References
Published online by Cambridge University Press: 02 February 2017
Summary
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
- Type
- Chapter
- Information
- Non-homogeneous Random WalksLyapunov Function Methods for Near-Critical Stochastic Systems, pp. 344 - 360Publisher: Cambridge University PressPrint publication year: 2016
References
[1] and , 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] , Excursions and local limit theorems for Bessel-like random walks, Electron. J. Probab. 16 (2011), no. 1, 1–44.Google Scholar
[3] , , and , The shape theorem for the frog model, Ann. Appl. Probab. 12 (2002), no. 2, 533–546.Google Scholar
[4] , Continuous-time Markov chains, Springer Series in Statistics: Probability and its Applications, Springer-Verlag, New York, 1991.
[5] , , and , Positive recurrence of processes associated to crystal growth models, Ann. Appl. Probab. 16 (2006), no. 3, 1059–1085.Google Scholar
[6] , Applied probability and queues, second ed., Applications of Mathematics (New York), vol. 51, Springer-Verlag, New York, 2003.
[7] and , Tails of passage-times and an application to stochastic processes with boundary reflection in wedges, Stochastic Process. Appl. 66 (1997), no. 1, 115–145.Google Scholar
[8] and , Asymptotic behaviour of stationary distributions for countable Markov chains, with some applications, Bernoull. 5 (1999), no. 3, 535–569.Google Scholar
[9] and , 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).Google Scholar
[10] , , and , 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.Google Scholar
[11] and , Branching processes, Dover Publications, Inc., Mineola, NY, 2004, reprint of the 1972 original.
[12] , , , and , 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).Google Scholar
[13] , Weighted sums of certain dependent random variables, Tohoku Math. J. 19 (1967), no. 2, 357–367.Google Scholar
[15] and , Random and restricted walks: theory and applications, Gordon and Breach, New York, 1970.
[16] , On convergence to +∞ in the law of large numbers, Ann. Math. Statist. 34 (1963), 219–222.Google Scholar
[17] , , , and , A mixture of the exclusion process and the voter model, Bernoull. 7 (2001), no. 1, 119–144.Google Scholar
[18] , Vertex-reinforced random walks and a conjecture of Pemantle, Ann. Probab. 25 (1997), no. 1, 361–392.Google Scholar
[19] , , and , On the range of the simple random walk bridge on groups, Electron. J. Probab. 12 (2007), no. 20, 591–612.Google Scholar
[20] , , and , A balanced excited random walk, C. R. Math. Acad. Sci. Paris 349 (2011), no. 7–8, 459–462.
[22] and , Central limit theorem for the excited random walk in dimension D ≥ 2, Electron. Comm. Probab. 12 (2007), 303–314 (electronic).Google Scholar
[23] , Random walks in biology, expanded ed., Princeton University Press, New Jersey, 1993.
[24] and , Self-similar scaling limits of Markov chains on the positive integers, Ann. Appl. Probab (2016).
[25] , Probability and measure, third ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1995.
[26] , Random walk on spheres, Z. Wahrscheinlichkeitstheorie und Verw. Gebiet. 22 (1972), 169–192.Google Scholar
[27] , On transient Markov processes with a countable number of states and stationary transition probabilities, Ann. Math. Statist. 26 (1955), 654–658.Google Scholar
[28] and , A remark on the coin tossing game, Ann. Math. Statist. 35 (1964), 1345–1347.Google Scholar
[29] , , , and , Pearson's random walk approach to evaluating interference generated by a group of converters, Appl. Math. Comput. 219 (2013), no. 12, 6437–6444.Google Scholar
[30] , 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] and , Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston Inc., Boston, MA, 1985.
[32] and , Spatial analysis of animals' movements using a correlated random walk model, J. Theoret. Biol. 131 (1988), 419–433.Google Scholar
[33] , Transient atomic Markov chains with a denumerable number of states, Ann. Math. Statist. 29 (1958), 212–218.Google Scholar
[34] and , Random walks on randomly oriented lattices, Markov Process. Related Field. 9 (2003), no. 3, 391–412.Google Scholar
[35] , The history of the random-walk problem: considerations on the interdisciplinarity in modern physics, Riv. Nuovo Cimento (2. 7 (1977), no. 3, 419–427.Google Scholar
[36] , Moments and distribution of the local time of a two-dimensional random walk, Stochastic Process. Appl. 117 (2007), no. 2, 262–270.Google Scholar
[37] , On three classical problems for Markov chains with continuous time parameters, J. Appl. Probab. 28 (1991), no. 2, 305–320.Google Scholar
[38] and , Non-ergodicity criteria for denumerable continuous time Markov processes, Oper. Res. Lett. 32 (2004), no. 6, 574–580.Google Scholar
[39] and , 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).Google Scholar
[40] , , and , Moments of randomly stopped sums, Ann. Math. Statist. 36 (1965), 789–799.Google Scholar
[41] and , Probability theory: Independence, interchangeability, martingales, third ed., Springer Texts in Statistics, Springer-Verlag, New York, 1997.
