Download PDF

stable random variable∗
CWoo†
2013-03-21 21:31:40
A real random variable X defined on a probability space (Ω, F, P ) is said to
be stable if
1. X is not trivial; that is, the range of the distribution function of X strictly
includes {0, 1}, and
2. given any positive integer n and X1 , . . . , Xn random variables, iid as X:
t
Sn := X1 + · · · + Xn = X.
In other words, there are real constants a, b such that Sn and aX + b have
the same distribution functions; Sn and X are of the same type.
Furthermore, X is strictly stable if X is stable and the b given above can
always be take as 0. In other words, X is strictly stable if Sn and X belong to
the same scale family.
A distribution function is said to be stable (strictly stable) if it is the distribution function of a stable (strictly stable) random variable.
Remarks.
• If X is stable, then aX + b is stable for any a, b ∈ R.
• If X and Y are independent, stable, and of the same type, then X + Y is
stable.
• X is stable iff for any independent X1 , X2 , identically distributed as X,
and any a, b ∈ R, there exist c, d ∈ R such that aX1 + bX2 and cX + d are
identically distributed.
• A stable distribution function is absolutely continuous and infinitely divisible.
Some common stable distribution functions are the normal distributions and
Cauchy distributions.
∗ hStableRandomVariablei
created: h2013-03-21i by: hCWooi version: h38584i Privacy
setting: h1i hDefinitioni h60E07i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1