The area of a spectrally positive stable process stopped at zero

A multiplicative identity in law for the area of a spectrally positive Lévy ∝-stable process stopped at zero is established. Extending that of Lefebvre for Brownian motion, it involves an inverse beta random variable and the square of a positive stable random variable. This simple identity makes it possible to study precisely the behaviour of the density at zero, which is Fréchet-like.

Some further results in infinite divisibility

Goldie (2), Steutel (8, 9), Kelker (4), Keilson and Steutel (3) and several others have studied the mixtures of certain distributions which are infinitely divisible. Recently Shanbhag and Sreehari (7) have proved that if Z is exponential with unit parameter and for 0 < α < 1, if Yx is a positive stable random variable with , t ≥ 0 and independent of Z, then for every 0 < α < 1Using this result, they have obtained several interesting results concerning stable random variables including some extensions of the results of the above authors. More recently, Williams (11) has used the same approach to show that if , where n is a positive integer ≥ 2, then is distributed as the product of n − 1 independent gamma random variables with index parameters α, 2α, …, (n − 1) α. Prior to these investigations, Zolotarev (12) had studied the problems of M-divisibility of stable laws.