[42] and , A note on Feller's strong law of large numbers, Ann. Probab. 14 (1986), no. 3, 1088–1094.Google Scholar
[43] and , 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] , On the maximum partial sums of sequences of independent random variables, Trans. Amer.Math. Soc. 64 (1948), 205–233.Google Scholar
[45] , Markov chains with stationary transition probabilities, Second ed. Die Grundlehren der mathematischenWissenschaften, Band 104, Springer-Verlag New York Inc., New York, 1967.
[46] , A course in probability theory, third ed., Academic Press Inc., San Diego, CA, 2001.
[47] and , On the distribution of values of sums of random variables, Mem. Amer.Math. Soc. 1951 (1951), no. 6, 12.Google Scholar
[49] , , and , Random walk models in biology, J. R. Soc. Interfac. 5 (2008), no. 25, 813–834.Google Scholar
[50] , Analysis of random walks, Studies in Probability, Optimization and Statistics, vol. 2, IOS Press, Amsterdam, 1992.
[51] , On the random walk with zero drifts in the first quadrant of R2, Comm. Statist. Stochastic Models 8 (1992), no. 3, 359–374.Google Scholar
[52] , , and , 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).Google Scholar
[53] , , , and , Random walk with barycentric self-interaction, J. Stat. Phys. 143 (2011), no. 5, 855–888.Google Scholar
[54] and , On multidimensional branching random walks in random environment, Ann. Probab. 35 (2007), no. 1, 68–114.Google Scholar
[55] and , Shape and local growth for multidimensional branching random walks in random environment, ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007), 273–299.Google Scholar
[56] , , , and , Billiards in a general domain with random reflections, Arch. Ration. Mech. Anal. 191 (2009), no. 3, 497–537.Google Scholar
[57] , , , and , Knudsen gas in a finite random tube: transport diffusion and first passage properties, J. Stat. Phys. 140 (2010), no. 5, 948–984.Google Scholar
[58] , , and , Lyapunov functions for random walks and strings in random environment, Ann. Probab. 26 (1998), no. 4, 1433–1445.Google Scholar
[59] and , Analysis of random walks using orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), no. 1–2, 387–399.Google Scholar
[61] , On the lower limits of maxima and minima of Wiener process and partial sums, Z. Wahrsch. Verw. Gebiet. 43 (1978), no. 3, 205–221.Google Scholar
[62] , , , and , On the local time of random walk on the 2-dimensional comb, Stochastic Process. Appl. 121 (2011), no. 6, 1290–1314.Google Scholar
[63] , , and , Joint asymptotic behavior of local and occupation times of random walk in higher dimension, Studia Sci.Math. Hungar. 44 (2007), no. 4, 535–563.Google Scholar
[64] , , and , Transient nearest neighbor random walk and Bessel process, J. Theoret. Probab. 22 (2009), no. 4, 992–1009.Google Scholar
[65] , , and , Transient nearest neighbor random walk on the line, J. Theoret. Probab. 22 (2009), no. 1, 100–122.Google Scholar
[66] , , and , 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.Google Scholar
[67] , , and , Random walk weakly attracted to a wall, J. Stat. Phys. 133 (2008), no. 2, 271–280.Google Scholar
[68] and , 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.Google Scholar
[69] and , Simple random walk on the line in random environment, Probab. Theory Relat. Field. 72 (1986), no. 2, 215–230.Google Scholar
[70] , 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] , , and , A note on transience versus recurrence for a branching random walk in random environment, J. Statist. Phys. 95 (1999), no. 3–4, 587–614.Google Scholar
[72] , , and , Potential analysis for positive recurrent Markov chains with asymptotically zero drift: power-type asymptotics, Stochastic Process. Appl. 123 (2013), no. 8, 3027–3051.Google Scholar
[73] and , 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.Google Scholar
[74] and , The strong law of large numbers when the first moment does not exist, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 586–587.Google Scholar
[75] , , , and , 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.Google Scholar
[76] , Éléments d'une théorie générale des chaines simples constantes de Markoff, Ann. Sci. Ècole Norm. Sup. (3. 57 (1940), 61–111.Google Scholar
[77] and , On the number of distinct sites visited by a random walk, Comm. Pure Appl. Math. 32 (1979), no. 6, 721–747.Google Scholar
[78] , Stochastic processes, John Wiley & Sons Inc., New York, 1953.
