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


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


Zbl 0403.60026
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