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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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