[79] , , , and , Practical drift conditions for subgeometric rates of convergence, Ann. Appl. Probab. 14 (2004), no. 3, 1353–1377.Google Scholar
[80] and , The transient behaviour of the simple random walk in the plane, J. Appl. Probab. 25 (1988), no. 1, 58–69.Google Scholar
[81] and , Random walks and electric networks, Carus Mathematical Monographs, vol. 22, Mathematical Association of America, Washington, DC, 1984.
[82] , 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] , Probability: theory and examples, fourth ed., Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010.
[84] and , 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] and , Stochastic interacting particle systems and nonlinear kinetic equations, Ann. Appl. Probab. 13 (2003), no. 3, 845–889.Google Scholar
[86] , Investigations on the theory of the Brownian movement, Dover Publications Inc., New York, 1956, Edited with notes by , translated by .
[87] , , and , How many matrices can be spectrally balanced simultaneously? Preprint. (2016).
[88] and , Some problems concerning the structure of random walk paths, Acta Math. Acad. Sci. Hungar. 11 (1960), 137–162.Google Scholar
[89] , The strong law of large numbers when the mean is undefined, Trans. Amer. Math. Soc. 185 (1973), 371–381.Google Scholar
[90] , Stochastic billiards on general tables, Ann. Appl. Probab. 11 (2001), no. 2, 419–437. References 349Google Scholar
[91] ', 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).Google Scholar
[92] ', 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).Google Scholar
[93] ', 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).Google Scholar
[95] , , and , 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.Google Scholar
[96] , , and , Topics in the constructive theory of countable Markov chains, Cambridge University Press, Cambridge, 1995.
[97] , A limit theorem for random variables with infinite moments, Amer. J. Math. 68 (1946), 257–262.Google Scholar
[98] , An introduction to probability theory and its applications. Vol. I, Third ed., John Wiley & Sons Inc., New York, 1968.
[99] , An introduction to probability theory and its applications. Vol. II., Second ed., John Wiley & Sons Inc., New York, 1971.
[100] , Shock fluctuations in asymmetric simple exclusion, Probab. Theory Related Field. 91 (1992), no. 1, 81–101.Google Scholar
[101] , 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] , , and , Microscopic structure of travelling waves in the asymmetric simple exclusion process, Ann. Probab. 19 (1991), no. 1, 226–244.Google Scholar
[103] , Criterion for ergodicity of homogeneous discrete Markov chains, Ukrain. Math. J. 41 (1989), 1223–1225, translated from Ukrain. Mat. Zh. 41 (1989) 1421–1422.Google Scholar
[105] , , and , An introduction to heavy-tailed and subexponential distributions, second ed., Springer Series in Operations Research and Financial Engineering, Springer, New York, 2013.
[106] and , 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.Google Scholar
[107] , Markoff chains with an enumerable number of states and a class of cascade processes, Math. Proc. Cambridge Philos. Soc. 47 (1951), 77–85.Google Scholar
[108] , On the stochastic matrices associated with certain queuing processes, Ann. Math. Statistic. 24 (1953), 355–360.Google Scholar
[109] , , and , Étude asymptotique d'une marche aléatoire centrifuge, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 2, 147–170.Google Scholar
[110] , , , and , Classification of Markov chains that describe the evolution of random strings, Uspekhi Mat. Nauk 50 (1995), no. 2(302), 5–24.
[111] , , and , Two-sided evolution of a random chain, Markov Process. Related Field. 1 (1995), no. 2, 281–316.Google Scholar
[112] , Comportement asymptotique des marches aléatoires associées aux polynomes de Gegenbauer et applications, Adv. Appl. Probab. 16 (1984), no. 2, 293–323.Google Scholar
[113] , , and , Localization for a random walk in slowly decreasing random potential, J. Stat. Phys. 150 (2013), no. 2, 285–298.Google Scholar
[114] , , , and , Anomalous recurrence properties of many-dimensional zero-drift random walks, Adv. Appl. Probab. 48 (2016), 99–118.Google Scholar
[115] and , Non-homogeneous random walks on a semi-infinite strip, Stochastic Process. Appl. 124 (2014), no. 10, 3179–3205.Google Scholar
[116] , Random polymer models, Imperial College Press, London, 2007.
