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Sur le passage de certaines marches aléatoires planes au-dessus d’une hyperbole équilatère. (On the crossing of certain planar random walks over an equilateral hyperbola). (French) Zbl 0701.60069
The generalized inverse Gaussian distribution $$\mu$$ ($$\lambda$$,a,b) has the density $2^{-1}(a/b)^{1/2}\exp (-2^{-1}(ax+bx^{- 1}))/K_{\lambda}((ab)^{1/2})\text{ on } (0,\infty),$ where $$K_{\lambda}$$ is the modified Bessel function of the third kind. From the Laplace-Stieltjes transform of $$\mu$$ ($$\lambda$$,a,b) given by O. Barndorff-Nielsen and C. Halgreen [Z. Wahrscheinlichkeitstheor. Verw. Geb. 38, 309-311 (1977; Zbl 0403.60026)] follows the convolution equation $(*)\quad \mu (\lambda,a,b)=\mu (-\lambda,a,b)*\gamma (\lambda,2/a),$ where $$\gamma$$ ($$\lambda$$,c) denotes the gamma distribution with parameter $$\lambda$$ and expectation $$\lambda$$ c. Let X(t), Y(t), $$t\geq 0$$, be two independent processes with independent increments where $$X(0)=Y(0)=0$$ and $$X(t+\tau)-X(t)$$ and $$Y(t+\tau)-Y(t)$$ have the distributions $$\gamma (\tau,\beta_ 1)$$ and $$\gamma (\tau,\beta_ 2)$$, respectively. Let N be the entrance time of the random walk $Z_ n=(X_ n,Y_ n)=(X(\alpha_ 1+n),Y(\alpha_ 2+n))$ into the set $$\{(x,y)\in {\mathbb{R}}^ 2_+|$$ $$xy>1\}$$ and let $$E_ 1$$ be the event $$\{$$ $$N\geq 1$$, $$X_ NY_{N-1}>1\}$$, meaning that the hyperbola $$xy=1$$ cuts the line segment between $$Z_{N-1}$$ and $$(X_ N,Y_{N-1}).$$
The author proves some curious independences relating to $$Z_{N-1}$$ and $$Z_ N$$. Example: When $$\alpha_ 2>\alpha_ 1+1$$ let $M_ 1=(Y^{- 1}_{N-1},Y_{N-1}),\quad M_ 2=(X(\alpha_ 2+N-1),(X(\alpha_ 2+N- 1))^{-1})$ be points on $$xy=1$$ and $$M=(M_{1x},M_{2y})$$, where $$A_ x$$ and $$A_ y$$ denote the x- and y-coordinate of $$A\in {\mathbb{R}}^ 2$$. Conditionally given $$E_ 1$$, the random variables N and $$Y_{N-1}$$ are independent, $$M_ y$$ and $$M_{1y}-M_ y$$ are independent and the distributions of $$Y_{N-1}$$, $$M_ y$$ and $$M_{1y}-M_ y$$ are $\mu (\alpha_ 2-\alpha_ 1,2\beta_ 2^{-1},2\beta_ 1^{-1}),\quad \mu (\alpha_ 1-\alpha_ 2,2\beta_ 2^{-1},2\beta_ 1^{-1})\text{ and } \gamma (\alpha_ 2-\alpha_ 1,\beta_ 2),$ respectively. This gives an a.s. realization of (*). Analogous results hold for $$\alpha_ 1>\alpha_ 2$$. Proofs lead to a number of lemmas on conditional distributions involving gamma and generalized inverse Gaussian distributions.
Reviewer: A.J.Stam

##### MSC:
 60G50 Sums of independent random variables; random walks 60E99 Distribution theory 60J75 Jump processes (MSC2010) 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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