Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-669899f699-swprf Total loading time: 0 Render date: 2025-04-25T16:20:26.877Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  24 April 2025

Denis Denisov ,
Dmitry Korshunov  and
Vitali Wachtel
Denis Denisov
Affiliation:
University of Manchester
Dmitry Korshunov
Affiliation:
Lancaster University
Vitali Wachtel
Affiliation:
Universität Bielefeld, Germany
Get access

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.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Markov Chains with Asymptotically Zero Drift
Lamperti's Problem
 , pp. 400 - 407
Publisher: Cambridge University Press
Print publication year: 2025

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

References

Abramson, N.. The ALOHA system – another alternative for computer communications. Proc. Fall Joint Computer Conf., AFIPS Press, 37:281–285, 1970.Google Scholar
Albrecher, H. and Asmussen, S.. Ruin Probabilities, 2nd Ed. World Scientific Publishing, Hackensack, NJ, 2010.Google Scholar
Albrecher, H., Constantinescu, C., Palmowski, Z., Regensburger, G., and Rosenkranz, M.. Exact and asymptotic results for insurance risk models with surplus-dependent premiums. SIAM J. Appl. Math., 73:47–66, 2013.CrossRefGoogle Scholar
Aldous, D.. Probability Approximations Via the Poisson Clumping Heuristic. Springer, New York, 1989.CrossRefGoogle Scholar
Alexander, K. S.. Excursions and local limit theorems for Bessel-like random walks. Electron. J. Probab., 16:1–44, 2011.CrossRefGoogle Scholar
Alexander, K. S. and Zygouras, N.. Quenched and annealed critical points in polymer pinning models. Commun. Math. Phys., 291:659–689, 2009.CrossRefGoogle Scholar
Asmussen, S.. Ruin Probabilities. World Scientific, Singapore, 2000.CrossRefGoogle Scholar
Asmussen, S.. Applied Probability and Queues. Springer, New York, 2003.Google Scholar
Aspandiiarov, S. and Iasnogorodski, R.. Asymptotic behaviour of stationary distributions for countable Markov chains, with some applications. Bernoulli, 5:535–569, 1999.CrossRefGoogle Scholar
Athreya, K. B., McDonald, D., and Ney, P.. Limit theorems for semi-Markov processes and renewal theory for Markov chains. Ann. Probab., 6:788–797, 1978.CrossRefGoogle Scholar
Babillot, M., Bougerol, P., and Elie, L.. The random difference equation Xn = AnXn−1 + Bn in the critical case. Ann. Probab., 25:478–493, 1997.CrossRefGoogle Scholar
Berger, Q.. Strong renewal theorems and local large deviations for multivariate random walks and renewals. Electron. J. Probab., 24:47 pp., 2019.CrossRefGoogle Scholar
Bertoin, J. and Doney, R. A.. On conditioning a random walk to stay nonnegative. Ann. Probab., 22:2152–2167, 1994.CrossRefGoogle Scholar
Bertoin, J. and Doney, R. A.. Some asymptotic results for transient random walks. Adv. Appl. Probab., 28:207–226, 1996.CrossRefGoogle Scholar
Bertoin, J. and Kortchemski, I.. Self-similar scaling limits of Markov chains on the positive integers. Ann. Appl. Probab., 26:2556–2595, 2016.CrossRefGoogle Scholar
Billingsley, P.. Convergence of Probability Measures. J. Wiley & Sons, New York, 1968.Google Scholar
Bingham, N., Goldie, C., and Teugels, J.. Regular Variation. Cambridge University Press, Cambridge, 1987.CrossRefGoogle Scholar
Blackwell, D.. A renewal theorem. Duke Math. J., 15:145–150, 1948.CrossRefGoogle Scholar
Blackwell, D.. Extension of a renewal theorem. Pacific J. Math., 3:315–320, 1953.CrossRefGoogle Scholar
