About a possible analytic approach for walks in the quarter plane with arbitrary big jumps. (Autour d’une approche analytique pour les marches à sauts arbitrairement grands dans le quart de plan.) (English. French summary) Zbl 1319.60090
Summary: In this note, we consider random walks in the quarter plane with arbitrary big jumps. We announce the extension to the class of models of the analytic approach of [G. Fayolle et al., Random walks in the quarter-plane. Algebraic methods, boundary value problems and applications. Berlin: Springer (1999; Zbl 0932.60002)], initially valid for walks with small steps in the quarter plane. New technical challenges arise, most of them being tackled in the framework of generalized boundary value problems on compact Riemann surfaces.
MSC:
| 60G50 | Sums of independent random variables; random walks |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 30F10 | Compact Riemann surfaces and uniformization |
Citations:
Zbl 0932.60002References:
| [1] | Bousquet-Mélou, M.; Mishna, M., Walks with small steps in the quarter plane, Contemp. Math., 520, 1-39 (2010) · Zbl 1209.05008 |
| [2] | Cohen, J. W.; Boxma, O. J., Boundary Value Problems in Queueing System Analysis (1983), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0515.60092 |
| [3] | Fayolle, G.; Iasnogorodski, R., Two coupled processors: the reduction to a Riemann-Hilbert problem, Z. Wahrscheinlichkeitstheor. Verw. Geb., 47, 325-351 (1979) · Zbl 0395.68032 |
| [4] | Fayolle, G.; Iasnogorodski, R.; Malyshev, V., Random Walks in the Quarter Plane (1999), Springer-Verlag: Springer-Verlag Berlin · Zbl 0932.60002 |
| [5] | Flajolet, P.; Sedgewick, R., Analytic Combinatorics (2009), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 1165.05001 |
| [6] | Malyshev, V., An analytical method in the theory of two-dimensional positive random walks, Sib. Math. J., 13, 1314-1329 (1972) · Zbl 0287.60072 |
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