Prove that $phi(t)= frac{1}{2} + frac{e^{t^2}}{4e^{t^2} - 2}$ defines characteristic function
2
$begingroup$
Let $phi : mathbb{R} rightarrow mathbb{R}$ be given by
$$phi(t) = frac{1}{2} + frac{e^{t^2}}{4e^{t^2} - 2}$$
Prove that $phi$ is a characteristic function.
My attempt:
I know that there are certain criteria that say that a function is a characteristic function of distribution, eg the Bochner criterion, but I do not know how to prove that a function is positively defined. I also have the Polya criterion, but here the condition that $phi (infty) = 0$ is not satisfied. Could You give me some hints?
probability-theory characteristic-functions
edited Dec 6 '18 at 15:26
Did
247k23223459
asked Dec 6 '18 at 15:13
Marcinek665Marcinek665
265110
$endgroup$
add a comment |
2
$begingroup$
Let $phi : mathbb{R} rightarrow mathbb{R}$ be given by
$$phi(t) = frac{1}{2} + frac{e^{t^2}}{4e^{t^2} - 2}$$
Prove that $phi$ is a characteristic function.
My attempt:
I know that there are certain criteria that say that a function is a characteristic function of distribution, eg the Bochner criterion, but I do not know how to prove that a function is positively defined. I also have the Polya criterion, but here the condition that $phi (infty) = 0$ is not satisfied. Could You give me some hints?
probability-theory characteristic-functions
edited Dec 6 '18 at 15:26
Did
247k23223459
asked Dec 6 '18 at 15:13
Marcinek665Marcinek665
265110
$endgroup$
4
$begingroup$
Low-tech approach: Expanding the geometric series, one gets $$varphi(t)=frac12+frac14frac1{1-frac12e^{-t^2}}=frac34+sum_{ngeqslant1}frac1{2^{n+2}}e^{-nt^2}$$ Now, use (and/or prove) the result that if every $varphi_n$ is a characteristic function and if $(p_n)$ are nonnegative with $sum p_n=1$ then $$varphi=sum_{ngeqslant0}p_nvarphi_n$$ is a characteristic function as well.
$endgroup$
– Did
Dec 6 '18 at 15:24
$begingroup$
@Did That approach needs slightly more care in this case because there are infinitely many of the $varphi_n$, but it works.
$endgroup$
– J.G.
Dec 6 '18 at 15:25
1
$begingroup$
@J.G. I do not know what you mean by "slightly more care" ("more" than what?) but indeed the result I mentioned holds, for infinitely many nonzero coefficients $p_n$ just like it holds for finitely many.
$endgroup$
– Did
Dec 6 '18 at 15:28
1
$begingroup$
@Did It's a trivial result when we take a linear combination of any finite number of cfs by induction, but the extension to infinitely many requires discussion of limits. In this case I'd take the pointwise limit of a sequence of pdfs from Fourier inversion, then Fourier-transform back to $varphi$.
$endgroup$
– J.G.
Dec 6 '18 at 15:35
1
$begingroup$
"Discussion of limits" Not necessarily, if one builds explicitely a random variable whose CF is $sumlimits_{n=0}^infty p_nvarphi_n$.
$endgroup$
– Did
Dec 6 '18 at 16:06
add a comment |
2
2
2
0
$begingroup$
Let $phi : mathbb{R} rightarrow mathbb{R}$ be given by
$$phi(t) = frac{1}{2} + frac{e^{t^2}}{4e^{t^2} - 2}$$
Prove that $phi$ is a characteristic function.
My attempt:
I know that there are certain criteria that say that a function is a characteristic function of distribution, eg the Bochner criterion, but I do not know how to prove that a function is positively defined. I also have the Polya criterion, but here the condition that $phi (infty) = 0$ is not satisfied. Could You give me some hints?
probability-theory characteristic-functions
edited Dec 6 '18 at 15:26
Did
247k23223459
asked Dec 6 '18 at 15:13
Marcinek665Marcinek665
265110
$endgroup$
Let $phi : mathbb{R} rightarrow mathbb{R}$ be given by
$$phi(t) = frac{1}{2} + frac{e^{t^2}}{4e^{t^2} - 2}$$
Prove that $phi$ is a characteristic function.