Bolthausen, E.. On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab., 3:480–485, 1976.Google Scholar
Borodin, A. N. and Salminen, P.. Handbook of Brownian Motion – Facts and Formulae, 2nd Ed. Birkhäuser, Basel, 2002.CrossRefGoogle Scholar
Borovkov, A. and Borovkov, K.. Asymptotic Analysis of Random Walks. Heavy-Tailed Distributions. Cambridge University Press, Cambridge, 2008.CrossRefGoogle Scholar
Borovkov, A. A.. Ergodicity and Stability of Stochastic Processes. John Wiley, Chichester, 1998.Google Scholar
Borovkov, A. A. and Korshunov, D.. Large-deviation probabilities for onedimensional Markov chains. Part 1: Stationary distributions. Theory Probab. Appl., 41:1–24, 1997.CrossRefGoogle Scholar
Borovkov, A. A., Fayolle, G., and Korshunov, D.. Transient phenomena for Markov chains and their applications. Adv. Appl. Probab., 24:322–342, 1992.CrossRefGoogle Scholar
Boxma, O. and Mandjes, M.. Affine storage and insurance risk models. Math. Oper. Res., 46:1282–1302, 2021.CrossRefGoogle Scholar
Brézis, H., Rosenkrantz, W., and Singer, B.. An extension of Khintchine’s estimate for large deviations to a class of Markov chains converging to a singular diffusion. Comm. Pure Appl. Math., 24:705–726, 1971.CrossRefGoogle Scholar
Brofferio, S. and Buraczewski, D.. On unbounded invariant measures of stochastic dynamical systems. Ann. Probab., 43:1456–1492, 2015.CrossRefGoogle Scholar
Brown, B. M.. Martingale central limit theorems. Ann. Math. Statist., 42:59–66, 1971.CrossRefGoogle Scholar
Bryn-Jones, A. and Doney, R. A.. A functional limit theorem for random walks conditioned to stay non-negative. J. Lond. Math. Soc. (2), 74:244–258, 2006.CrossRefGoogle Scholar
Buraczewski, D., Damek, E., and Mikosch, T.. Stochastic Models with Power-Law Tails. The Equation X = AX + B. Springer, Switzerland, 2016.CrossRefGoogle Scholar
Caravenna, F. and Chaumont, L.. Invariance principles for random walks conditioned to stay positive. Ann. Inst. Henri Poincaré Probab. Statist., 44:170–190, 2008.CrossRefGoogle Scholar
Caravenna, F. and Doney, R.. Local large deviations and the strong renewal theorem. Electron. J. Probab., 24:48 pp., 2019.CrossRefGoogle Scholar
Cherny, A. S. and Engelbert, H.-J.. Singular Stochastic Differential Equations. Springer, Berlin, 2005.CrossRefGoogle Scholar
Choquet, G. and Deny, J.. Sur l’équation de convolution μ = μ * σ. C. R. Acad. Sci. Paris Série A, 250:799–801, 1960.Google Scholar
Chung, K. L. and Walsh, J. B.. Markov Processes, Brownian Motion, and Time Symmetry. Springer, 2005.CrossRefGoogle Scholar
Cox, D. R. and Smith, W. L.. A direct proof of a fundamental theorem of renewal theory. Scand. Actuarial J., 36:139–150, 1953.CrossRefGoogle Scholar
Cramér, H.. Collective Risk Theory. Esselte, Stockholm, 1955.Google Scholar
Csáki, E., Földes, A., and Révész, P.. Transient nearest neighbor random walk and Bessel process. J. Theoret. Probab., 22:992–1009, 2009.CrossRefGoogle Scholar
Czarna, I., Pérez, J.-L., Rolski, T., and Yamazaki, K.. Fluctuation theory for leveldependent Lévy risk processes. Stochastic Process. Appl., 129:5406–5449, 2019.CrossRefGoogle Scholar
De Coninck, J., Dunlop, F., and Huilett, T.. Random walk weakly attracted to a wall. J. Stat. Phys., 133:271–280, 2008.CrossRefGoogle Scholar
Denisov, D., Korshunov, D., and Wachtel, V.. Potential analysis for positive recurrent Markov chains with asymptotically zero drift: power-type asymptotics. Stochastic Process. Appl., 123:3027–3051, 2013.CrossRefGoogle Scholar
