User:A.M.R./Stability (probability)

If one has a number of random variates that are "in the family" and any linear combination of these variates are also "in the family", then the the family is said to be stable.

Given a particular distribution, we can define a family of such distributions all related by a shift and scale. Suppose we pick a probability density function $$f(x)$$ as the "standard" distribution. Then -


 * $$f(x;\mu,c) = f\left(\frac{x-\mu}{c}\right)\,$$

a member of the family, is just $$f(x)$$ multiplied by the scale factor $$c$$ and shifted by the shift factor $$\mu$$.


 * $$X \sim \textrm{Fam}(\mu,c)$$

means that the distribution function for $$X$$ is a member of the family "Fam".

Suppose,
 * $$X_1 \sim \textrm{Fam}(\mu_1,c_1)$$
 * $$X_2 \sim \textrm{Fam}(\mu_2,c_2)$$

and,
 * $$Y=aX_1+bX_2.\,$$

The distribution family of $$X_1$$ and $$X_2$$ is said to be stable if the distribution function for $$Y$$ is a member of that same family to within a constant. In other words there exists some $$\mu$$ and $$c$$ such that


 * $$Y \sim \textrm{Fam}(\mu,c)+K$$

where $$K$$ is a constant. If the constant $$K$$ is always zero for any values of $$\mu_1$$, $$\mu_2$$, $$c_1$$ and $$c_2$$, then the distribution family is said to be strictly stable.