Re: Does there exist POSITIVE stable distributions with "alpha=1" (Feller's notations)
- From: "Zogman" <alex-math@xxxxxxx>
- Date: 4 Jun 2006 03:57:15 -0700
Herman Rubin писал(а):
In article <1149078133.839270.234310@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Zogman <alex-math@xxxxxxx> wrote:
Does there exist POSITIVE stable distributions with "alpha=1"
(Feller's notations)
i.e.
x_1+...+x_n=x+g_n,
where g_n =g n log(n), where g - constant.
Where can i find the information about the case "alpha=1", which
is exceptional ?
Many Thanks in advance,
PS
Cauchy distribution is of this type, but it is not POsitive.
I follow the notations of Feller vol.2 chapter 6.
An infinitely divisible distribution is a limit of
shifted compound Poisson distributions plus a normal
term plus a "bias" term; in terms of the logarithm of
the characteristic function,
f^(t) = bit - vt^2/2 + \int [(exp(itx) - 1)dL(x) - itdB(x)],
where L is the Levy measure, and B is any measure for which
the integral exists; changing B changes b. For the measure
to be in the positive reals, one must have a representation
of the special form b >= 0, v = 0, L has mass only on the
positive reals, and B=0. For a non-normal stable law, v = 0
and dL(x) = h(sgn(x))dx/x^(alpha+1), so a positive stable law
can only exist if h(-1) = 0 and it is possible that B = 0.
The integral then can only converge at 0 if alpha < 1.
So, am i understand right that positive distributions
exists iff
alpha <1 ?
.
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