Denisov, D., Korshunov, D., and Wachtel, V.. Markov chains on : analysis of stationary measure via harmonic functions approach. Queueing Syst., 91: 265–295, 2019.CrossRefGoogle Scholar
Denisov, D., Korshunov, D., and Wachtel, V.. Renewal theory for transient Markov chains with asymptotically zero drift. Trans. Amer. Math. Soc., 373:7253–7286, 2020.CrossRefGoogle Scholar
Denisov, D., Gotthardt, N., Korshunov, D., and Wachtel, V.. Probabilistic approach to risk processes with level-dependent premium rate. Insur. Math. Econ., 118:142–156, 2024.CrossRefGoogle Scholar
Denisov, D. E.. On the existence of a regularly varying majorant of an integrable monotone function. Math. Notes, 76:129–133, 2006.CrossRefGoogle Scholar
Dette, H.. First return probabilities of birth and death chains and associated orthogonal polynomials. Proc. Amer. Math. Soc., 129:1805–1815, 2001.CrossRefGoogle Scholar
Dharmadhikari, S. W. and Jogdeo, K.. Bounds on moments of certain random variables. Ann. Math. Statist., 40:1506–1509, 1969.CrossRefGoogle Scholar
Doney, R. A.. An analogue of the renewal theorem in higher dimensions. Proc. London Math. Soc., 16:669–684, 1966.CrossRefGoogle Scholar
Doney, R. A.. Local behaviour of first passage probabilities. Probab. Theory Related Fields, 152:559–588, 2012.CrossRefGoogle Scholar
Doney, R. A. and Jones, E. M.. Large deviation results for random walk conditioned to stay positive. Electron. Commun. Probab., paper no. 38:11 pp., 2012.Google Scholar
Doob, J. L.. Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France, 85:431–458, 1957.CrossRefGoogle Scholar
Doob, J. L.. Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag, New York, 1984.CrossRefGoogle Scholar
Duraj, J. and Wachtel, V.. Invariance principles for random walks in cones. Stochastic Process. Appl., 130:3920–3942, 2020.CrossRefGoogle Scholar
Durrett, R.. Conditioned limit theorems for some null recurrent Markov chains. Ann. Probab., 6:798–828, 1978.CrossRefGoogle Scholar
Durrett, R.. Probability – Theory and Examples, 5th Ed. Cambridge University Press, Cambridge, 2019.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C., and Mikosch, T.. Modelling Extremal Events for Insurance and Finance. Springer, Berlin, 1997.CrossRefGoogle Scholar
Erdös, P., Feller, W., and Pollard, H.. A property of power series with positive coefficients. Bull. Amer. Math. Soc., 55:201–204, 1949.CrossRefGoogle Scholar
Erickson, K. B.. Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc., 151:263–291, 1970.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G.. Markov Processes. Characterization and Convergence. J. Wiley & Sons, New York, 1986.CrossRefGoogle Scholar
Fayolle, G., Malyshev, V. A., and Menshikov, M. V.. Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
Feller, W.. On the integral equation of renewal theory. Ann. Math. Statist., 12: 243–267, 1941.CrossRefGoogle Scholar
Feller, W.. An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York, 1971.Google Scholar
Feller, W. and Orey, S.. A renewal theorem. J. Math. Mech., 10:619–624, 1961.Google Scholar
Flajolet, P. and Sedgewick, R.. Analytic Combinatorics. Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Foley, R. D. and McDonald, D. R.. Constructing a harmonic function for an irreducible nonnegative matrix with convergence parameter R > 1. Bull. London Math. Soc., 44:533–544, 2012.CrossRefGoogle Scholar
