[1] M., Abramowitz and I. A., Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, U.S. Government Printing Office, Washington, D.C., 1964.
[2] K. S., Alexander, Excursions and local limit theorems for Bessel-like random walks, Electron. J. Probab. 16 (2011), no. 1, 1–44.
[3] O. S. M., Alves, F. P., Machado, and S. Yu., Popov, The shape theorem for the frog model, Ann. Appl. Probab. 12 (2002), no. 2, 533–546.
[4] W. J., Anderson, Continuous-time Markov chains, Springer Series in Statistics: Probability and its Applications, Springer-Verlag, New York, 1991.
[5] E. D., Andjel, M. V., Menshikov, and V. V., Sisko, Positive recurrence of processes associated to crystal growth models, Ann. Appl. Probab. 16 (2006), no. 3, 1059–1085.
[6] S., Asmussen, Applied probability and queues, second ed., Applications of Mathematics (New York), vol. 51, Springer-Verlag, New York, 2003.
[7] S., Aspandiiarov and R., Iasnogorodski, Tails of passage-times and an application to stochastic processes with boundary reflection in wedges, Stochastic Process. Appl. 66 (1997), no. 1, 115–145.
[8] S., Aspandiiarov and R., Iasnogorodski, Asymptotic behaviour of stationary distributions for countable Markov chains, with some applications, Bernoull. 5 (1999), no. 3, 535–569.
[9] S., Aspandiiarov and R., Iasnogorodski, General criteria of integrability of functions of passage-times for nonnegative stochastic processes and their applications, Theory Probab. Appl. 43 (1999), 343–369, translated from Teor. Veroyatnost. i Primenen. 43 (1998) 509–539 (in Russian).
[10] S., Aspandiiarov, R., Iasnogorodski, and M., Menshikov, Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant, Ann. Probab. 24 (1996), no. 2, 932–960.
[11] K. B., Athreya and P. E., Ney, Branching processes, Dover Publications, Inc., Mineola, NY, 2004, reprint of the 1972 original.
[12] O., Aydogmus, A. P., Ghosh, S., Ghosh, and A., Roitershtein, Colored maximal branching processes, Theory Probab. Appl. 59 (2015), no. 4, 663–672, translated from Teor. Veroyatnost. i Primenen. 59 (2014) 790–800 (in Russian).
[13] K., Azuma, Weighted sums of certain dependent random variables, Tohoku Math. J. 19 (1967), no. 2, 357–367.
[14] L., Bachelier, Théorie de la spéculation, Ann. Sci. Ècole Norm. Sup. 17 (1900), no. 3, 21–86.
[15] M. N., Barber and B. W., Ninham, Random and restricted walks: theory and applications, Gordon and Breach, New York, 1970.
[16] L. E., Baum, On convergence to +∞ in the law of large numbers, Ann. Math. Statist. 34 (1963), 219–222.
[17] V., Belitsky, P. A., Ferrari, M. V., Menshikov, and S. Yu., Popov, A mixture of the exclusion process and the voter model, Bernoull. 7 (2001), no. 1, 119–144.
[18] M., Benaim, Vertex-reinforced random walks and a conjecture of Pemantle, Ann. Probab. 25 (1997), no. 1, 361–392.
[19] I., Benjamini, R., Izkovsky, and H., Kesten, On the range of the simple random walk bridge on groups, Electron. J. Probab. 12 (2007), no. 20, 591–612.
[20] I., Benjamini, G., Kozma, and B., Schapira, A balanced excited random walk, C. R. Math. Acad. Sci. Paris 349 (2011), no. 7–8, 459–462.
[21] I., Benjamini and D. B., Wilson, Excited random walk, Electron. Comm. Probab. 8 (2003), 86–92 (electronic).
[22] J., Bérard and A., Ramírez, Central limit theorem for the excited random walk in dimension D ≥ 2, Electron. Comm. Probab. 12 (2007), 303–314 (electronic).
[23] H. C., Berg, Random walks in biology, expanded ed., Princeton University Press, New Jersey, 1993.
[24] J., Bertoin and I., Kortchemski, Self-similar scaling limits of Markov chains on the positive integers, Ann. Appl. Probab (2016).
[25] P., Billingsley, Probability and measure, third ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1995.
[26] N. H., Bingham, Random walk on spheres, Z. Wahrscheinlichkeitstheorie und Verw. Gebiet. 22 (1972), 169–192.
[27] D., Blackwell, On transient Markov processes with a countable number of states and stationary transition probabilities, Ann. Math. Statist. 26 (1955), 654–658.
[28] D., Blackwell and D., Freedman, A remark on the coin tossing game, Ann. Math. Statist. 35 (1964), 1345–1347.
[29] J., Bojarski, R., Smolenski, A., Kempski, and P., Lezynski, Pearson's random walk approach to evaluating interference generated by a group of converters, Appl. Math. Comput. 219 (2013), no. 12, 6437–6444.
[30] P., Bougerol, Oscillation des produits de matrices aléatoires dont l'exposant de Lyapounov est nul, Lyapunov exponents (Bremen, 1984), Lecture Notes in Math., vol. 1186, Springer, Berlin, 1986, pp. 27–36.
[31] P., Bougerol and J., Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston Inc., Boston, MA, 1985.
[32] P., Bovet and S., Benhamou, Spatial analysis of animals' movements using a correlated random walk model, J. Theoret. Biol. 131 (1988), 419–433.
[33] L., Breiman, Transient atomic Markov chains with a denumerable number of states, Ann. Math. Statist. 29 (1958), 212–218.
[34] M., Campanino and D., Petritis, Random walks on randomly oriented lattices, Markov Process. Related Field. 9 (2003), no. 3, 391–412.
[35] B., Carazza, The history of the random-walk problem: considerations on the interdisciplinarity in modern physics, Riv. Nuovo Cimento (2. 7 (1977), no. 3, 419–427.
[36] J., Cerny, Moments and distribution of the local time of a two-dimensional random walk, Stochastic Process. Appl. 117 (2007), no. 2, 262–270.
[37] M.-F., Chen, On three classical problems for Markov chains with continuous time parameters, J. Appl. Probab. 28 (1991), no. 2, 305–320.
[38] B. D., Choi and B., Kim, Non-ergodicity criteria for denumerable continuous time Markov processes, Oper. Res. Lett. 32 (2004), no. 6, 574–580.
[39] P.-L., Chou and R. Z., Khas'minskiĭ, The method of Lyapunov function for the analysis of the absorption and explosion of Markov chains, Probl. Inf. Transm. 47 (2011), no. 3, 232–250, translated from Problemy Peredachi Informatsii 47 (2011) 19–38 (in Russian).
