Extreme value distributions of mixing two sequences with different MDA's

Area of Science:

• Probability Theory
• Extreme Value Theory
• Stochastic Processes

Background:

• Independent random variables Xi and Yi follow distribution functions FX(x) and FY(x).
• A combined sequence Z(i,n) is formed by mixing Xi and Yi with probability pn.
• The study focuses on the extreme value distribution of this triangular array of random variables.

Purpose of the Study:

• To analyze the extreme value distribution GZ(x) of the mixed random variables Z(i,n).
• To identify new forms of extreme value distributions arising from such mixtures.
• To investigate the properties of these new distributions, particularly max-stability.

Main Methods:

• Analysis of a triangular array of independent and identically distributed (i.i.d.) random variables Z(i,n).
• Derivation of the extreme value distribution GZ(x) for the combined sequences.
• Investigation of conditions under which FX(x) and FY(x) belong to the same Maximum Domain of Attraction (MDA).

Main Results:

• A novel extreme value distribution form, LambdaA(Rhox)Lambda(x) (0 < Rho < 1), was discovered.
• This new distribution is shown to be not max-stable.
• The existence of GZ(x) as mixture forms of different extreme value distribution types is precluded.

Conclusions:

• The study reveals a new non-max-stable extreme value distribution under specific conditions.
• The findings clarify the behavior of extreme values in mixed independent random sequences.
• The results contribute to a deeper understanding of extreme value theory for complex random structures.