engleski

On the result of Doney

Let X denote a spectrally positive stable process of index $\alpha \in (1, 2)$ whose Lévy measure has density $c x^{; ; -\alpha-1}; ; , x > 0$ and let $S = sup_{; ; 0 \leq t \leq 1}; ; X_t$. Doney [4] proved that the density of S say s behaves as $s(x) \sim c x^{; ; -\alpha}; ; $ 􀀀 as $ \to \infty$. The proof given was nearly four pages long. Here, we: i) give a shorter and a more general proof of the same result ; ii) derive the first known closed form expressions for $s(x)$ and the corresponding cumulative distribution function ; iii) derive the order of the remainder in the asymptotic expansion for $s(x)$.

Asymptotic behavior; Stable process; Wright generalized hypergeometric Psi function

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