My attempt:
I know that there are certain criteria that say that a function is a characteristic function of distribution, eg the Bochner criterion, but I do not know how to prove that a function is positively defined. I also have the Polya criterion, but here the condition that $phi (infty) = 0$ is not satisfied. Could You give me some hints?
probability-theory characteristic-functions
probability-theory characteristic-functions
edited Dec 6 '18 at 15:26
Did
247k23223459
asked Dec 6 '18 at 15:13
Marcinek665Marcinek665
265110
edited Dec 6 '18 at 15:26
Did
247k23223459
asked Dec 6 '18 at 15:13
Marcinek665Marcinek665
265110
edited Dec 6 '18 at 15:26
Did
247k23223459
edited Dec 6 '18 at 15:26
Did
247k23223459
edited Dec 6 '18 at 15:26
Did
247k23223459
247k23223459
asked Dec 6 '18 at 15:13
Marcinek665Marcinek665
265110
asked Dec 6 '18 at 15:13
Marcinek665Marcinek665
265110
asked Dec 6 '18 at 15:13
Marcinek665Marcinek665
265110
265110
4
$begingroup$
Low-tech approach: Expanding the geometric series, one gets $$varphi(t)=frac12+frac14frac1{1-frac12e^{-t^2}}=frac34+sum_{ngeqslant1}frac1{2^{n+2}}e^{-nt^2}$$ Now, use (and/or prove) the result that if every $varphi_n$ is a characteristic function and if $(p_n)$ are nonnegative with $sum p_n=1$ then $$varphi=sum_{ngeqslant0}p_nvarphi_n$$ is a characteristic function as well.
$endgroup$
– Did
Dec 6 '18 at 15:24
$begingroup$
@Did That approach needs slightly more care in this case because there are infinitely many of the $varphi_n$, but it works.
$endgroup$
– J.G.
Dec 6 '18 at 15:25
1
$begingroup$
@J.G. I do not know what you mean by "slightly more care" ("more" than what?) but indeed the result I mentioned holds, for infinitely many nonzero coefficients $p_n$ just like it holds for finitely many.
$endgroup$
– Did
Dec 6 '18 at 15:28
1
$begingroup$
@Did It's a trivial result when we take a linear combination of any finite number of cfs by induction, but the extension to infinitely many requires discussion of limits. In this case I'd take the pointwise limit of a sequence of pdfs from Fourier inversion, then Fourier-transform back to $varphi$.
$endgroup$
– J.G.
Dec 6 '18 at 15:35
1
$begingroup$
"Discussion of limits" Not necessarily, if one builds explicitely a random variable whose CF is $sumlimits_{n=0}^infty p_nvarphi_n$.
$endgroup$
– Did
Dec 6 '18 at 16:06
add a comment |
4
$begingroup$
Low-tech approach: Expanding the geometric series, one gets $$varphi(t)=frac12+frac14frac1{1-frac12e^{-t^2}}=frac34+sum_{ngeqslant1}frac1{2^{n+2}}e^{-nt^2}$$ Now, use (and/or prove) the result that if every $varphi_n$ is a characteristic function and if $(p_n)$ are nonnegative with $sum p_n=1$ then $$varphi=sum_{ngeqslant0}p_nvarphi_n$$ is a characteristic function as well.
$endgroup$
– Did
Dec 6 '18 at 15:24
$begingroup$
@Did That approach needs slightly more care in this case because there are infinitely many of the $varphi_n$, but it works.
$endgroup$
– J.G.
Dec 6 '18 at 15:25
1
$begingroup$
@J.G. I do not know what you mean by "slightly more care" ("more" than what?) but indeed the result I mentioned holds, for infinitely many nonzero coefficients $p_n$ just like it holds for finitely many.
$endgroup$
– Did
Dec 6 '18 at 15:28
1
$begingroup$
@Did It's a trivial result when we take a linear combination of any finite number of cfs by induction, but the extension to infinitely many requires discussion of limits. In this case I'd take the pointwise limit of a sequence of pdfs from Fourier inversion, then Fourier-transform back to $varphi$.
$endgroup$
– J.G.
Dec 6 '18 at 15:35
1
$begingroup$
"Discussion of limits" Not necessarily, if one builds explicitely a random variable whose CF is $sumlimits_{n=0}^infty p_nvarphi_n$.
$endgroup$
– Did
Dec 6 '18 at 16:06
4
4
$begingroup$
Low-tech approach: Expanding the geometric series, one gets $$varphi(t)=frac12+frac14frac1{1-frac12e^{-t^2}}=frac34+sum_{ngeqslant1}frac1{2^{n+2}}e^{-nt^2}$$ Now, use (and/or prove) the result that if every $varphi_n$ is a characteristic function and if $(p_n)$ are nonnegative with $sum p_n=1$ then $$varphi=sum_{ngeqslant0}p_nvarphi_n$$ is a characteristic function as well.
