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Looking for a name if there is one

Posted: February 7th, 2019, 6:04 pm
by Alan
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.

Re: Looking for a name if there is one

Posted: February 7th, 2019, 8:23 pm
by Cuchulainn
Truncated CF?

Re: Looking for a name if there is one

Posted: February 7th, 2019, 8:31 pm
by FaridMoussaoui
Truncated is used for conditonal distributions (for example in operational risk).

Re: Looking for a name if there is one

Posted: February 7th, 2019, 9:04 pm
by Cuchulainn
Modified CF = p X Heaviside function?

Re: Looking for a name if there is one

Posted: February 7th, 2019, 9:18 pm
by Alan
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.

Re: Looking for a name if there is one

Posted: February 7th, 2019, 9:32 pm
by ppauper
it looks a little like the formula for the fourier coefficients in a fourier sum

Re: Looking for a name if there is one

Posted: February 7th, 2019, 9:46 pm
by Alan
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.  

Re: Looking for a name if there is one

Posted: February 8th, 2019, 9:52 am
by oislah
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?

Re: Looking for a name if there is one

Posted: February 8th, 2019, 1:08 pm
by Cuchulainn
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

Re: Looking for a name if there is one

Posted: February 8th, 2019, 2:08 pm
by FaridMoussaoui
CHF is our Swiss Franc. CF is the the characteristic function.

Re: Looking for a name if there is one

Posted: February 8th, 2019, 2:10 pm
by Cuchulainn
CHF is our Swiss Franc. CF is the the characteristic function.
I'll give you 2 CFS for your CHF.

Re: Looking for a name if there is one

Posted: February 8th, 2019, 2:15 pm
by FaridMoussaoui
sorry for my ignorance. what is CFS?

Re: Looking for a name if there is one

Posted: February 8th, 2019, 2:17 pm
by Cuchulainn
sorry for my ignorance. what is CFS?
It's the plural of "CF".

Re: Looking for a name if there is one

Posted: February 8th, 2019, 2:43 pm
by oislah

Re: Looking for a name if there is one

Posted: February 8th, 2019, 5:19 pm
by Cuchulainn
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.