[40] Y. S., Chow, H., Robbins, and H., Teicher, Moments of randomly stopped sums, Ann. Math. Statist. 36 (1965), 789–799.
[41] Y. S., Chow and H., Teicher, Probability theory: Independence, interchangeability, martingales, third ed., Springer Texts in Statistics, Springer-Verlag, New York, 1997.
[42] Y. S., Chow and C.-H., Zhang, A note on Feller's strong law of large numbers, Ann. Probab. 14 (1986), no. 3, 1088–1094.
[43] F., Chung and L., Lu, Complex graphs and networks, CBMS Regional Conference Series in Mathematics, vol. 107, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2006.
[44] K. L., Chung, On the maximum partial sums of sequences of independent random variables, Trans. Amer.Math. Soc. 64 (1948), 205–233.
[45] K. L., Chung, Markov chains with stationary transition probabilities, Second ed. Die Grundlehren der mathematischenWissenschaften, Band 104, Springer-Verlag New York Inc., New York, 1967.
[46] K. L., Chung, A course in probability theory, third ed., Academic Press Inc., San Diego, CA, 2001.
[47] K. L., Chung and W. H. J., Fuchs, On the distribution of values of sums of random variables, Mem. Amer.Math. Soc. 1951 (1951), no. 6, 12.
[48] P., Clifford and A., Sudbury, A model for spatial conflict, Biometrik. 60 (1973), 581–588.
[49] E. A., Codling, M. J., Plank, and S., Benhamou, Random walk models in biology, J. R. Soc. Interfac. 5 (2008), no. 25, 813–834.
[50] J. W., Cohen, Analysis of random walks, Studies in Probability, Optimization and Statistics, vol. 2, IOS Press, Amsterdam, 1992.
[51] J. W., Cohen, On the random walk with zero drifts in the first quadrant of R2, Comm. Statist. Stochastic Models 8 (1992), no. 3, 359–374.
[52] F., Comets, M. V., Menshikov, and S. Yu., Popov, One-dimensional branching random walk in a random environment: a classification, Markov Process. Related Field. 4 (1998), no. 4, 465–477, I Brazilian School in Probability (Rio de Janeiro, 1997).
[53] F., Comets, M. V., Menshikov, S., Volkov, and A. R., Wade, Random walk with barycentric self-interaction, J. Stat. Phys. 143 (2011), no. 5, 855–888.
[54] F., Comets and S., Popov, On multidimensional branching random walks in random environment, Ann. Probab. 35 (2007), no. 1, 68–114.
[55] F., Comets and S., Popov, Shape and local growth for multidimensional branching random walks in random environment, ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007), 273–299.
[56] F., Comets, S., Popov, G. M., Schütz, and M., Vachkovskaia, Billiards in a general domain with random reflections, Arch. Ration. Mech. Anal. 191 (2009), no. 3, 497–537.
[57] F., Comets, S., Popov, G. M., Schütz, and M., Vachkovskaia, Knudsen gas in a finite random tube: transport diffusion and first passage properties, J. Stat. Phys. 140 (2010), no. 5, 948–984.
[58] F., Comets, M., Menshikov, and S., Popov, Lyapunov functions for random walks and strings in random environment, Ann. Probab. 26 (1998), no. 4, 1433–1445.
[59] P., Coolen-Schrijner and E. A. van, Doorn, Analysis of random walks using orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), no. 1–2, 387–399.
[60] E., Crane, N., Georgiou, S., Volkov, A. R., Wade, and R. J., Waters, The simple harmonic urn, Ann. Probab. 39 (2011), no. 6, 2119–2177.
[61] E., Csáki, On the lower limits of maxima and minima of Wiener process and partial sums, Z. Wahrsch. Verw. Gebiet. 43 (1978), no. 3, 205–221.
[62] E., Csáki, M., Csörgʺo, A., Földes, and P., Révész, On the local time of random walk on the 2-dimensional comb, Stochastic Process. Appl. 121 (2011), no. 6, 1290–1314.
[63] E., Csáki, A., Földes, and P., Révész, Joint asymptotic behavior of local and occupation times of random walk in higher dimension, Studia Sci.Math. Hungar. 44 (2007), no. 4, 535–563.
[64] E., Csáki, A., Földes, and P., Révész, Transient nearest neighbor random walk and Bessel process, J. Theoret. Probab. 22 (2009), no. 4, 992–1009.
[65] E., Csáki, A., Földes, and P., Révész, Transient nearest neighbor random walk on the line, J. Theoret. Probab. 22 (2009), no. 1, 100–122.
[66] E., Csáki, P., Révész, and J., Rosen, Functional laws of the iterated logarithm for local times of recurrent random walks on Z2, Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 4, 545–563.
[67] J. De, Coninck, F., Dunlop, and T., Huillet, Random walk weakly attracted to a wall, J. Stat. Phys. 133 (2008), no. 2, 271–280.
[68] D. De, Blassie and R., Smits, The influence of a power law drift on the exit time of Brownian motion from a half-line, Stochastic Process. Appl. 117 (2007), no. 5, 629–654.
[69] P., Deheuvels and P., Révész, Simple random walk on the line in random environment, Probab. Theory Relat. Field. 72 (1986), no. 2, 215–230.
[70] F. den, Hollander, Random polymers, Lecture Notes in Mathematics, vol. 1974, Springer-Verlag, Berlin, 2009, Lectures from the 37th Probability Summer School held in Saint-Flour, 2007.
[71] F. den, Hollander, M. V., Menshikov, and S. Yu., Popov, A note on transience versus recurrence for a branching random walk in random environment, J. Statist. Phys. 95 (1999), no. 3–4, 587–614.
[72] D., Denisov, D., Korshunov, and V., Wachtel, Potential analysis for positive recurrent Markov chains with asymptotically zero drift: power-type asymptotics, Stochastic Process. Appl. 123 (2013), no. 8, 3027–3051.
[73] D. E., Denisov and S. G., Foss, On transience conditions for Markov chains and random walks, Siberian Math. J. 44 (2003), no. 1, 44–57, translated from Sibirsk. Mat. Zh. 44 (2003) 53–68, in Russian.
[74] C., Derman and H., Robbins, The strong law of large numbers when the first moment does not exist, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 586–587.
[75] B., Derrida, S., Goldstein, J. L., Lebowitz, and E. R., Speer, Shift equivalence of measures and the intrinsic structure of shocks in the asymmetric simple exclusion process, J. Statist. Phys. 93 (1998), no. 3–4, 547–571.
[76] W., Doeblin, Éléments d'une théorie générale des chaines simples constantes de Markoff, Ann. Sci. Ècole Norm. Sup. (3. 57 (1940), 61–111.