[117] , Centrally biased discrete random walk, Quart. J. Math. Oxford Ser. (2. 7 (1956), 144–152.Google Scholar
[118] and , Extended Watson integrals for the cubic lattices, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1800–1801.Google Scholar
[119] , Limit distributions for random walks in a random environment, Dokl. Akad. Nauk SSS. 271 (1983), no. 1, 25–29.Google Scholar
[120] , , and , Asymptotic behaviour of critical controlled branching processes with random control functions, J. Appl. Probab. 42 (2005), no. 2, 463–477.Google Scholar
[121] , An integral test for the rate of escape of d-dimensional random walk, Ann. Probab. 11 (1983), no. 4, 953–961.Google Scholar
[122] and , On a stopping rule and the central limit theorem, Ann. Math. Statist. 38 (1967), 1915–1917.Google Scholar
[123] , Probability: A graduate course, Springer, 2005.
[124] , An almost sure invariance principle for the range of random walks, Stochastic Process. Appl. 78 (1998), no. 2, 131–143.Google Scholar
[125] and , Conditions for integrability of Markov chains, J. Appl. Probab. 32 (1995), no. 2, 541–547.Google Scholar
[126] and , A note on feedback queues with bulk service, J. Assoc. Comput. Mach. 19 (1972), 727–733.Google Scholar
[127] , First passage and recurrence distributions, Trans. Amer. Math. Soc. 73 (1952), 471–486.Google Scholar
[128] , A strong law for the maximum cumulative sum of independent random variables, Comm. Pure Appl. Math. 18 (1965), 109–127.Google Scholar
[129] , Jr and , Recurrence-time moments in random walks, Pacific J. Math. 3 (1953), 127–136.Google Scholar
[130] , Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30.Google Scholar
[131] , Random walk models for animal behavior, Statistical Ecology: vol. 2 (, , and , eds.), Penn State University Press, University Park, PA, 1971, pp. 1–12.
[132] and , Ergodic theorems for weakly interacting infinite systems and the voter model, Ann. Probabilit. 3 (1975), no. 4, 643–663.Google Scholar
[133] , , , and , Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips, Electron. J. Probab. 17 (2012), article 59.Google Scholar
[134] and , Long-time behaviour in a model of microtubule growth, Adv. Appl. Probab. 42 (2010), no. 1, 268–291.Google Scholar
[135] , , and , Excursions and path functionals for stochastic processes with asymptotically zero drifts, Stochastic Process. Appl. 123 (2013), no. 6, 1891–1921.Google Scholar
[136] , , and , Random walk in mixed random environment without uniform ellipticity, Proc. Stekov. Inst. Math. 282 (2013), 106–123.Google Scholar
[137] and , Large deviations view points for heavy-tailed random walks, J. Theoret. Probab. 17 (2004), no. 3, 761–768.Google Scholar
[138] and , The limits of Sinai's simple random walk in random environment, Ann. Probab. 26 (1998), no. 4, 1477–1521.Google Scholar
[139] , Random walks and random environments. Vol. 1, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1995.
[140] , Random walk with long-range interaction with a barrier and its dual: exact results, J. Comput. Appl. Math. 233 (2010), no. 10, 2449–2467.Google Scholar
[141] and , Maxima of partial sums of independent random variables, Z.Wahrscheinlichkeitstheorie und Verw. Gebiet. 27 (1973), 141–151.Google Scholar
[142] and , Polynomial convergence rates of Markov chains, Ann. Appl. Probab. 12 (2002), no. 1, 224–247.Google Scholar
[143] , On the theory of fluctuations in the decay of sound, J. Acoust. Soc. Amer. 11 (1940), 324–332.Google Scholar
[144] , The noisy voter-exclusion process, Stochastic Process. Appl. 115 (2005), no. 12, 1979–2005.Google Scholar
[145] , Practical stability of difference equations, Automation and Remote Contro. 9 (1967), 1404–1407, translated from Avtomatika i Telemekhanika 9 (1967) 172–175 (in Russian).Google Scholar
[146] , 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).Google Scholar
[147] , 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).Google Scholar
[148] , 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).Google Scholar
[149] , 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).Google Scholar
[150] , Foundations of modern probability, second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002.
[151] , A sufficient condition for nonergodicity of a Markov chain, IEEE Trans. Inform. Theor. 25 (1979), no. 4, 470–471.Google Scholar
[152] and , Representation of a class of stochastic processes, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 387–391.Google Scholar
[154] , , and , Limit theorems for one-dimensional diffusions and random walks in random environments, Probab. Theor. Relat. Field. 80 (1989), no. 4, 501–541.Google Scholar
[155] , , and , On the asymptotic behaviour of discrete time stochastic growth processes, Ann. Probab. 15 (1987), no. 1, 305–343.Google Scholar
[156] , Markovian functions of a Markov chain, Sankhya Ser.. 44 (1982), no. 3, 372–379.Google Scholar
[158] , Some problems in the theory of queues, J. Roy. Statist. Soc. Ser. B. 13 (1951), 151–173; discussion: 173–185.Google Scholar
[159] , On recurrence and transience of growth models, J. Appl. Probab. 23 (1986), no. 3, 614–625.Google Scholar
[160] , Asymptotic distribution for stochastic difference equations, Stochastic Process. Appl. 40 (1992), no. 1, 15–28.Google Scholar
[161] and , Sharp conditions for nonexplosions and explosions in Markov jump processes, Ann. Probab. 23 (1995), no. 1, 268–272.Google Scholar
[162] and , 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] , The limit points of a normalized random walk, Ann. Math. Statist. 41 (1970), 1173–1205.Google Scholar
[164] , The limit distribution of Sinai's random walk in random environment, Phys. A 138 (1986), no. 1–2, 299–309.