Foss, S., Korshunov, D., and Zachary, S.. An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd Ed. Springer, New York, 2013.CrossRefGoogle Scholar
Foster, F. G.. On the stochastic matrices associated with certain queueing processes. Ann. Math. Statist., 24:355–360, 1953.CrossRefGoogle Scholar
Foster, J.. A limit theorem for a branching process with state-dependent immigration. Ann. Math. Statist., 42:1773–1776, 1971.CrossRefGoogle Scholar
Fuk, D. X. and Nagaev, S. V.. Probabilistic inequalities for sums of independent random variables. Theor. Probab. Appl., 16:643–660, 1971.CrossRefGoogle Scholar
Gaenssler, P., Strobel, J., and Stute, W.. On central limit theorems for martingale triangular arrays. Acta Mathematica Acad. Sci. Hungaricae, 31:205–216, 1978.CrossRefGoogle Scholar
Garsia, A. and Lamperti, J.. A discrete renewal theorem with infinite mean. Comment. Math. Helv., 37:221–234, 1962.CrossRefGoogle Scholar
Goldie, C. M.. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab., 1:126–166, 1991.CrossRefGoogle Scholar
Gradshteyn, I. S. and Ryzhik, I. M.. Table of Integrals, Series, and Products, 7th Ed. Elsevier/Academic Press, Amsterdam, 2007.Google Scholar
Guibourg, D. and Hervé, L.. Multidimensional renewal theory in the non-centered case. Application to strongly ergodic Markov chains. Potential Anal., 38: 471–497, 2013.CrossRefGoogle Scholar
Guivarc’h, Y., Keane, M., and Roynette, B.. Marches Aléatoires sur les Groupes de Lie (in French). Springer, Berlin–New York, 1977.CrossRefGoogle Scholar
Harris, T. E.. First passage and recurrence distributions. Trans. Amer. Math. Soc., 73:471–486, 1952.CrossRefGoogle Scholar
Hitczenko, P. and Wesolowski, J.. Renorming divergent perpetuities. Bernoulli, 17:880–894, 2011.CrossRefGoogle Scholar
Hodges, J. L. and Rosenblatt, M. Jr. Recurrence-time moments in random walks. Pacific J. Math., 3:127–136, 1953.CrossRefGoogle Scholar
Hoffman-Jørgensen, J. and Pisier, G.. The law of large numbers and the central limit theorem in Banach spaces. Ann. Probab., 4:587–599, 1976.CrossRefGoogle Scholar
Höpfner, R.. On some classes of population-size-dependent Galton–Watson processes. J. Appl. Probab., 22:25–36, 1985.CrossRefGoogle Scholar
Höpfner, R.. Some results on population-size-dependent Galton–Watson processes. J. Appl. Probab., 23:297–306, 1986.CrossRefGoogle Scholar
Hryniv, O., Menshikov, M. V., and Wade, A. R.. Excursions and path functionals for stochastic processes with asymptotically zero drift. Stochastic Process. Appl., 123:1891–1921, 2013.CrossRefGoogle Scholar
Huillet, T.. Random walk with long-range interaction with a barrier and its dual: exact results. J. Comput. Appl. Math., 233:2449–2467, 2010.CrossRefGoogle Scholar
Iglehart, D.. Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab., 2:608–619, 1974.CrossRefGoogle Scholar
Karlin, S. and McGregor, J.. Random walks. Illinois J. Math., 3:66–81, 1959.CrossRefGoogle Scholar
Karlin, S. and Taylor, H. M.. A First Course in Stochastic Processes, 2nd Ed. Academic Press, New York, 1975.Google Scholar
Karlin, S. and Taylor, H. M.. A Second Course in Stochastic Processes. Academic Press, New York, 1981.Google Scholar
Kaverin, S. V.. A refinement of limit theorems for critical branching processes with an emigration. Theory Probab. Appl., 35:574–580, 1990.CrossRefGoogle Scholar