[77] M. D., Donsker and S. R. S., Varadhan, On the number of distinct sites visited by a random walk, Comm. Pure Appl. Math. 32 (1979), no. 6, 721–747.
[78] J. L., Doob, Stochastic processes, John Wiley & Sons Inc., New York, 1953.
[79] R., Douc, G., Fort, E., Moulines, and P., Soulier, Practical drift conditions for subgeometric rates of convergence, Ann. Appl. Probab. 14 (2004), no. 3, 1353–1377.
[80] D. Y., Downham and S. B., Fotopoulos, The transient behaviour of the simple random walk in the plane, J. Appl. Probab. 25 (1988), no. 1, 58–69.
[81] P. G., Doyle and J. L., Snell, Random walks and electric networks, Carus Mathematical Monographs, vol. 22, Mathematical Association of America, Washington, DC, 1984.
[82] R., Durrett, Ten lectures on particle systems, Lectures on probability theory (Saint-Flour, 1993), Lecture Notes in Math., vol. 1608, Springer, Berlin, 1995, pp. 97–201.
[83] R., Durrett, Probability: theory and examples, fourth ed., Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010.
[84] A., Dvoretzky and P., Erdʺos, Some problems on random walk in space, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950 (Berkeley and Los Angeles), University of California Press, 1951, pp. 353–367.
[85] A., Eibeck and W., Wagner, Stochastic interacting particle systems and nonlinear kinetic equations, Ann. Appl. Probab. 13 (2003), no. 3, 845–889.
[86] A., Einstein, Investigations on the theory of the Brownian movement, Dover Publications Inc., New York, 1956, Edited with notes by R., Fürth, translated by A. D., Cowper.
[87] R., Eldan, F., Nazarov, and Y., Peres, How many matrices can be spectrally balanced simultaneously? Preprint. (2016).
[88] P., Erdʺos and S. J., Taylor, Some problems concerning the structure of random walk paths, Acta Math. Acad. Sci. Hungar. 11 (1960), 137–162.
[89] K. B., Erickson, The strong law of large numbers when the mean is undefined, Trans. Amer. Math. Soc. 185 (1973), 371–381.
[90] S. N., Evans, Stochastic billiards on general tables, Ann. Appl. Probab. 11 (2001), no. 2, 419–437. References 349
[91] A. M., Fal', Recurrence times for certain Markov random walks, Ukrainian Math. J. 23 (1971), 676–681, translated from Ukrain. Mat. Zh. 23 (1971) 824–830 (in Russian).
[92] A. M., Fal', Generalization of the results obtained by R. L., Dobrushin for additive functionals of a Markov random walk, Theory Probab. Appl. 22 (1978), 569–571, translated from Teor. Veroyatnost. i Primenen. 22 (1977) 582–584 (in Russian).
[93] A. M., Fal', Certain limit theorems for an elementary Markov random walk, Ukrainian Math. J. 33 (1981), 433–435, translated from Ukrain. Mat. Zh. 33 (1981) 564–566 (in Russian).
[94] E. F., Fama, The behavior of stock-market prices, J. Busines. 38 (1965), 34–105.
[95] G., Fayolle, V. A., Malyshev, and M. V., Men'shikov, Random walks in a quarter plane with zero drifts. I. Ergodicity and null recurrence, Ann. Inst. H. Poincaré Probab. Statist. 28 (1992), no. 2, 179–194.
[96] G., Fayolle, V. A., Malyshev, and M. V., Menshikov, Topics in the constructive theory of countable Markov chains, Cambridge University Press, Cambridge, 1995.
[97] W., Feller, A limit theorem for random variables with infinite moments, Amer. J. Math. 68 (1946), 257–262.
[98] W., Feller, An introduction to probability theory and its applications. Vol. I, Third ed., John Wiley & Sons Inc., New York, 1968.
[99] W., Feller, An introduction to probability theory and its applications. Vol. II., Second ed., John Wiley & Sons Inc., New York, 1971.
[100] P. A., Ferrari, Shock fluctuations in asymmetric simple exclusion, Probab. Theory Related Field. 91 (1992), no. 1, 81–101.
[101] P. A., Ferrari, Shocks in one-dimensional processes with drift, Probability and phase transition (Cambridge, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 420, Kluwer Acad. Publ., Dordrecht, 1994, pp. 35–48.
[102] P. A., Ferrari, C., Kipnis, and E., Saada, Microscopic structure of travelling waves in the asymmetric simple exclusion process, Ann. Probab. 19 (1991), no. 1, 226–244.
[103] Yu. P., Filonov, Criterion for ergodicity of homogeneous discrete Markov chains, Ukrain. Math. J. 41 (1989), 1223–1225, translated from Ukrain. Mat. Zh. 41 (1989) 1421–1422.
[104] L., Flatto, A problem on random walk, Quart. J. Math. Oxford Ser. (2. 9 (1958), 299–300.
[105] S., Foss, D., Korshunov, and S., Zachary, An introduction to heavy-tailed and subexponential distributions, second ed., Springer Series in Operations Research and Financial Engineering, Springer, New York, 2013.
[106] S. G., Foss and D. E., Denisov, On transience conditions for Markov chains, Siberian Math. J. 42 (2001), no. 2, 364–371, translated from Sibirsk. Mat. Zh. 42 (2001) 425–433, in Russian.
[107] F. G., Foster, Markoff chains with an enumerable number of states and a class of cascade processes, Math. Proc. Cambridge Philos. Soc. 47 (1951), 77–85.
[108] F. G., Foster, On the stochastic matrices associated with certain queuing processes, Ann. Math. Statistic. 24 (1953), 355–360.
[109] J.-D., Fouks, E., Lesigne, and M., Peigné, Étude asymptotique d'une marche aléatoire centrifuge, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 2, 147–170.
[110] A. S., Gaĭrat, V. A., Malyshev, M.V., Men'shikov, and K. D., Pelikh, Classification of Markov chains that describe the evolution of random strings, Uspekhi Mat. Nauk 50 (1995), no. 2(302), 5–24.
[111] A. S., Gajrat, V. A., Malyshev, and A. A., Zamyatin, Two-sided evolution of a random chain, Markov Process. Related Field. 1 (1995), no. 2, 281–316.
[112] L., Gallardo, Comportement asymptotique des marches aléatoires associées aux polynomes de Gegenbauer et applications, Adv. Appl. Probab. 16 (1984), no. 2, 293–323.
[113] C., Gallesco, S., Popov, and G. M., Schütz, Localization for a random walk in slowly decreasing random potential, J. Stat. Phys. 150 (2013), no. 2, 285–298.