[165] and , Two renewal theorems for general random walks tending to infinity, Probab. Theor. Related Field. 106 (1996), no. 1, 1–38.Google Scholar
[166] and , Random walks crossing power law boundaries, Studia Sci. Math. Hungar. 34 (1998), no. 1–3, 219–252.Google Scholar
[167] , On the stability of the trajectory of Markov processes, J. Appl. Math. Mech. 26 (1962), 1554–1565.Google Scholar
[168] , 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] and , Tests for nonergodicity of denumerable continuous time Markov processes, Comput. Math. Appl. 55 (2008), no. 6, 1310–1321.Google Scholar
[173] , Stochastic difference equations and generalized gamma distributions, Ann. Probab. 17 (1989), no. 1, 178–188.Google Scholar
[174] , Introduction to stochastic calculus with applications, second ed., Imperial College Press, London, 2005.
[175] , Random walk on the quadrant, Stochastic Process. Appl. 26 (1987), 228.
[176] , Random walk in the quadrant, Ph.D. thesis, University of Amsterdam, 1996.
[177] and , Escape time for a random walk from an orthant, Stochastic Process. Appl. 35 (1990), no. 1, 1–9.Google Scholar
[178] , Path integrals in quantum mechanics, statistics, polymer physics, and financial markets, fourth ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
[179] , Kinetic theory of gases: some modern aspects, Methuen's Monographs on Physical Subjects, Methuen, London, 1952.
[180] , Moments of stationaryMarkov chains with asymptotically zero drift, Sibirsk. Mat. Zh. 52 (2011), no. 4, 829–840.Google Scholar
[181] and , Excited random walks: results, methods, open problems, Bull. Inst. Math. Acad. Sin. (N.S.. 8 (2013), no. 1, 105–157.Google Scholar
[182] and , A class of Markov chains with no spectral gap, Proc. Amer. Math. Soc. 141 (2013), no. 12, 4317–4326.Google Scholar
[183] , Random walk in a one-dimensional random medium, Teor. Verojatnost. i Primenen. 18 (1973), 406–408.Google Scholar
[184] , , and , Excited random walk with periodic cookies, Ann. Inst. H. Poincaré Probab. Statist. 52 (2016).Google Scholar
[185] , 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.Google Scholar
[186] and , 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] , On the stability of stochastic dynamical systems, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 8–12.Google Scholar
[188] , Finite time stochastic stability and the analysis of tracking systems, IEEE Trans. Automatic Control AC-11 (1966), 219–227.Google Scholar
[189] , Asymptotic growth of controlled Galton–Watson processes, Ann. Probab. 13 (1985), no. 4, 1157–1178.Google Scholar
[190] , Criteria for the recurrence or transience of stochastic process. I., J. Math. Anal. Appl. 1 (1960), 314–330.Google Scholar
[192] , Criteria for stochastic processes. II. Passage-time moments, J. Math. Anal. Appl. 7 (1963), 127–145.Google Scholar
[193] , Maximal branching processes and ‘long-range percolation’, J. Appl. Probab. 7 (1970), 89–98.Google Scholar
[194] , Remarks on maximal branching processes, Teor. Verojatnost. i Primenen. 17 (1972), 46–54.Google Scholar
[195] and , Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010.
[197] , Coupling the simple exclusion process, Ann. Probabilit. 4 (1976), no. 3, 339–356.Google Scholar
[198] , Interacting particle systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276, Springer-Verlag, New York, 1985.
[199] , Stochastic interacting systems: contact, voter and exclusion processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 324, Springer-Verlag, Berlin, 1999.
[200] , Probability theory. I, fourth ed., Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, Vol. 45.
[201] , Probability theory. I, fourth ed., Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, Vol. 45.
[202] (ed.), Markov chains and Monte Carlo calculations in polymer science, Marcel Dekker, New York, 1970.
