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Alan
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### Looking for a name if there is one

Let $p(x)$ be a probability density function (pdf) for a random variable with support on $R = (-\infty,\infty)$. Then $F(x) = \int_{-\infty}^x p(y) \, dy$ is the (cumulative) distribution function, and $\phi(z) = \int_R e^{i z x} p(x) \, dx$ is the characteristic function.

But, does anybody know a standard name for $G(a,b; z) \equiv \int_a^b e^{i z x} p(x) \, dx$? Basically, it is the "Fourier transform of the pdf restricted to an interval", which is a mouthful. If not, I am thinking of calling it simply a "restricted characteristic function", but would welcome other naming suggestions if I indeed have to make up a name.

Cuchulainn
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### Re: Looking for a name if there is one

Truncated CF?
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FaridMoussaoui
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### Re: Looking for a name if there is one

Truncated is used for conditonal distributions (for example in operational risk).

Cuchulainn
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### Re: Looking for a name if there is one

Modified CF = p X Heaviside function?
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Alan
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### Re: Looking for a name if there is one

Truncated CF?
Excellent! You just reminded me of something I embarrassingly forgot. I used this construction before in my last book (pg. 481). Got it from J. H. B. Kemperman
"A Wiener-Hopf type method for a general random walk with a two-sided boundary". (Ann. Math. Statistics, 1963).  I called it, slightly adapting from him, the "truncation-by-A" operation on transforms of measures, where to use his (and my) notation:

$[\hat{\mu}]^A = [\hat{\mu}]^A(z) \equiv \int_A e^{i z x} \, \mu(dx)$

So, with 'A' the interval $(a,b)$, that's it.

A senior moment -- how soon we forget our own stuff!

Thanks, guys.
Last edited by Alan on February 7th, 2019, 9:36 pm, edited 3 times in total.

ppauper
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### Re: Looking for a name if there is one

it looks a little like the formula for the fourier coefficients in a fourier sum

Alan
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### Re: Looking for a name if there is one

Yes, a little. But typically with Fourier coefs, finite "A" would be the whole space (or a period of a periodic function), and the 'z' would be restricted to discrete values. In my case, A is an arbitrary subset of the whole line and the legal 'z' are chosen from a continuous set: either the whole real line or a complex-valued strip.

oislah
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### Re: Looking for a name if there is one

by change of variable (of tangent type), this looks like a particular instance of  a Fourier Integral Operator : https://en.wikipedia.org/wiki/Fourier_integral_operator
doesn't it?

Cuchulainn
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### Re: Looking for a name if there is one

by change of variable (of tangent type), this looks like a particular instance of  a Fourier Integral Operator : https://en.wikipedia.org/wiki/Fourier_integral_operator
doesn't it?
More precisely,

CHF is the Fourier transform of the probability density function (AFAIR Laplace proved this). Nice to know of course.

But I think the central issue (Alan's book page 481) is that the interval is now of compact support and not infinite.

edit: this Borel measure is really the restriction of a Borel measure to a Borel set A.

is the Fourier transform of the probability density function
Last edited by Cuchulainn on February 8th, 2019, 2:19 pm, edited 1 time in total.
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FaridMoussaoui
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### Re: Looking for a name if there is one

CHF is our Swiss Franc. CF is the the characteristic function.

Cuchulainn
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### Re: Looking for a name if there is one

CHF is our Swiss Franc. CF is the the characteristic function.
I'll give you 2 CFS for your CHF.
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Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
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FaridMoussaoui
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### Re: Looking for a name if there is one

sorry for my ignorance. what is CFS?

Cuchulainn
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### Re: Looking for a name if there is one

sorry for my ignorance. what is CFS?
It's the plural of "CF".
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Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
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oislah
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### Re: Looking for a name if there is one

Cuchulainn
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### Re: Looking for a name if there is one

This assumes that the function is compactly supported, yes? Which is too strong because in the current case probability density function only decays going to infinity.
I had a quick look at Kemperman and that addresses the problem imo.
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