[114] N., Georgiou, M. V., Menshikov, A., Mijatovíc, and A. R., Wade, Anomalous recurrence properties of many-dimensional zero-drift random walks, Adv. Appl. Probab. 48 (2016), 99–118.
[115] N., Georgiou and A. R., Wade, Non-homogeneous random walks on a semi-infinite strip, Stochastic Process. Appl. 124 (2014), no. 10, 3179–3205.
[116] G., Giacomin, Random polymer models, Imperial College Press, London, 2007.
[117] J., Gillis, Centrally biased discrete random walk, Quart. J. Math. Oxford Ser. (2. 7 (1956), 144–152.
[118] M. L., Glasser and I. J., Zucker, Extended Watson integrals for the cubic lattices, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1800–1801.
[119] A. O., Golosov, Limit distributions for random walks in a random environment, Dokl. Akad. Nauk SSS. 271 (1983), no. 1, 25–29.
[120] M., González, M., Molina, and I. del, Puerto, Asymptotic behaviour of critical controlled branching processes with random control functions, J. Appl. Probab. 42 (2005), no. 2, 463–477.
[121] P. S., Griffin, An integral test for the rate of escape of d-dimensional random walk, Ann. Probab. 11 (1983), no. 4, 953–961.
[122] R. F., Gundy and D., Siegmund, On a stopping rule and the central limit theorem, Ann. Math. Statist. 38 (1967), 1915–1917.
[123] A., Gut, Probability: A graduate course, Springer, 2005.
[124] Y., Hamana, An almost sure invariance principle for the range of random walks, Stochastic Process. Appl. 78 (1998), no. 2, 131–143.
[125] K., Hamza and F. C., Klebaner, Conditions for integrability of Markov chains, J. Appl. Probab. 32 (1995), no. 2, 541–547.
[126] C. M., Harris and P. G., Marlin, A note on feedback queues with bulk service, J. Assoc. Comput. Mach. 19 (1972), 727–733.
[127] T. E., Harris, First passage and recurrence distributions, Trans. Amer. Math. Soc. 73 (1952), 471–486.
[128] W. M., Hirsch, A strong law for the maximum cumulative sum of independent random variables, Comm. Pure Appl. Math. 18 (1965), 109–127.
[129] J. L., Hodges, Jr and M., Rosenblatt, Recurrence-time moments in random walks, Pacific J. Math. 3 (1953), 127–136.
[130] W., Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30.
[131] P., Holgate, Random walk models for animal behavior, Statistical Ecology: vol. 2 (G., Patil, E., Pielou, and W., Walters, eds.), Penn State University Press, University Park, PA, 1971, pp. 1–12.
[132] R. A., Holley and T. M., Liggett, Ergodic theorems for weakly interacting infinite systems and the voter model, Ann. Probabilit. 3 (1975), no. 4, 643–663.
[133] O., Hryniv, I. M., MacPhee, M. V., Menshikov, and A. R., Wade, Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips, Electron. J. Probab. 17 (2012), article 59.
[134] O., Hryniv and M., Menshikov, Long-time behaviour in a model of microtubule growth, Adv. Appl. Probab. 42 (2010), no. 1, 268–291.
[135] O., Hryniv, M. V., Menshikov, and A. R., Wade, Excursions and path functionals for stochastic processes with asymptotically zero drifts, Stochastic Process. Appl. 123 (2013), no. 6, 1891–1921.
[136] O., Hryniv, M. V., Menshikov, and A. R., Wade, Random walk in mixed random environment without uniform ellipticity, Proc. Stekov. Inst. Math. 282 (2013), 106–123.
[137] Y., Hu and H., Nyrhinen, Large deviations view points for heavy-tailed random walks, J. Theoret. Probab. 17 (2004), no. 3, 761–768.
[138] Y., Hu and Z., Shi, The limits of Sinai's simple random walk in random environment, Ann. Probab. 26 (1998), no. 4, 1477–1521.
[139] B. D., Hughes, Random walks and random environments. Vol. 1, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1995.
[140] T., Huillet, Random walk with long-range interaction with a barrier and its dual: exact results, J. Comput. Appl. Math. 233 (2010), no. 10, 2449–2467.
[141] N. C., Jain and W. E., Pruitt, Maxima of partial sums of independent random variables, Z.Wahrscheinlichkeitstheorie und Verw. Gebiet. 27 (1973), 141–151.
[142] S. F., Jarner and G. O., Roberts, Polynomial convergence rates of Markov chains, Ann. Appl. Probab. 12 (2002), no. 1, 224–247.
[143] R. C., Jones, On the theory of fluctuations in the decay of sound, J. Acoust. Soc. Amer. 11 (1940), 324–332.
[144] P., Jung, The noisy voter-exclusion process, Stochastic Process. Appl. 115 (2005), no. 12, 1979–2005.
[145] V. V., Kalashnikov, Practical stability of difference equations, Automation and Remote Contro. 9 (1967), 1404–1407, translated from Avtomatika i Telemekhanika 9 (1967) 172–175 (in Russian).
[146] V. V., Kalashnikov, The use of Lyapunov's method in the solution of queueing theory problems, Engrg. Cybernetics (1968), no. 5, 77–84, translated from Izv. Akad. Nauk SSSR Tehn. Kibernet. 1968 89–95 (in Russian).
[147] V. V., Kalashnikov, Reliability analysis by Lyapunov's method, Engrg. Cybernetics (1970), no. 2, 257–269, translated from Izv. Akad. Nauk SSSR Tehn. Kibernet. 1970 65–76 (in Russian).
[148] V. V., Kalashnikov, Analysis of ergodicity of queueing systems by means of the direct method of Lyapunov, Automation and Remote Contro. 32 (1971), 559–566, translated from Avtomatika i Telemekhanika 32 (1971) 46–54 (in Russian).
[149] V. V., Kalashnikov, The property of γ -reflexivity for Markov sequences, Soviet Math. Dokl. 14 (1973), 1869–1873, translated from Dokl. Akad. Nauk SSSR 213 (1973) 1243–1246 (in Russian).
[150] O., Kallenberg, Foundations of modern probability, second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002.
[151] M., Kaplan, A sufficient condition for nonergodicity of a Markov chain, IEEE Trans. Inform. Theor. 25 (1979), no. 4, 470–471.
[152] S., Karlin and J., McGregor, Representation of a class of stochastic processes, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 387–391.
[153] S., Karlin and J., McGregor, Random walks, Illinois J. Math. 3 (1959), 66–81.
[154] K., Kawazu, Y., Tamura, and H., Tanaka, Limit theorems for one-dimensional diffusions and random walks in random environments, Probab. Theor. Relat. Field. 80 (1989), no. 4, 501–541.
