=0}^{infty} frac{(is)^m}{m!} sigma_{k}^m frac{Gamma(beta + m)}{Gamma(beta)}$$
then I know that the left hand side is the characteristic function of a Gamma distributed random variable with shape parameter $beta$ and scale parameter $sigma_k$, since the right hand-side is.
My Question: is it possible to show that there exists a $sigma_k$ so that $(star)$ is true? Is there some form of intermediate value theorem that I can appeal to here? Note that the sequence $sigma_{0,k}, sigma_{1,k}, ...$ is a decreasing sequence.
characteristic-functions gamma-distribution
asked Dec 22 '18 at 7:50
SEJSEJ
184
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add a comment |
$begingroup$
I have the following characteristic function
$$sum_{m=0}^{infty} frac{(is)^m}{m!} sigma_{m,k} frac{Gamma(beta + m)}{Gamma(beta)},$$
where $i$ is the imaginary unit, $beta>0$, $Gamma(cdot)$ is the Gamma function and
$$sigma_{m,k} equiv frac{Gamma(beta+alpha)Gamma(beta+k)}{Gamma(beta + m)Gamma(beta + alpha + k)},$$
where $k$ can be any non-negative integer and $alpha > 0$ is another parameter. If there exists a $sigma_k > 0$ (a positive constant that does not depend on $m$, but can depend on $k$), such that
$$(star) qquad qquad sum_{m=0}^{infty} frac{(is)^m}{m!} sigma_{m,k} frac{Gamma(beta + m)}{Gamma(beta)} = sum_{m=0}^{infty} frac{(is)^m}{m!} sigma_{k}^m frac{Gamma(beta + m)}{Gamma(beta)}$$
then I know that the left hand side is the characteristic function of a Gamma distributed random variable with shape parameter $beta$ and scale parameter $sigma_k$, since the right hand-side is.
My Question: is it possible to show that there exists a $sigma_k$ so that $(star)$ is true? Is there some form of intermediate value theorem that I can appeal to here? Note that the sequence $sigma_{0,k}, sigma_{1,k}, ...$ is a decreasing sequence.
characteristic-functions gamma-distribution
asked Dec 22 '18 at 7:50
SEJSEJ
184
$endgroup$
add a comment |
$begingroup$
I have the following characteristic function
$$sum_{m=0}^{infty} frac{(is)^m}{m!} sigma_{m,k} frac{Gamma(beta + m)}{Gamma(beta)},$$
where $i$ is the imaginary unit, $beta>0$, $Gamma(cdot)$ is the Gamma function and
$$sigma_{m,k} equiv frac{Gamma(beta+alpha)Gamma(beta+k)}{Gamma(beta + m)Gamma(beta + alpha + k)},$$
where $k$ can be any non-negative integer and $alpha > 0$ is another parameter. If there exists a $sigma_k > 0$ (a positive constant that does not depend on $m$, but can depend on $k$), such that
$$(star) qquad qquad sum_{m=0}^{infty} frac{(is)^m}{m!} sigma_{m,k} frac{Gamma(beta + m)}{Gamma(beta)} = sum_{m=0}^{infty} frac{(is)^m}{m!} sigma_{k}^m frac{Gamma(beta + m)}{Gamma(beta)}$$
then I know that the left hand side is the characteristic function of a Gamma distributed random variable with shape parameter $beta$ and scale parameter $sigma_k$, since the right hand-side is.
My Question: is it possible to show that there exists a $sigma_k$ so that $(star)$ is true? Is there some form of intermediate value theorem that I can appeal to here? Note that the sequence $sigma_{0,k}, sigma_{1,k}, ...$ is a decreasing sequence.
characteristic-functions gamma-distribution
asked Dec 22 '18 at 7:50
SEJSEJ
184
$endgroup$
I have the following characteristic function
$$sum_{m=0}^{infty} frac{(is)^m}{m!} sigma_{m,k} frac{Gamma(beta + m)}{Gamma(beta)},$$
where $i$ is the imaginary unit, $beta>0$, $Gamma(cdot)$ is the Gamma function and
$$sigma_{m,k} equiv frac{Gamma(beta+alpha)Gamma(beta+k)}{Gamma(beta + m)Gamma(beta + alpha + k)},$$
where $k$ can be any non-negative integer and $alpha > 0$ is another parameter. If there exists a $sigma_k > 0$ (a positive constant that does not depend on $m$, but can depend on $k$), such that
$$(star) qquad qquad sum_{m=0}^{infty} frac{(is)^m}{m!} sigma_{m,k} frac{Gamma(beta + m)}{Gamma(beta)} = sum_{m=0}^{infty} frac{(is)^m}{m!} sigma_{k}^m frac{Gamma(beta + m)}{Gamma(beta)}$$
then I know that the left hand side is the characteristic function of a Gamma distributed random variable with shape parameter $beta$ and scale parameter $sigma_k$, since the right hand-side is.
My Question: is it possible to show that there exists a $sigma_k$ so that $(star)$ is true? Is there some form of intermediate value theorem that I can appeal to here? Note that the sequence $sigma_{0,k}, sigma_{1,k}, ...$ is a decreasing sequence.
characteristic-functions gamma-distribution
characteristic-functions gamma-distribution
asked Dec 22 '18 at 7:50
SEJSEJ
184
asked Dec 22 '18 at 7:50
SEJSEJ
184
edited Dec 22 '18 at 13:22
SEJ
asked Dec 22 '18 at 7:50
SEJSEJ
184
asked Dec 22 '18 at 7:50
SEJSEJ
184
asked Dec 22 '18 at 7:50
SEJSEJ
184
184
add a comment |
add a comment |
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