Stability postulate

In probability theory, to obtain a nondegenerate limiting distribution for extremes of samples, it is necessary to "reduce" the actual greatest value by applying a linear transformation with coefficients that depend on the sample size.

If $X_1,X_2,\dots ,X_n$ are independent random variables with common probability density function $\mathbb{P}(X_j=x)\equiv f_X(x),$

then the cumulative distribution function $F_{Y_n}$ for $Y_n\equiv \max \{X_1,\dots ,X_n\}$ is given by the simple relation

$F_{Y_n}(y)={[F_X(y)]}^n.$

If there is a limiting distribution for the distribution of interest, the stability postulate states that the limiting distribution must be for some sequence of transformed or "reduced" values, such as $(a_n{Y}_n+b_n),$ where $a_n,b_n$ may depend on n but not on x. This equation was obtained by Maurice René Fréchet and also by Ronald Fisher.

Only three possible distributions

To distinguish the limiting cumulative distribution function from the "reduced" greatest value from $F(x),$ we will denote it by $G(y).$ It follows that $G(y)$ must satisfy the functional equation

${\left[G\left(y\right)\right]}^n=G(a_{n}y+b_n).$

Boris Vladimirovich Gnedenko has shown there are no other distributions satisfying the stability postulate other than the following three:

• Gumbel distribution for the minimum stability postulate
  • If $X_i=\text{Gumbel}(\mu ,\beta )$ and $Y\equiv \min \{X_1,\dots ,X_n\}$ then $Y\sim a_{n}X+b_n,$
    where $a_n=1$ and $b_n=\beta \log n;$
  • In other words, $Y\sim \mathsf{\text{𝖦𝗎𝗆𝖻𝖾𝗅}}(\mu -\beta \log n,\beta ).$

• Weibull distribution (extreme value) for the maximum stability postulate
  • If $X_i=\mathsf{\text{𝖶𝖾𝗂𝖻𝗎𝗅𝗅}}(\mu ,\sigma )$ and $Y\equiv \max \{X_1,\dots ,X_n\}$ then $Y\sim a_{n}X+b_n,$
    where $a_n=1$ and $b_n=\sigma \log \left(\frac{1}{n}\right);$
  • In other words, $Y\sim \mathsf{\text{𝖶𝖾𝗂𝖻𝗎𝗅𝗅}}(\mu -\sigma \log \left(\frac{1}{n}\right),\sigma ).$

• Fréchet distribution for the maximum stability postulate
  • If $X_i=\mathsf{\text{𝖥𝗋𝖾𝖼𝗁𝖾𝗍}}(\alpha ,s,m)$ and $Y\equiv \max \{X_1,\dots ,X_n\}$ then $Y\sim a_{n}X+b_n,$
    where $a_n=n^{-\frac{1}{\alpha }}$ and $b_n=m(1-n^{-\frac{1}{\alpha }});$
  • In other words, $Y\sim \mathsf{\text{𝖥𝗋𝖾𝖼𝗁𝖾𝗍}}(\alpha ,n^{\frac{1}{\alpha }}s,m).$