[155] G., Keller, G., Kersting, and U., Rösler, On the asymptotic behaviour of discrete time stochastic growth processes, Ann. Probab. 15 (1987), no. 1, 305–343.
[156] F. P., Kelly, Markovian functions of a Markov chain, Sankhya Ser.. 44 (1982), no. 3, 372–379.
[157] J. H. B., Kemperman, The oscillating random walk, Stochastic Process. Appl. 2 (1974), 1–29.
[158] D. G., Kendall, Some problems in the theory of queues, J. Roy. Statist. Soc. Ser. B. 13 (1951), 151–173; discussion: 173–185.
[159] G., Kersting, On recurrence and transience of growth models, J. Appl. Probab. 23 (1986), no. 3, 614–625.
[160] G., Kersting, Asymptotic distribution for stochastic difference equations, Stochastic Process. Appl. 40 (1992), no. 1, 15–28.
[161] G., Kersting and F. C., Klebaner, Sharp conditions for nonexplosions and explosions in Markov jump processes, Ann. Probab. 23 (1995), no. 1, 268–272.
[162] G., Kersting and F. C., Klebaner, Explosions in Markov processes and submartingale convergence, Athens Conference on Applied Probability and Time Series Analysis, Vol. I (1995), Lecture Notes in Statist., vol. 114, Springer, New York, 1996, pp. 127–136.
[163] H., Kesten, The limit points of a normalized random walk, Ann. Math. Statist. 41 (1970), 1173–1205.
[164] H., Kesten, The limit distribution of Sinai's random walk in random environment, Phys. A 138 (1986), no. 1–2, 299–309.
[165] H., Kesten and R. A., Maller, Two renewal theorems for general random walks tending to infinity, Probab. Theor. Related Field. 106 (1996), no. 1, 1–38.
[166] H., Kesten and R. A., Maller, Random walks crossing power law boundaries, Studia Sci. Math. Hungar. 34 (1998), no. 1–3, 219–252.
[167] R. Z., Khas'minskii, On the stability of the trajectory of Markov processes, J. Appl. Math. Mech. 26 (1962), 1554–1565.
[168] R. Z., Khas'minskii, Stochastic stability of differential equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, vol. 7, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, translated and updated version of the original Russian edition, Nauka, Moscow, 1969.
[169] B., Kim and I., Lee, Tests for nonergodicity of denumerable continuous time Markov processes, Comput. Math. Appl. 55 (2008), no. 6, 1310–1321.
[170] J. F. C., Kingman, The ergodic behaviour of random walks, Biometrik. 48 (1961), 391–396.
[171] J. F. C., Kingman, Two similar queues in parallel, Ann. Math. Statist. 32 (1961), 1314–1323.
[172] J. F. C., Kingman, Random walks with spherical symmetry, Acta Math. 109 (1963), 11–53.
[173] F. C., Klebaner, Stochastic difference equations and generalized gamma distributions, Ann. Probab. 17 (1989), no. 1, 178–188.
[174] F. C., Klebaner, Introduction to stochastic calculus with applications, second ed., Imperial College Press, London, 2005.
[175] L. A. Klein, Haneveld, Random walk on the quadrant, Stochastic Process. Appl. 26 (1987), 228.
[176] L. A. Klein, Haneveld, Random walk in the quadrant, Ph.D. thesis, University of Amsterdam, 1996.
[177] L. A. Klein, Haneveld and A. O., Pittenger, Escape time for a random walk from an orthant, Stochastic Process. Appl. 35 (1990), no. 1, 1–9.
[178] H., Kleinert, Path integrals in quantum mechanics, statistics, polymer physics, and financial markets, fourth ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
[179] M., Knudsen, Kinetic theory of gases: some modern aspects, Methuen's Monographs on Physical Subjects, Methuen, London, 1952.
[180] D. A., Korshunov, Moments of stationaryMarkov chains with asymptotically zero drift, Sibirsk. Mat. Zh. 52 (2011), no. 4, 829–840.
[181] E., Kosygina and M. P. W., Zerner, Excited random walks: results, methods, open problems, Bull. Inst. Math. Acad. Sin. (N.S.. 8 (2013), no. 1, 105–157.
[182] Y., Kovchegov and N., Michalowski, A class of Markov chains with no spectral gap, Proc. Amer. Math. Soc. 141 (2013), no. 12, 4317–4326.
[183] M. V., Kozlov, Random walk in a one-dimensional random medium, Teor. Verojatnost. i Primenen. 18 (1973), 406–408.
[184] G., Kozma, T., Orenshtein, and I., Shinkar, Excited random walk with periodic cookies, Ann. Inst. H. Poincaré Probab. Statist. 52 (2016).
[185] V. M., Kruglov, A strong law of large numbers for pairwise independent identically distributed random variables with infinite means, Statist. Probab. Lett. 78 (2008), no. 7, 890–895.
[186] I., Kurkova and K., Raschel, Passage time from four to two blocks of opinions in the voter model and walks in the quarter plane, Queueing Syst. 74 (2013), no. 2–3, 219–234.
[187] H. J., Kushner, On the stability of stochastic dynamical systems, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 8–12.
[188] H. J., Kushner, Finite time stochastic stability and the analysis of tracking systems, IEEE Trans. Automatic Control AC-11 (1966), 219–227.
[189] P., Küster, Asymptotic growth of controlled Galton–Watson processes, Ann. Probab. 13 (1985), no. 4, 1157–1178.
[190] J., Lamperti, Criteria for the recurrence or transience of stochastic process. I., J. Math. Anal. Appl. 1 (1960), 314–330.
[191] J., Lamperti, A new class of probability limit theorems, J. Math. Mech. 11 (1962), 749–772.
[192] J., Lamperti, Criteria for stochastic processes. II. Passage-time moments, J. Math. Anal. Appl. 7 (1963), 127–145.
[193] J., Lamperti, Maximal branching processes and ‘long-range percolation’, J. Appl. Probab. 7 (1970), 89–98.
[194] J., Lamperti, Remarks on maximal branching processes, Teor. Verojatnost. i Primenen. 17 (1972), 46–54.
[195] G. F., Lawler and V., Limic, Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010.
[196] F. W., Leysieffer, Functions of finite Markov chains, Ann. Math. Statist. 38 (1967), 206–212.
[197] T. M., Liggett, Coupling the simple exclusion process, Ann. Probabilit. 4 (1976), no. 3, 339–356.
[198] T. M., Liggett, Interacting particle systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276, Springer-Verlag, New York, 1985.
[199] T. M., Liggett, Stochastic interacting systems: contact, voter and exclusion processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 324, Springer-Verlag, Berlin, 1999.
