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Probability measures, Lévy measures and analyticity in time. (English) Zbl 1162.60013

Let \(U\) be the Lévy measure of an infinitely divisible law in \(\mathbb{R}^d\) with the associated Lévy process \(\{X_t, t\geq0\}\) and let \(P\{dx;t\}\) denote the law of \(X_t\). This paper deals with the problem of calculating \(P\{dx;t\}\) from \(U\). When \(d=1,\) and both \(P\{dx;t\}\) and \(U(dx)\) are concentrated on \(R_{>0}=(0,\infty),\) it is convenient to give the form \[ P\left((x, \infty), t \right)=\sum^{\infty}_{n=1}\frac{t^n}{n!}U_n\left((x,\infty)\right) \] where \(U(dx)\) are, in general, signed measures and \(U_1=U.\) If \(P\{dx;t\}\) and \(U(x)\) are absolutely continuous with densities \(p(x,t)\) and \(u(x),\) \[ p\left(x, t \right)=\sum^{\infty}_{n=1}\frac{1}{n!}u_n(x). \] The question is how the coefficients \(u_n\) may be calculated from \(u.\) Three methods are provided. The first one is based on the approximation of compound Poisson distributions involving, as the final step, a limiting operation \[ u_n(x) =\sum^n_{n=1}(-1)^{n-k}\binom n k {c}(\varepsilon)^{n-k}u_{\varepsilon}^{*k}(x). \] Here \(u_{\varepsilon}\) (x) is an approximation of the Lévy density that corresponds to a compound Poisson process with intensity \(c(\varepsilon)\). The second one uses derivatives of convolution integrals of the upper tail integral of the Lévy measure and the third method applies the analytic continuation of the Lévy density to a complex cone and contour integration. Chapter 2 consists of a number of initial remarks. Namely, at first the motivating example: the positive \(\alpha\)- stable distribution is considered and all the above mentioned methods are applied. Section 2.2. contains
Theorem 1. If \(U_{0\varepsilon}(x)=1\), \(U_{n\varepsilon}(x)=-\int\limits^{\infty}_xu_{n\varepsilon}(y)\,dy\), \(n\geq 1\) and \[ \lim_{\varepsilon\rightarrow 0}\int (1\wedge x)| u_\varepsilon (x) -u(x)| dx =0, \] then \[ P(t;x) = \lim_{\varepsilon\rightarrow 0}\sum_{n\geq 0} U_{n\varepsilon}(x)\frac{t^n}{n!} \] pointwise for each \(x\in \mathbb{R}_{>0}\) and \(t>0\).
Further, formulas for \(u_{n\varepsilon}\) are calculated, and cancelation of singularities in the convergence of \(u_{n\varepsilon}\) to the function \(u_n\) is discussed. Section 2.5 contains miscellaneous further points being in touch with the problems or methods used in this paper. Chapter 3 contains the main results. The following issues are analyzed:
(i). Does \(u_{n\varepsilon}(x)\) converge, as \(\varepsilon\rightarrow 0?\) If so, a more direct method is found to compute \(\lim u_n(x)\) from \(U(x)\).
(ii) If we have convergence, is \(p(x,t)\) in fact n-times differentiable in \(t\geq 0\) and, if so, is \(u_n (x)\) the \(n\)-th derivative
(iii) If the answer in (ii) is positive for all \(n\geq 1,\) do we have a convergent Taylor expansion of \(p(x,t)\) at \(t=0?\) Is \(p(x, t)\) an entire function in \(t\in \mathbb{C}\)?
Chapter 4 deals with, as examples, with the positive \(\alpha\)- stable law, the gamma distribution, the inverse Gaussian distribution and on the whole \(\mathbb{R},\) the Meixner and the normal inverse Gaussian distribution. As a bivariate example, the inverse Gaussian-normal inverse Gaussian law is considered. The auxiliary lemmas are proved in the Appendix. The estimates for the cumulant function, convolutions and Laplace transforms, are obtained.
The volume of the paper is 25 pages, the list of references contains 28 positions.

MSC:

60G51 Processes with independent increments; Lévy processes
60A10 Probabilistic measure theory
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