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From: OsherD on 26 Apr 2010 13:09
From Osher Doctorow

Look at:

1) "Levy distribution", Wikipedia (online).
2) "Stable distribution", Wikipedia (online).
3) "Landau distribution", Wikipedia (online).
4) "Dirac delta function," Wikipedia (online).

It is well known that:

5) The variance of Stable distributions is infinity except when a
parameter alpha is 2. The Gaussian/normal distribution has alpha = 2,
but the Levy distribution has alpha = 1/2 and another parameter beta =
1, while the Cauchy distribution has alpha = 1 and beta = 0, and the
Landau distribution has alpha = beta = 1.

6) Stable distributions do not have analytically representable
probability density functions (pdfs) but they have characteristic
functions E(exp(itX)) = phi given by:

7) phi(t; u, c, alpha, beta) = exp[f - g(1 - h)). f = itu, f = |ct|
^alpha, h = i beta sgn(t) PHI
where PHI = tan(pi alpha/2) except for alpha = 1 PHI = (-2/pi)log|t|,
beta is in [-1, 1], u real.

We also have:

8) The mean of Stable distributions is undefined when alpha < = 1 (for
example, for the Cauchy, Levy, Landau distributions).

These are all inconsistent with the Heisenberg Uncertainty Principle,
as is the 0 moment property (see the above papers). Note that the
Dirac Delta function is not technically a probability density function
or cumulative distribution function but either a measure or a
"generalized distribution", but the terminology is correct as a
limiting case.

Physical applications are numerous, and I will try to discuss them
later.

Osher Doctorow


From: OsherD on 26 Apr 2010 13:18
From Osher Doctorow

1) The Levy distribution is the length of the path followed by a
photon in a turbid medium, the statistics of solar flares, as well as
the time of hitting a single point not equal to the starting point 0
by Brownian motion, among other things.

2) The Landau distribution describes fluctuations of energy loss of
charged particles passing through thin layers of matter, etc.

Osher Doctorow
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