[200] M., Loève, Probability theory. I, fourth ed., Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, Vol. 45.
[201] M., Loève, Probability theory. I, fourth ed., Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, Vol. 45.
[202] G. G., Lowry (ed.), Markov chains and Monte Carlo calculations in polymer science, Marcel Dekker, New York, 1970.
[203] F. P., Machado, M. V., Menshikov, and S. Yu., Popov, Recurrence and transience of multitype branching random walks, Stochastic Process. Appl. 91 (2001), no. 1, 21–37.
[204] F. P., Machado and S. Yu., Popov, One-dimensional branching random walks in a Markovian random environment, J. Appl. Probab. 37 (2000), no. 4, 1157–1163.
[205] F. P., Machado and S. Yu., Popov, Branching random walk in random environment on trees, Stochastic Process. Appl. 106 (2003), no. 1, 95–106.
[206] I., MacPhee, M., Menshikov, D., Petritis, and S., Popov, A Markov chain model of a polling system with parameter regeneration, Ann. Appl. Probab. 17 (2007), no. 5-6, 1447–1473.
[207] I., MacPhee, M. V., Menshikov, and M., Vachkovskaia, Dynamics of the non-homogeneous supermarket model, Stoch. Model. 28 (2012), no. 4, 533–556.
[208] I. M., MacPhee and M. V., Menshikov, Critical random walks on two-dimensional complexes with applications to polling systems, Ann. Appl. Probab. 13 (2003), no. 4, 1399–1422.
[209] I. M., MacPhee, M. V., Menshikov, S., Popov, and S., Volkov, Periodicity in the transient regime of exhaustive polling systems, Ann. Appl. Probab. 16 (2006), no. 4, 1816–1850. References 355
[210] I. M., MacPhee, M. V., Menshikov, S., Volkov, and A. R., Wade, Passage-time moments and hybrid zones for the exclusion-voter model, Bernoull. 16 (2010), no. 4, 1312–1342.
[211] I. M., MacPhee, M. V., Menshikov, and A. R., Wade, Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift, Markov Process. Related Field. 16 (2010), no. 2, 351–388.
[212] I. M., MacPhee, M. V., Menshikov, and A. R., Wade, Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts, J. Theoret. Probab. 26 (2013), 1–30.
[213] V. A., Malyšev and M. V., Men'šikov, Ergodicity, continuity, and analyticity of countable Markov chains, Trans.MoscowMath. Soc. 39 (1979), 1–48, translated from Trudy Moskov. Mat. Obshch. 39 (1979) 3–48 (in Russian).
[214] V. A., Malyshev, Classification of two-dimensional positive random walks and almost linear semimartingales, Soviet Math. Dokl. 13 (1972), 136–139, translated from Dokl. Akad. Nauk SSSR 202 (1972) 526–528 (in Russian).
[215] V. A., Malyshev, Random grammars, Uspekhi Mat. Nauk
53 (1998), no. 2(320), 107–134.
[216] X., Mao, Stochastic differential equations and applications, second ed., Horwood Publishing Limited, Chichester, 2008.
[217] M. B., Marcus and J., Rosen, Logarithmic averages for the local times of recurrent random walks and Lévy processes, Stochastic Process. Appl. 59 (1995), no. 2, 175–184.
[218] J. G., Mauldon, On non-dissipative Markov chains, Proc. Cambridge Philos. Soc. 53 (1957), 825–835.
[219] W. H., McCrea and F. J. W., Whipple, Random paths in two and three dimensions, Proc. Roy. Soc. Edinburg. 60 (1940), 281–298.
[220] M., Menshikov and D., Petritis, Explosion, implosion, and moments of passage times for continuous-time Markov chains: A semimartingale approach, Stochastic Process. Appl. 124 (2014), no. 7, 2388–2414.
[221] M., Menshikov and S., Popov, On range and local time of many-dimensional submartingales, J. Theoret. Probab. 27 (2014), no. 2, 601–617.
[222] M., Menshikov, S., Popov, A., Ramírez, and M., Vachkovskaia, On a general many-dimensional excited random walk, Ann. Probab. 40 (2012), no. 5, 2106–2130.
[223] M., Menshikov and S., Volkov, Urn-related random walk with drift ρxα/tβ, Electron. J. Probab. 13 (2008), no. 31, 944–960.
[224] M., Menshikov and R. J., Williams, Passage-time moments for continuous non-negative stochastic processes and applications, Adv. Appl. Probab. 28 (1996), no. 3, 747–762.
[225] M., Menshikov and S., Zuyev, Polling systems in the critical regime, Stochastic Process. Appl. 92 (2001), no. 2, 201–218.
[226] M. V., Menshikov, Ergodicity and transience conditions for random walks in the positive octant of space, Soviet Math. Dokl. 15 (1974), 1118–1121, translated from Dokl. Akad. Nauk SSSR 217 (1974) 755–758 (in Russian).
[227] M. V., Menshikov, Martingale approach for Markov processes in random environment and branching Markov chains, Resenha. 3 (1997), no. 2, 159–171. 356 References
[228] M. V., Menshikov, I. M., Asymont, and R., Iasnogorodskii, Markov processes with asymptotically zero drifts, Probl. Inf. Transm. 31 (1995), 248–261, translated from Problemy Peredachi Informatsii 31 (1995) 60–75 (in Russian).
[229] M. V., Menshikov, D., Petritis, and A. R., Wade, Heavy-tailed random walks on complexes of half-lines, Preprint. (2016).
[230] M. V., Menshikov and S. Yu., Popov, Exact power estimates for countable Markov chains, Markov Process. Related Field. 1 (1995), no. 1, 57–78.
[231] M. V., Menshikov, S. Yu., Popov, V., Sisko, and M., Vachkovskaia, On a many-dimensional random walk in a rarefied random environment, Markov Process. Related Field. 10 (2004), no. 1, 137–160.
[232] M. V., Menshikov, M., Vachkovskaia, and A. R., Wade, Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains, J. Stat. Phys. 132 (2008), no. 6, 1097–1133.
[233] M. V., Menshikov and S. E., Volkov, Branching Markov chains: qualitative characteristics, Markov Process. Related Field. 3 (1997), no. 2, 225–241.
[234] M. V., Menshikov and A. R., Wade, Random walk in random environment with asymptotically zero perturbation, J. Eur. Math. Soc. (JEMS. 8 (2006), no. 3, 491–513.
[235] M. V., Menshikov and A. R., Wade, Logarithmic speeds for one-dimensional perturbed random walks in random environments, Stochastic Process. Appl. 118 (2008), no. 3, 389–416.