[203] , , and , Recurrence and transience of multitype branching random walks, Stochastic Process. Appl. 91 (2001), no. 1, 21–37.Google Scholar
[204] and , One-dimensional branching random walks in a Markovian random environment, J. Appl. Probab. 37 (2000), no. 4, 1157–1163.Google Scholar
[205] and , Branching random walk in random environment on trees, Stochastic Process. Appl. 106 (2003), no. 1, 95–106.Google Scholar
[206] , , , and , A Markov chain model of a polling system with parameter regeneration, Ann. Appl. Probab. 17 (2007), no. 5-6, 1447–1473.Google Scholar
[207] , , and , Dynamics of the non-homogeneous supermarket model, Stoch. Model. 28 (2012), no. 4, 533–556.Google Scholar
[208] and , Critical random walks on two-dimensional complexes with applications to polling systems, Ann. Appl. Probab. 13 (2003), no. 4, 1399–1422.Google Scholar
[209] , , , and , Periodicity in the transient regime of exhaustive polling systems, Ann. Appl. Probab. 16 (2006), no. 4, 1816–1850. References 355Google Scholar
[210] , , , and , Passage-time moments and hybrid zones for the exclusion-voter model, Bernoull. 16 (2010), no. 4, 1312–1342.Google Scholar
[211] , , and , Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift, Markov Process. Related Field. 16 (2010), no. 2, 351–388.Google Scholar
[212] , , and , Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts, J. Theoret. Probab. 26 (2013), 1–30.Google Scholar
[213] and , 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).Google Scholar
[214] , 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).Google Scholar
[216] , Stochastic differential equations and applications, second ed., Horwood Publishing Limited, Chichester, 2008.
[217] and , Logarithmic averages for the local times of recurrent random walks and Lévy processes, Stochastic Process. Appl. 59 (1995), no. 2, 175–184.Google Scholar
[218] , On non-dissipative Markov chains, Proc. Cambridge Philos. Soc. 53 (1957), 825–835.Google Scholar
[219] and , Random paths in two and three dimensions, Proc. Roy. Soc. Edinburg. 60 (1940), 281–298.Google Scholar
[220] and , Explosion, implosion, and moments of passage times for continuous-time Markov chains: A semimartingale approach, Stochastic Process. Appl. 124 (2014), no. 7, 2388–2414.Google Scholar
[221] and , On range and local time of many-dimensional submartingales, J. Theoret. Probab. 27 (2014), no. 2, 601–617.Google Scholar
[222] , , , and , On a general many-dimensional excited random walk, Ann. Probab. 40 (2012), no. 5, 2106–2130.Google Scholar
[223] and , Urn-related random walk with drift ρxα/tβ, Electron. J. Probab. 13 (2008), no. 31, 944–960.Google Scholar
[224] and , Passage-time moments for continuous non-negative stochastic processes and applications, Adv. Appl. Probab. 28 (1996), no. 3, 747–762.Google Scholar
[225] and , Polling systems in the critical regime, Stochastic Process. Appl. 92 (2001), no. 2, 201–218.Google Scholar
[226] , 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).Google Scholar
[227] , Martingale approach for Markov processes in random environment and branching Markov chains, Resenha. 3 (1997), no. 2, 159–171. 356 References
Google Scholar
[228] , , and , Markov processes with asymptotically zero drifts, Probl. Inf. Transm. 31 (1995), 248–261, translated from Problemy Peredachi Informatsii 31 (1995) 60–75 (in Russian).Google Scholar
[229] , , and , Heavy-tailed random walks on complexes of half-lines, Preprint. (2016).
[230] and , Exact power estimates for countable Markov chains, Markov Process. Related Field. 1 (1995), no. 1, 57–78.Google Scholar
[231] , , , and , On a many-dimensional random walk in a rarefied random environment, Markov Process. Related Field. 10 (2004), no. 1, 137–160.Google Scholar
[232] , , and , Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains, J. Stat. Phys. 132 (2008), no. 6, 1097–1133.Google Scholar
[233] and , Branching Markov chains: qualitative characteristics, Markov Process. Related Field. 3 (1997), no. 2, 225–241.Google Scholar
[234] and , Random walk in random environment with asymptotically zero perturbation, J. Eur. Math. Soc. (JEMS. 8 (2006), no. 3, 491–513.Google Scholar
[235] and , Logarithmic speeds for one-dimensional perturbed random walks in random environments, Stochastic Process. Appl. 118 (2008), no. 3, 389–416.Google Scholar
[236] and , Rate of escape and central limit theorem for the supercritical Lamperti problem, Stochastic Process. Appl. 120 (2010), no. 10, 2078–2099.Google Scholar
[237] and , Recurrence of edge-reinforced random walk on a two-dimensional graph, Ann. Probab. 37 (2009), no. 5, 1679–1714.Google Scholar
[238] , , and , Necessary and sufficient conditions for recurrence and transience of Markov chains, in terms of inequalities, J. Appl. Probab. 15 (1978), no. 4, 848–851.Google Scholar
[239] and , Markov chains and stochastic stability, second ed., Cambridge University Press, Cambridge, 2009, With a prologue by .
