MY MATEX NOTES

          In a previous post we showed how useful Fourier Transform may be in computing convolution of symmetric R.V.s. The Laplace tranform plays an analogous role for the convolution of one sided non-negative R.V.s. In this post We will show it´s usefulness in handling with the convolution of Levy-Smirnov R.V.s., an alpha stable distribution. Laplace transform: Recall the Laplace transform pair (1) (2) Now recall  from the previous post the convolution of two non-negative R.V.s If We take the it´s Laplace transform We obtain If and are i.i.d we obtain Similarly for the sum of three R.V.s Taking it´s Laplace transform Keeping on this process We may obtain (3) By equation (2) We Obtain (4) Levy-Smirnov distribution:      The Levy-Smirnov distribution is a continuous probability distribution for a non-negative random variable. It belongs to the family of alpha-stable distributions. Like all stable distributions ...

Today we will prove the following two amazing integrals (1) (2) In order to prove (1), first recall the following results: (3) (4) Then, (5) If we let and in (5), we get (6) Where We used the result We can rewrite (6) as (7) If we now integrate (7) w.r. to z we have: The evaluation of the constant is a beautiful exercise per se. Fortunately relying on the previous estabilished Vardi´s integral We may accomplish it easily. Setting in the last equation, the L.H.S. becomes Where we used the Vardi´s integral proved here : And The R.H.S. becomes Equating L.H.S. and R.H.S. we conclude that And finally (8) Or (9) Appendix Recall Legendre Duplication Formula for the Gamma Function Letting