[236] M. V., Menshikov and A. R., Wade, Rate of escape and central limit theorem for the supercritical Lamperti problem, Stochastic Process. Appl. 120 (2010), no. 10, 2078–2099.
[237] F., Merkl and S. W. W., Rolles, Recurrence of edge-reinforced random walk on a two-dimensional graph, Ann. Probab. 37 (2009), no. 5, 1679–1714.
[238] J.-F., Mertens, E., Samuel-Cahn, and S., Zamir, Necessary and sufficient conditions for recurrence and transience of Markov chains, in terms of inequalities, J. Appl. Probab. 15 (1978), no. 4, 848–851.
[239] S., Meyn and R. L., Tweedie, Markov chains and stochastic stability, second ed., Cambridge University Press, Cambridge, 2009, With a prologue by Peter W., Glynn.
[240] R. G., MillerJr., Foster's Markov chain theorems in continuous time, Tech. Report 88, Applied Mathematics and Statistics Laboratory, Stanford University, 1963.
[241] P. A. P., Moran, An introduction to probability theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1984, Corrected reprint of the 1968 original.
[242] M. D., Moustafa, Input-output Markov processes, Proc. Konin. Neder. Akad. Wetens. A. Math. Sci. 19 (1957), 112–118.
[243] J. R., Norris, Markov chains, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2, Cambridge University Press, Cambridge, 1998.
[244] R., Nossal, Stochastic aspects of biological locomotion, J. Stat. Phys. 30 (1983), 391–400.
[245] R. J., Nossal and G. H., Weiss, A generalized Pearson random walk allowing for bias, J. Stat. Phys. 10 (1974), 245–253. References 357
[246] E., Nummelin, General irreducible Markov chains and nonnegative operators, Cambridge Tracts in Mathematics, vol. 83, Cambridge University Press, Cambridge, 1984.
[247] S., Orey, Lecture notes on limit theorems for Markov chain transition probabilities, Van Nostrand Reinhold Co., London-New York-Toronto, 1971, Van Nostrand Reinhold Mathematical Studies, No. 34.
[248] R. A., Orwoll and W. H., Stockmayer, Stochastic models for chain dynamics, Adv. Chem. Phys. 15 (1969), 305–324.
[249] V. I., Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, TrudyMoskov. Mat. Obš?c. 19 (1968), 179–210.
[250] A. G., Pakes, Some conditions for ergodicity and recurrence of Markov chains, Operations Res. 17 (1969), 1058–1061.
[251] A. G., Pakes, Some remarks on a one-dimensional skip-free process with repulsion, J. Austral. Math. Soc. Ser. A
30 (1980/81), no. 1, 107–128.
[252] E., Parzen, Stochastic processes, Classics in Applied Mathematics, vol. 24, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999, Reprint of the 1962 original.
[253] K., Pearson, The problem of the random walk, Nature 72 (1905), 342.
[254] K., Pearson and J., Blakeman, A mathematical theory of random migration, Drapers' Company Research Memoirs Biometric Series, Dulau and co., London, 1906.
[255] R., Pemantle, A survey of random processes with reinforcement, Probab. Surv. 4 (2007), 1–79.
[256] R., Pemantle and S., Volkov, Vertex-reinforced random walk on Z has finite range, Ann. Probab. 27 (1999), no. 3, 1368–1388.
[257] Y., Peres, S., Popov, and P., Sousi, On recurrence and transience of self-interacting random walks, Bull. Braz. Math. Soc. 44 (2013), no. 4, 1–27.
[258] R. G., Pinsky, Positive harmonic functions and diffusion, Cambridge Studies in Advanced Mathematics, vol. 45, Cambridge University Press, Cambridge, 1995.
[259] G., Polya, Ü ber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Math. Ann. 84 (1921), no. 1–2, 149–160.
[260] N. N., Popov, The rate of convergence for countable Markov chains, Theor. Probability Appl. 24 (1979), no. 2, 401–405, translated from Teor. Verojatnost. i Primenen. 24 (1979) 395–399 (in Russian).
[261] W. E., Pruitt, The rate of escape of random walk, Ann. Probab. 18 (1990), no. 4, 1417–1461.
[262] O., Raimond and B., Schapira, On some generalized reinforced random walk on integers, Electron. J. Probab. 14 (2009), no. 60, 1770–1789.
[263] C., Rau, Sur le nombre de points visités par une marche aléatoire sur un amas infini de percolation, Bull. Soc. Math. Franc. 135 (2007), no. 1, 135–169.
[264] Lord Rayleigh, On the resultant of a large number of vibrations of the same pitch and of arbitrary phase, Phil. Mag. Ser.. 10 (1880), 73–78.
[265] J. A. F., Regnault, Calcul des chances et philosophie de la bourse, Mallet- Bachelier/Castel, Paris, 1863.
[266] P. H. F., Reimberg and L. R., Abramo, Random flights through spaces of different dimensions, J. Math. Phys. 56 (2015), no. 1, 013512.
[267] S. I., Resnick, A probability path, Birkhäuser Boston Inc., Boston, MA, 1999.
[268] G. E. H., Reuter, Competition processes, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, CA, 1961, pp. 421–430.
[269] Pál, Révész, Random walk in random and non-random environments, third ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
[270] D., Revuz, Markov chains, second ed., North-Holland Mathematical Library, vol. 11, North-Holland Publishing Co., Amsterdam, 1984.
[271] P., Robert, Stochastic networks and queues, Applications of Mathematics, vol. 52, Springer-Verlag, Berlin, 2003.
[272] L. C. G., Rogers and J. W., Pitman, Markov functions, Ann. Probab. 9 (1981), no. 4, 573–582.
[273] B. A., Rogozin and S. G., Foss, The recurrence of an oscillating random walk, Theor. Probability Appl. 23 (1978), no. 1, 155–162, translated from Teor. Verojatnost. i Primenen. 23 (1978) 161–169 (in Russian).
[274] M., Rosenblatt, Functions of a Markov process that are Markovian, J. Math. Mech. 8 (1959), 585–596.
[275] W. A., Rosenkrantz, A local limit theorem for a certain class of random walks, Ann. Math. Statist. 37 (1966), 855–859.
[276] N., Sandríc, Recurrence and transience property for a class of Markov chains, Bernoulli 19 (2013), no. 5B, 2167–2199.
[277] N., Sandríc, Recurrence and transience criteria for two cases of stable-like Markov chains, J. Theoret. Probab. 27 (2014), 754–788.
[278] K.-I., Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999, translated from the 1990 Japanese original, Revised by the author.
[279] S., Schumacher, Diffusions with random coefficients, Ph.D. thesis, University of California, Los Angeles, 1984.