[240] Jr., Foster's Markov chain theorems in continuous time, Tech. Report 88, Applied Mathematics and Statistics Laboratory, Stanford University, 1963.
[241] , An introduction to probability theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1984, Corrected reprint of the 1968 original.
[242] , Input-output Markov processes, Proc. Konin. Neder. Akad. Wetens. A. Math. Sci. 19 (1957), 112–118.Google Scholar
[243] , Markov chains, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2, Cambridge University Press, Cambridge, 1998.
[244] , Stochastic aspects of biological locomotion, J. Stat. Phys. 30 (1983), 391–400.Google Scholar
[245] and , A generalized Pearson random walk allowing for bias, J. Stat. Phys. 10 (1974), 245–253. References 357Google Scholar
[246] , General irreducible Markov chains and nonnegative operators, Cambridge Tracts in Mathematics, vol. 83, Cambridge University Press, Cambridge, 1984.
[247] , 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] and , Stochastic models for chain dynamics, Adv. Chem. Phys. 15 (1969), 305–324.Google Scholar
[249] , A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, TrudyMoskov. Mat. Obš?c. 19 (1968), 179–210.Google Scholar
[250] , Some conditions for ergodicity and recurrence of Markov chains, Operations Res. 17 (1969), 1058–1061.Google Scholar
[251] , Some remarks on a one-dimensional skip-free process with repulsion, J. Austral. Math. Soc. Ser. A
30 (1980/81), no. 1, 107–128.Google Scholar
[252] , Stochastic processes, Classics in Applied Mathematics, vol. 24, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999, Reprint of the 1962 original.
[253] , The problem of the random walk, Nature 72 (1905), 342.
[254] and , A mathematical theory of random migration, Drapers' Company Research Memoirs Biometric Series, Dulau and co., London, 1906.
[255] , A survey of random processes with reinforcement, Probab. Surv. 4 (2007), 1–79.Google Scholar
[256] and , Vertex-reinforced random walk on Z has finite range, Ann. Probab. 27 (1999), no. 3, 1368–1388.Google Scholar
[257] , , and , On recurrence and transience of self-interacting random walks, Bull. Braz. Math. Soc. 44 (2013), no. 4, 1–27.Google Scholar
[258] , Positive harmonic functions and diffusion, Cambridge Studies in Advanced Mathematics, vol. 45, Cambridge University Press, Cambridge, 1995.
[259] , Ü ber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Math. Ann. 84 (1921), no. 1–2, 149–160.Google Scholar
[260] , 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).Google Scholar
[262] and , On some generalized reinforced random walk on integers, Electron. J. Probab. 14 (2009), no. 60, 1770–1789.Google Scholar
[263] , 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.Google Scholar
[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] , Calcul des chances et philosophie de la bourse, Mallet- Bachelier/Castel, Paris, 1863.
[266] and , Random flights through spaces of different dimensions, J. Math. Phys. 56 (2015), no. 1, 013512.Google Scholar
[267] , A probability path, Birkhäuser Boston Inc., Boston, MA, 1999.
[268] , Competition processes, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, CA, 1961, pp. 421–430.
[269] , Random walk in random and non-random environments, third ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
[270] , Markov chains, second ed., North-Holland Mathematical Library, vol. 11, North-Holland Publishing Co., Amsterdam, 1984.
[271] , Stochastic networks and queues, Applications of Mathematics, vol. 52, Springer-Verlag, Berlin, 2003.
[273] and , 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).Google Scholar
[274] , Functions of a Markov process that are Markovian, J. Math. Mech. 8 (1959), 585–596.Google Scholar
[275] , A local limit theorem for a certain class of random walks, Ann. Math. Statist. 37 (1966), 855–859.Google Scholar
[276] , Recurrence and transience property for a class of Markov chains, Bernoulli 19 (2013), no. 5B, 2167–2199.
[277] , Recurrence and transience criteria for two cases of stable-like Markov chains, J. Theoret. Probab. 27 (2014), 754–788.Google Scholar
[278] , 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] , Diffusions with random coefficients, Ph.D. thesis, University of California, Los Angeles, 1984.
