Re: why an arbitrary shaped waveform when convoleved

On Mar 13, 11:28 pm, "bharat pathak" <bha...@xxxxxxxxxxx> wrote:
Hello,

why an arbitrary shaped waveform when convolved with itself
couple of times results in a gaussian looking pulse? what is
the significance of such phenomena? Also can this be classified
under the realms of central limit theorem?

yes, it *is* what the central limit theorem is built upon. but there
are caveats, such as the Cauchy r.v.

p(x) = (1/pi)/(1 + x^2)

it looks like a Gaussian bell, but it clearly is not if you examine
the "tails" and how fast they go to zero.

you can convolve this with itself until it's skin is raw and bleeding
and it won't become any more gaussian.

even though it's a legit random variable (the integral of the curve is
1), this r.v. has no mean (unless you hand wave a limits of
integration argument like we do for evaluating the Hilbert transform -
1/x functions don't like to integrate to infinity) and an infinite
variance. i think if the r.v. has a finite variance (and then i think
it would have to have a well defined and finite mean), *then* if you
convolve it with itself repeatedly (and normalize the scaling), it
becomes more gaussian.

r b-j
.

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