[280] S., Schumacher, Diffusions with random coefficients, Particle systems, random media and large deviations (Brunswick, Maine, 1984), Contemp. Math., vol. 41, Amer. Math. Soc., Providence, RI, 1985, pp. 351–356.
[281] L. I., Sennott, P. A., Humblet, and R. L., Tweedie, Mean drifts and the nonergodicity of Markov chains, Oper. Res. 31 (1983), no. 4, 783–789.
[282] V., Shcherbakov and S., Volkov, Stability of a growth process generated by monomer filling with nearest-neighbour cooperative effects, Stochastic Process. Appl. 120 (2010), no. 6, 926–948.
[283] L. A., Shepp, Symmetric random walk, Trans. Amer. Math. Soc. 104 (1962), 144–153.
[284] L. A., Shepp, Recurrent random walks with arbitrarily large steps, Bull. Amer. Math. Soc. 70 (1964), 540–542.
[285] T., Shiga, A., Shimizu, and T., Soshi, Passage-time moments for positively recurrent Markov chains, Nagoya Math. J. 162 (2001), 169–185.
[286] A. N., Shiryaev, Probability, second ed., Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1996, translated from the first (1980) Russian edition by R. P., Boas.
[287] M. F., Shlesinger and B. J., West (eds.), Random walks and their applications in the physical and biological sciences, American Institute of Physics, New York, 1984.
[288] Ya. G., Sinaĭ, The limiting behavior of a one-dimensional random walk in a random medium, Theor. Probability Appl. 27 (1983), 256–268, translated from Teor. Veroyatnost. i Primenen. 27 (1982) 247–258 (in Russian).
[289] A., Singh, A slow transient diffusion in a drifted stable potential, J. Theoret. Probab. 20 (2007), no. 2, 153–166.
[290] P. E., Smouse, S., Focardi, P. R., Moorcroft, J. G., Kie, J. D., Forester, and J. M., Morales, Stochastic modelling of animal movement, Phil. Trans. Roy. Soc. Ser. B Biol. Sci. 365 (2010), 2201–2211.
[291] F., Solomon, Random walks in a random environment, Ann. Probab. 3 (1975), 1–31.
[292] F., Spitzer, Renewal theorems for Markov chains, Proc. 5th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, Calif., 1967, pp. 311–320.
[293] F., Spitzer, Principles of random walk, second ed., Springer-Verlag, New York, 1976, Graduate Texts in Mathematics, Vol. 34.
[294] A., Stannard and P., Coles, Random-walk statistics and the spherical harmonic representation of cosmic microwave background maps, Mon. Not. Roy. Astron. Soc. 364 (2005), no. 3, 929–933.
[295] W. F., Stout, Almost sure convergence, Academic Press, 1974, Probability and Mathematical Statistics, Vol. 24.
[296] D. W., Stroock, An introduction to Markov processes, Graduate Texts in Mathematics, vol. 230, Springer-Verlag, Berlin, 2005.
[297] A., Sturm and J. M., Swart, Tightness of voter model interfaces, Electron. Commun. Probab. 13 (2008), 165–174.
[298] R., Syski, Exit time from a positive quadrant for a two-dimensional Markov chain, Comm. Statist. Stochastic Model. 8 (1992), no. 3, 375–395.
[299] G. J., Székely, On the asymptotic properties of diffusion processes, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 17 (1974), 69–71 (1975).
[300] A.-S., Sznitman, Topics in random walks in random environment, School and Conference on Probability Theory, ICTP Lect. Notes, XVII, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 203–266 (electronic).
[301] P., Tarrès, Vertex-reinforced random walk on Z eventually gets stuck on five points, Ann. Probab. 32 (2004), no. 3B, 2650–2701.
[302] R. L., Tweedie, Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space, Stochastic Processes Appl. 3 (1975), no. 4, 385–403.
[303] R. L., Tweedie, Sufficient conditions for regularity, recurrence and ergodicity of Markov processes, Math. Proc. Cambridge Philos. Soc. 78 (1975), 125–136.
[304] R. L., Tweedie, Criteria for classifying general Markov chains, Adv. in Appl. Probab. 8 (1976), no. 4, 737–771.
[305] R. L., Tweedie, Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes, J. Appl. Probab. 18 (1981), no. 1, 122–130.
[306] R. van der, Hofstad and M., Holmes, Monotonicity for excited random walk in high dimensions, Probab. Theory Related Fields
147 (2010), no. 1–2, 333–348.
[307] Y., Velenik, Localization and delocalization of random interfaces, Probab. Surv. 3 (2006), 112–169.
[308] A. Yu., Veretennikov, On the rate of convergence for infinite server Erlang– Sevastyanov's problem, Queueing Syst. 76 (2014), no. 2, 181–203.
[309] A. Yu., Veretennikov and G. A., Zverkina, Simple proof of Dynkin's formula for single-server systems and polynomial convergence rates, Markov Process. Related Field. 20 (2014), no. 3, 479–504.
[310] M., Voit, Central limit theorems for random walks on N that are associated with orthogonal polynomials, J. Multivariate Anal. 34 (1990), no. 2, 290–322.
[311] M., Voit, Strong laws of large numbers for random walks associated with a class of one-dimensional convolution structures, Monatsh. Math. 113 (1992), no. 1, 59–74.
[312] S., Volkov, Vertex-reinforced random walk on arbitrary graphs, Ann. Probab. 29 (2001), no. 1, 66–91.
[313] W., Wagner, Explosion phenomena in stochastic coagulation-fragmentation models, Ann. Appl. Probab. 15 (2005), no. 3, 2081–2112.
[314] G., H.Weiss, Random walks and their applications, Amer. Sci. 71 (1983), 65–71.
[315] G. H., Weiss and R. J., Rubin, Random walks: theory and selected applications, Adv. Chem. Phys. 52 (1983), 363–505.
[316] W. M., Wonham, Liapunov criteria for weak stochastic stability, J. Differential Equation. 2 (1966), 195–207.
[317] W. M., Wonham, A Liapunov method for the estimation of statistical averages, J. Differential Equation. 2 (1966), 365–377.
[318] S., Zachary, On two-dimensional Markov chains in the positive quadrant with partial spatial homogeneity, Markov Process. Related Field. 1 (1995), no. 2, 267–280.
[319] O., Zeitouni, Random walks in random environment, Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1837, Springer, Berlin, 2004, pp. 189–312.
[320] O., Zeitouni, Random walks in random environments, J. Phys. A 39 (2006), no. 40, R433–R464.
[321] M. P. W., Zerner, Recurrence and transience of excited random walks on Zd and strips, Electron. Comm. Probab. 11 (2006), 118–128 (electronic).