[280] , 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] , , and , Mean drifts and the nonergodicity of Markov chains, Oper. Res. 31 (1983), no. 4, 783–789.Google Scholar
[282] and , Stability of a growth process generated by monomer filling with nearest-neighbour cooperative effects, Stochastic Process. Appl. 120 (2010), no. 6, 926–948.Google Scholar
[284] , Recurrent random walks with arbitrarily large steps, Bull. Amer. Math. Soc. 70 (1964), 540–542.Google Scholar
[285] , , and , Passage-time moments for positively recurrent Markov chains, Nagoya Math. J. 162 (2001), 169–185.Google Scholar
[286] , Probability, second ed., Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1996, translated from the first (1980) Russian edition by .
[287] and (eds.), Random walks and their applications in the physical and biological sciences, American Institute of Physics, New York, 1984.
[288] , 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).Google Scholar
[289] , A slow transient diffusion in a drifted stable potential, J. Theoret. Probab. 20 (2007), no. 2, 153–166.Google Scholar
[290] , , , , , and , Stochastic modelling of animal movement, Phil. Trans. Roy. Soc. Ser. B Biol. Sci. 365 (2010), 2201–2211.Google Scholar
[292] , 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] , Principles of random walk, second ed., Springer-Verlag, New York, 1976, Graduate Texts in Mathematics, Vol. 34.
[294] and , Random-walk statistics and the spherical harmonic representation of cosmic microwave background maps, Mon. Not. Roy. Astron. Soc. 364 (2005), no. 3, 929–933.Google Scholar
[295] , Almost sure convergence, Academic Press, 1974, Probability and Mathematical Statistics, Vol. 24.
[296] , An introduction to Markov processes, Graduate Texts in Mathematics, vol. 230, Springer-Verlag, Berlin, 2005.
[297] and , Tightness of voter model interfaces, Electron. Commun. Probab. 13 (2008), 165–174.Google Scholar
[298] , Exit time from a positive quadrant for a two-dimensional Markov chain, Comm. Statist. Stochastic Model. 8 (1992), no. 3, 375–395.Google Scholar
[299] , On the asymptotic properties of diffusion processes, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 17 (1974), 69–71 (1975).Google Scholar
[300] , 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] , Vertex-reinforced random walk on Z eventually gets stuck on five points, Ann. Probab. 32 (2004), no. 3B, 2650–2701.Google Scholar
[302] , Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space, Stochastic Processes Appl. 3 (1975), no. 4, 385–403.Google Scholar
[303] , Sufficient conditions for regularity, recurrence and ergodicity of Markov processes, Math. Proc. Cambridge Philos. Soc. 78 (1975), 125–136.Google Scholar
[304] , Criteria for classifying general Markov chains, Adv. in Appl. Probab. 8 (1976), no. 4, 737–771.Google Scholar
[305] , Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes, J. Appl. Probab. 18 (1981), no. 1, 122–130.Google Scholar
[306] and , Monotonicity for excited random walk in high dimensions, Probab. Theory Related Fields
147 (2010), no. 1–2, 333–348.Google Scholar
[307] , Localization and delocalization of random interfaces, Probab. Surv. 3 (2006), 112–169.Google Scholar
[308] , On the rate of convergence for infinite server Erlang– Sevastyanov's problem, Queueing Syst. 76 (2014), no. 2, 181–203.Google Scholar
[309] and , Simple proof of Dynkin's formula for single-server systems and polynomial convergence rates, Markov Process. Related Field. 20 (2014), no. 3, 479–504.Google Scholar
[310] , Central limit theorems for random walks on N that are associated with orthogonal polynomials, J. Multivariate Anal. 34 (1990), no. 2, 290–322.Google Scholar
[311] , 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.Google Scholar
[312] , Vertex-reinforced random walk on arbitrary graphs, Ann. Probab. 29 (2001), no. 1, 66–91.Google Scholar
[313] , Explosion phenomena in stochastic coagulation-fragmentation models, Ann. Appl. Probab. 15 (2005), no. 3, 2081–2112.Google Scholar
[315] and , Random walks: theory and selected applications, Adv. Chem. Phys. 52 (1983), 363–505.Google Scholar
[316] , Liapunov criteria for weak stochastic stability, J. Differential Equation. 2 (1966), 195–207.Google Scholar
[317] , A Liapunov method for the estimation of statistical averages, J. Differential Equation. 2 (1966), 365–377.Google Scholar
[318] , On two-dimensional Markov chains in the positive quadrant with partial spatial homogeneity, Markov Process. Related Field. 1 (1995), no. 2, 267–280.Google Scholar
[319] , Random walks in random environment, Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1837, Springer, Berlin, 2004, pp. 189–312.
[321] , Recurrence and transience of excited random walks on Zd and strips, Electron. Comm. Probab. 11 (2006), 118–128 (electronic).Google Scholar