A mixture representation of the Linnik distribution
Ostrovskii, I. V.
Statistics and Probability Letters
61 - 64
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Linnik distribution with the characteristic function φα(t) = 1/(1 + |t|α), 0 < α < 2, is shown to possess the following property. Let Xα,Xβ be random variables possessing the Linnik distribution with parameters α and β respectively (0 < α < β ≤ 2). Denote by Yαβ an independent of Xβ non-negative random variable with the density (formula presented) Then Xα =̇ Xβ Yαβ, where =̇ denotes the equality in the sense of distributions. Infinite divisibility of mixtures of Linnik distributions with respect to the parameter α and scale is obtained as a corollary.