Geodesic
Random Walks in the~Positive Quadrant.~III. Constants in an integral and a~local theorem
Matematičeskie trudy, Tome 4 (2001) no. 1, pp. 68-93.

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In this article, we obtain precise formulas for constants in the local and integral theorems proven in [1, 2]. We also propose a version of the integral theorem which complements the main result of [2] and give a probabilistic interpretation of a solution to an integral equation in the positive quadrant. 
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A. A. Mogul'skii; B. A. Rogozin. Random Walks in the~Positive Quadrant.~III. Constants in an integral and a~local theorem. Matematičeskie trudy, Tome 4 (2001) no. 1, pp. 68-93. http://geodesic.mathdoc.fr/item/MT_2001_4_1_a4/

[1] Borovkov A. A., Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya, Nauka, M., 1972 | MR

[2] Mogulskii A. A., Rogozin B. A., “Sluchainye bluzhdaniya v polozhitelnom kvadrante. I: Lokalnye teoremy”, Mat. trudy, 2:2 (1999), 57–97 | MR

[3] Mogulskii A. A., Rogozin B. A., “Sluchainye bluzhdaniya v polozhitelnom kvadrante. II: Integralnaya teorema”, Mat. trudy, 3:1 (2000), 48–118 | MR

[4] Petrov V. V., Summy nezavisimykh sluchainykh velichin, Nauka, M., 1972 | MR

[5] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, t. II, Mir, M., 1967

[6] Doney R. A., “On the asymptotic behavior of the first passage times for transient random walk”, Probab. Theory Related Fields, 81:2 (1989), 239–246 | DOI | MR | Zbl

[7] Shimura M., “Exit probability of two-dimensional random walk from quadrant”, Proc. Japan Acad. Ser. A Math. Sci., 75:3 (1999), 39–42 | DOI | MR | Zbl