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A property of the generalized inverse Gaussian distribution with some applications. (English) Zbl 0536.60022
The author considers the generalized inverse Gaussian distribution (GIGD) called $$N^{-1}(\lambda,\kappa,\psi)$$, with probability density $$F'(x)=(\psi /\kappa)^{\lambda /2}(2K_{\lambda}\sqrt{\kappa\psi})^{-1}x^{\lambda -1}\cdot \exp(-(\kappa x^{-1}+\psi x)/2)$$, $$(x>0)$$, where $$K_{\lambda}(\cdot)$$ is the modified Bessel function of third kind, and parameters obey either $$\lambda>0,\kappa \geq 0,\psi>0$$, or $$\lambda =0,\kappa>0,\psi>0$$, or $$\lambda<0$$, $$\kappa>0$$, $$\psi\geq 0$$. Using the Laplace-Stieltjes transform of F: $$f(s)=\int^{\infty}_{0}e^{-sx}F'(x)dx=K_{\lambda}(\omega \cdot(1+2s/\psi)^{\frac{1}{2}})/(1+2s/\psi)^{\lambda /2}K_{\lambda}(\omega)$$ with $$\omega =\sqrt{\kappa \psi}$$, he proves that if F is $$N^{-1}(\lambda,\kappa,\psi)$$ with $$\lambda<0$$, $$\kappa>0$$, $$\psi\geq 0$$, then it belongs to the class $$S(\gamma)$$, $$\gamma\geq 0$$, of distribution functions F on $$[0,\infty [$$, introduced by J. Chover, P. Ney, and S. Wainger [Ann. Probab. 1, 663-673 (1973; Zbl 0387.60097)] and defined by following conditions: $$\lim_{x\to \infty}[1-F^{(2)}(x)]/[1-F(x)]=c<\infty$$, $$\lim_{x\to \infty}[1-F(x-y)]/[1-F(x)]=e^{\gamma \cdot y}$$ for all y real, and $$c/2=f(-\gamma)=\int^{\infty}_{0}e^{\gamma \cdot x}dF(x)<\infty$$, where $$F^{(2)}(x)$$ denotes the convolution of F with itself. This asymptotic convolution property of GIGD with $$\lambda<0$$ proves useful in estimating tails of distribution functions based on them.
The result is applied to collective risk theory by considering a compound Poisson process $$X(t)=\sum^{N(t)}_{k=1}A_ k$$ with $$\{$$ N(t)$$| t\geq 0\}$$ following a Poisson process with parameter $$\nu>0$$, and supposing the distribution of the independent claim sizes $$A_ 1$$, $$A_ 2,..$$. to be the same $$N^{-1}(\lambda,\kappa,\psi)$$ with $$\lambda<0$$; asymptotic estimates are thus derived for the probability of ruin $$1- R(x)=1-P\{X(t)\leq x+c\cdot t,\forall t\geq 0\}$$ with given constants $$x>0$$, $$c>0$$. Finally, the author discusses further applications of his main theorem to other stochastic models, as concerning subcritical age- dependent branching processes, renewal theory, random walk, lifetime distributions.
Reviewer: M.P.Geppert

##### MSC:
 60E05 Probability distributions: general theory 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 60K10 Applications of renewal theory (reliability, demand theory, etc.) 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60K05 Renewal theory
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