Why Lévy α-stable distributions lack general closed-form expressions for arbitrary α

Area of Science:

• Probability Theory
• Complex Analysis
• Mathematical Physics

Background:

• Lévy α-stable distributions are fundamental in probability theory but lack general closed-form expressions.
• Understanding their analytic properties has been a long-standing challenge.

Purpose of the Study:

• To investigate the complex analytic continuation of symmetric Lévy α-stable distributions.
• To clarify the conditions under which these distributions exhibit analytic properties.

Main Methods:

• Complex analytic continuation of Lévy α-stable distributions.
• Holomorphic and meromorphic function analysis.
• Domain coloring visualization techniques.

Main Results:

• Proved that Lévy α-stable distributions are holomorphic on the complex plane for 1<α≤2.
• Demonstrated that these distributions are not meromorphic for 0<α<1.
• Visualized the complete complex analytic structure of these distributions.

Conclusions:

• The study confirms the absence of general closed-form expressions in elementary functions for Lévy α-stable distributions.
• The findings provide deeper insights into the analytic nature of these distributions across the complex plane.