Characteristic function and point masses
Let $X$ be some random variable with characteristic function $hat{X}$. Show that $X$ has no point masses if and only if
$$
lim_{Tto +infty}frac1{2T}int_{-T}^Tleftlvert hat{X}(t)rightrvert^2mathrm dt=0.
$$
I think I have to consider the probability $P(X − Y = 0)$ where $Y$ is independent of $X$ and $X=Y$ in distribution, but I don't know how proceed.
random-variables characteristic-functions
edited Dec 10 '18 at 11:13
Davide Giraudo
125k16150260
asked Nov 28 '18 at 17:54
Francesca BallatoreFrancesca Ballatore
186
add a comment |
Let $X$ be some random variable with characteristic function $hat{X}$. Show that $X$ has no point masses if and only if
$$
lim_{Tto +infty}frac1{2T}int_{-T}^Tleftlvert hat{X}(t)rightrvert^2mathrm dt=0.
$$
I think I have to consider the probability $P(X − Y = 0)$ where $Y$ is independent of $X$ and $X=Y$ in distribution, but I don't know how proceed.
random-variables characteristic-functions
edited Dec 10 '18 at 11:13
Davide Giraudo
125k16150260
asked Nov 28 '18 at 17:54
Francesca BallatoreFrancesca Ballatore
186
What is your definition of a point mass?
– Paul
Nov 28 '18 at 17:58
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. So I think that the point masses that I have to consider in the exercise are those that have the probability exactly equal to some value.
– Francesca Ballatore
Nov 28 '18 at 18:05
Did you try to work with the inversion formula for characteristic functions?
– Davide Giraudo
Dec 12 '18 at 10:16
add a comment |
Let $X$ be some random variable with characteristic function $hat{X}$. Show that $X$ has no point masses if and only if
$$
lim_{Tto +infty}frac1{2T}int_{-T}^Tleftlvert hat{X}(t)rightrvert^2mathrm dt=0.
$$
I think I have to consider the probability $P(X − Y = 0)$ where $Y$ is independent of $X$ and $X=Y$ in distribution, but I don't know how proceed.
random-variables characteristic-functions
edited Dec 10 '18 at 11:13
Davide Giraudo
125k16150260
asked Nov 28 '18 at 17:54
Francesca BallatoreFrancesca Ballatore
186
Let $X$ be some random variable with characteristic function $hat{X}$. Show that $X$ has no point masses if and only if
$$
lim_{Tto +infty}frac1{2T}int_{-T}^Tleftlvert hat{X}(t)rightrvert^2mathrm dt=0.
$$
I think I have to consider the probability $P(X − Y = 0)$ where $Y$ is independent of $X$ and $X=Y$ in distribution, but I don't know how proceed.
random-variables characteristic-functions
random-variables characteristic-functions
edited Dec 10 '18 at 11:13
Davide Giraudo
125k16150260
asked Nov 28 '18 at 17:54
Francesca BallatoreFrancesca Ballatore
186
edited Dec 10 '18 at 11:13
Davide Giraudo
125k16150260
asked Nov 28 '18 at 17:54
Francesca BallatoreFrancesca Ballatore
186
edited Dec 10 '18 at 11:13
Davide Giraudo
125k16150260
edited Dec 10 '18 at 11:13
Davide Giraudo
125k16150260
edited Dec 10 '18 at 11:13
Davide Giraudo
125k16150260
125k16150260
asked Nov 28 '18 at 17:54
Francesca BallatoreFrancesca Ballatore
186
asked Nov 28 '18 at 17:54
Francesca BallatoreFrancesca Ballatore
186
asked Nov 28 '18 at 17:54
Francesca BallatoreFrancesca Ballatore
186
186
What is your definition of a point mass?
– Paul
Nov 28 '18 at 17:58
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. So I think that the point masses that I have to consider in the exercise are those that have the probability exactly equal to some value.
– Francesca Ballatore
Nov 28 '18 at 18:05
Did you try to work with the inversion formula for characteristic functions?
– Davide Giraudo
Dec 12 '18 at 10:16
add a comment |
What is your definition of a point mass?
– Paul
Nov 28 '18 at 17:58
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. So I think that the point masses that I have to consider in the exercise are those that have the probability exactly equal to some value.
– Francesca Ballatore
Nov 28 '18 at 18:05
Did you try to work with the inversion formula for characteristic functions?
– Davide Giraudo
Dec 12 '18 at 10:16
What is your definition of a point mass?
– Paul
Nov 28 '18 at 17:58
What is your definition of a point mass?
– Paul
Nov 28 '18 at 17:58
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. So I think that the point masses that I have to consider in the exercise are those that have the probability exactly equal to some value.
– Francesca Ballatore
Nov 28 '18 at 18:05
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. So I think that the point masses that I have to consider in the exercise are those that have the probability exactly equal to some value.
– Francesca Ballatore
Nov 28 '18 at 18:05
Did you try to work with the inversion formula for characteristic functions?
– Davide Giraudo
Dec 12 '18 at 10:16
Did you try to work with the inversion formula for characteristic functions?
– Davide Giraudo
Dec 12 '18 at 10:16
add a comment |
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What is your definition of a point mass?
– Paul
Nov 28 '18 at 17:58
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. So I think that the point masses that I have to consider in the exercise are those that have the probability exactly equal to some value.
– Francesca Ballatore
Nov 28 '18 at 18:05
Did you try to work with the inversion formula for characteristic functions?
– Davide Giraudo
Dec 12 '18 at 10:16