$endgroup$
– Did
Dec 6 '18 at 15:24
$begingroup$
Low-tech approach: Expanding the geometric series, one gets $$varphi(t)=frac12+frac14frac1{1-frac12e^{-t^2}}=frac34+sum_{ngeqslant1}frac1{2^{n+2}}e^{-nt^2}$$ Now, use (and/or prove) the result that if every $varphi_n$ is a characteristic function and if $(p_n)$ are nonnegative with $sum p_n=1$ then $$varphi=sum_{ngeqslant0}p_nvarphi_n$$ is a characteristic function as well.
$endgroup$
– Did
Dec 6 '18 at 15:24
$begingroup$
@Did That approach needs slightly more care in this case because there are infinitely many of the $varphi_n$, but it works.
$endgroup$
– J.G.
Dec 6 '18 at 15:25
$begingroup$
@Did That approach needs slightly more care in this case because there are infinitely many of the $varphi_n$, but it works.
$endgroup$
– J.G.
Dec 6 '18 at 15:25
1
1
$begingroup$
@J.G. I do not know what you mean by "slightly more care" ("more" than what?) but indeed the result I mentioned holds, for infinitely many nonzero coefficients $p_n$ just like it holds for finitely many.
$endgroup$
– Did
Dec 6 '18 at 15:28
$begingroup$
@J.G. I do not know what you mean by "slightly more care" ("more" than what?) but indeed the result I mentioned holds, for infinitely many nonzero coefficients $p_n$ just like it holds for finitely many.
$endgroup$
– Did
Dec 6 '18 at 15:28
1
1
$begingroup$
@Did It's a trivial result when we take a linear combination of any finite number of cfs by induction, but the extension to infinitely many requires discussion of limits. In this case I'd take the pointwise limit of a sequence of pdfs from Fourier inversion, then Fourier-transform back to $varphi$.
$endgroup$
– J.G.
Dec 6 '18 at 15:35
$begingroup$
@Did It's a trivial result when we take a linear combination of any finite number of cfs by induction, but the extension to infinitely many requires discussion of limits. In this case I'd take the pointwise limit of a sequence of pdfs from Fourier inversion, then Fourier-transform back to $varphi$.
$endgroup$
– J.G.
Dec 6 '18 at 15:35
1
1
$begingroup$
"Discussion of limits" Not necessarily, if one builds explicitely a random variable whose CF is $sumlimits_{n=0}^infty p_nvarphi_n$.
$endgroup$
– Did
Dec 6 '18 at 16:06
$begingroup$
"Discussion of limits" Not necessarily, if one builds explicitely a random variable whose CF is $sumlimits_{n=0}^infty p_nvarphi_n$.
$endgroup$
– Did
Dec 6 '18 at 16:06
add a comment |
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4
$begingroup$
Low-tech approach: Expanding the geometric series, one gets $$varphi(t)=frac12+frac14frac1{1-frac12e^{-t^2}}=frac34+sum_{ngeqslant1}frac1{2^{n+2}}e^{-nt^2}$$ Now, use (and/or prove) the result that if every $varphi_n$ is a characteristic function and if $(p_n)$ are nonnegative with $sum p_n=1$ then $$varphi=sum_{ngeqslant0}p_nvarphi_n$$ is a characteristic function as well.
$endgroup$
– Did
Dec 6 '18 at 15:24
$begingroup$
@Did That approach needs slightly more care in this case because there are infinitely many of the $varphi_n$, but it works.
$endgroup$
– J.G.
Dec 6 '18 at 15:25
1
$begingroup$
@J.G. I do not know what you mean by "slightly more care" ("more" than what?) but indeed the result I mentioned holds, for infinitely many nonzero coefficients $p_n$ just like it holds for finitely many.
$endgroup$
– Did
Dec 6 '18 at 15:28
1
$begingroup$
@Did It's a trivial result when we take a linear combination of any finite number of cfs by induction, but the extension to infinitely many requires discussion of limits. In this case I'd take the pointwise limit of a sequence of pdfs from Fourier inversion, then Fourier-transform back to $varphi$.
$endgroup$
– J.G.
Dec 6 '18 at 15:35
1
$begingroup$
"Discussion of limits" Not necessarily, if one builds explicitely a random variable whose CF is $sumlimits_{n=0}^infty p_nvarphi_n$.
$endgroup$
– Did
Dec 6 '18 at 16:06