The Calculation of the Density and Distribution Functions of Strictly Stable Laws
Abstract
:1. Introduction
2. Auxiliary Results
- 1.
- , , , the contour has the form and ;
- 2.
- , , , the contour has the form and ;
- 3.
- , the contour has the form and ;
- 4.
- , , the contour has the form and ;
- 5.
- , , , the contour has the form and ;
- 6.
- , , , the contour has the form and ;
- 7.
- , , , the contour has the form and ;
- 8.
- , , , the contour has the form and ;
- 9.
- , , , the contour has the form and .Here, and
- 1.
- Every contour starts in the point .
- 2.
- None of the contours Γ intersects the lines of the cut.
- 3.
- Moving from the point along the contour Γ we let it tend to infinity but in such a way that starting from some place all points have values of arguments within the limits:
- 1.
- if , then
- 2.
- if , then
3. Main Results
- 1.
- If and for any values
- 2.
- If , then for any and
- 3.
- If , then for any and any values x
- 1.
- 2.
- If , then for any and any x
- 3.
- If , then for any admissible α and θ
4. The Calculation of the Density and Distribution Function of a Stable Law
5. Conclusions
Funding
Acknowledgments
Conflicts of Interest
References
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Saenko, V. The Calculation of the Density and Distribution Functions of Strictly Stable Laws. Mathematics 2020, 8, 775. https://doi.org/10.3390/math8050775
Saenko V. The Calculation of the Density and Distribution Functions of Strictly Stable Laws. Mathematics. 2020; 8(5):775. https://doi.org/10.3390/math8050775
Chicago/Turabian StyleSaenko, Viacheslav. 2020. "The Calculation of the Density and Distribution Functions of Strictly Stable Laws" Mathematics 8, no. 5: 775. https://doi.org/10.3390/math8050775