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By Brewer M. J.

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4. 1. f is a Bernstein function if and only if the mapping x → e−tf (x) is completely monotone for all t ≥ 0. 2. f is a Bernstein function if and only if it has the representation ∞ f (x) = a + bx + (1 − e−yx )λ(dy), 0 ∞ for all x > 0 where a, b ≥ 0 and 0 (y ∧ 1)λ(dy) < ∞. 3. g is completely monotone if and only if there exists a measure µ on [0, ∞) for which ∞ g(x) = e−xy µ(dy). 61-72. To interpret this theorem, first consider the case a = 0. 5), we see that there is a one to one correspondence between Bernstein functions for which limx→0 f (x) = 0 and Laplace exponents of subordinators.

Xd ) ∈ Rd , then xα = xα 1 · · · xd . Now we define Schwartz space S (Rd , C) to be the linear space of all f ∈ C ∞ (Rd , C) for which sup |xβ Dα f (x)| < ∞, x∈Rd for all multi-indices α and β. Note that Cc∞ (Rd , C) ⊂ S (Rd , C) and the 2 “Gaussian function” x → e−|x| is in S (Rd , C). S (Rd , C) is dense in C0 (Rd , C) p d and in L (R , C) for all 1 ≤ p < ∞. These statements remain true when C is replaced by R. ||N , N ∈ N ∪ {0}} where for each f ∈ S (Rd , C), ||f ||N = max sup (1 + |x|2 )N |Dα f (x)|.

The relatively slow decay of the 14 David Applebaum tails for non-Gaussian stable laws makes them ideally suited for modelling a wide range of interesting phenomena, some of which exhibit “long-range dependence”. The generalisation of stability to random vectors is straightforward - just replace X1 , . . 7 extends directly. 7, the L´evy measure takes the form c ν(dx) = dx d+α |x| where c > 0. g. 83. We can generalise the definition of stable random variables if we weaken the conditions on the random variables (Y (n), n ∈ N) in the general central limit problem by requiring these to be independent, but no longer necessarily identically distributed.

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