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Zbl 0701.60069
Vallois, P.
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)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 25, No.4, 443-456 (1989). ISSN 0246-0203

The generalized inverse Gaussian distribution $\mu$ ($\lambda$,a,b) has the density $$ 2\sp{-1}(a/b)\sp{1/2}\exp (-2\sp{-1}(ax+bx\sp{- 1}))/K\sb{\lambda}((ab)\sp{1/2})\text{ on } (0,\infty), $$ where $K\sb{\lambda}$ is the modified Bessel function of the third kind. From the Laplace-Stieltjes transform of $\mu$ ($\lambda$,a,b) given by {\it O. Barndorff-Nielsen} and {\it 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), $t\ge 0$, 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 $\gamma (\tau,\beta\sb 1)$ and $\gamma (\tau,\beta\sb 2)$, respectively. Let N be the entrance time of the random walk $$ Z\sb n=(X\sb n,Y\sb n)=(X(\alpha\sb 1+n),Y(\alpha\sb 2+n)) $$ into the set $\{(x,y)\in {\bbfR}\sp 2\sb+\vert$ $xy>1\}$ and let $E\sb 1$ be the event $\{$ $N\ge 1$, $X\sb NY\sb{N-1}>1\}$, meaning that the hyperbola $xy=1$ cuts the line segment between $Z\sb{N-1}$ and $(X\sb N,Y\sb{N-1}).$ \par The author proves some curious independences relating to $Z\sb{N-1}$ and $Z\sb N$. Example: When $\alpha\sb 2>\alpha\sb 1+1$ let $$ M\sb 1=(Y\sp{- 1}\sb{N-1},Y\sb{N-1}),\quad M\sb 2=(X(\alpha\sb 2+N-1),(X(\alpha\sb 2+N- 1))\sp{-1}) $$ be points on $xy=1$ and $M=(M\sb{1x},M\sb{2y})$, where $A\sb x$ and $A\sb y$ denote the x- and y-coordinate of $A\in {\bbfR}\sp 2$. Conditionally given $E\sb 1$, the random variables N and $Y\sb{N-1}$ are independent, $M\sb y$ and $M\sb{1y}-M\sb y$ are independent and the distributions of $Y\sb{N-1}$, $M\sb y$ and $M\sb{1y}-M\sb y$ are $$ \mu (\alpha\sb 2-\alpha\sb 1,2\beta\sb 2\sp{-1},2\beta\sb 1\sp{-1}),\quad \mu (\alpha\sb 1-\alpha\sb 2,2\beta\sb 2\sp{-1},2\beta\sb 1\sp{-1})\text{ and } \gamma (\alpha\sb 2-\alpha\sb 1,\beta\sb 2), $$ respectively. This gives an a.s. realization of (*). Analogous results hold for $\alpha\sb 1>\alpha\sb 2$. Proofs lead to a number of lemmas on conditional distributions involving gamma and generalized inverse Gaussian distributions.
[A.J.Stam]

MSC 2000:: *60G50 Sums of independent random variables
60E99 Distribution theory in probability theory
60J75 Jump processes
60B15 Probability measures on groups

Keywords: hitting point; inverse Gaussian distribution; gamma distribution; entrance time

Citations: Zbl 0403.60026

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