Keller, G., Kersting, G., and Rösler, U.. On the asymptotic behaviour of discrete time stochastic growth processes. Ann. Probab., 15:305–343, 1987.CrossRefGoogle Scholar
Kemperman, J. H. B.. The oscillating random walk. Stochastic Process. Appl., 2: 1–29, 1974.CrossRefGoogle Scholar
Kersting, G.. On recurrence and transience of growth models. J. Appl. Probab., 23:614–625, 1986.CrossRefGoogle Scholar
Kersting, G.. A law of large numbers for stochastic difference equations. Stochastic Process. Appl., 40:1–13, 1992.CrossRefGoogle Scholar
Kersting, G.. Asymptotic Γ-distribution for stochastic difference equations. Stochastic Process. Appl., 40:15–28, 1992.CrossRefGoogle Scholar
Kesten, H.. Random difference equations and renewal theory for products of random matrices. Acta Math., 131:207–248, 1973.CrossRefGoogle Scholar
Kesten, H.. Renewal theory for functionals of a Markov chain with general state space. Ann. Probab., 2:355–386, 1974.CrossRefGoogle Scholar
Kesten, H. and Spitzer, F.. Ratio theorems for random walks I. J. Anal. Math., 11: 285–322, 1963.CrossRefGoogle Scholar
Klebaner, F. C.. On population size dependent branching processes. Adv. Appl. Probab., 16:30–55, 1984.CrossRefGoogle Scholar
Klebaner, F. C.. Stochastic difference equations and generalized gamma distributions. Ann. Probab., 17:178–188, 1989.CrossRefGoogle Scholar
Klüppelberg, C. and Pergamenchtchikov, S.. Renewal theory for functionals of a Markov chain with compact state space. Ann. Probab., 31:2270–2300, 2003.CrossRefGoogle Scholar
Korshunov, D. A.. Tightness and continuity of a family of invariant measures for Markov chains depending on a parameter. Siberian Math. J., 37:730–746, 1996.CrossRefGoogle Scholar
Korshunov, D. A.. On distribution tail of the maximum of a random walk. Stochastic Process. Appl., 72:97–103, 1997.CrossRefGoogle Scholar
Korshunov, D. A.. Limit theorems for general Markov chains. Siberian Math. J., 42:301–316, 2001.CrossRefGoogle Scholar
Korshunov, D. A.. One-dimensional asymptotically homogeneous Markov chains: Cramér transform and large deviation probabilities. Siberian Adv. Math., 14(4):30–70, 2004.Google Scholar
Korshunov, D. A.. The key renewal theorem for a transient Markov chain. J. Theoret. Probab., 21:234–245, 2008.CrossRefGoogle Scholar
Korshunov, D. A.. Moments for stationary Markov chains with asymptotically zero drift. Siberian Math. J., 52:655–664, 2011.CrossRefGoogle Scholar
Korshunov, D. A.. A look at perpetuities via asymptotically homogeneous in space Markov chains. arXiv: 1603.08410, 2016.Google Scholar
Kosygina, E. and Mountford, T.. Limit laws of transient excited random walks on integers. Ann. Inst. Henri Poincaré Probab. Stat., 47:575–600, 2011.CrossRefGoogle Scholar
Kozlov, M. V.. The asymptotic behavior of the probability of non-extinction of critical branching processes in a random environment. (In Russian. English summary). Teor. Verojatnost. i Primenen., 21:813–825, 1976.Google Scholar
Lam, S. S. and Kleinrock, L.. Packet switching in a multiaccess broadcast channel: dynamic control procedures. IEEE Trans. Commun., 24:891–904, 1975.CrossRefGoogle Scholar
Lamperti, J.. Criteria for the recurrence or transience of stochastic processes I. J. Math. Anal. Appl., 1:314–330, 1960.CrossRefGoogle Scholar
Lamperti, J.. A new class of probability limit theorems. J. Math. Mech., 11: 749–772, 1962.Google Scholar
Lamperti, J.. Criteria for stochastic processes ii: passage time moments. J. Math. Anal. Appl., 7:127–145, 1963.CrossRefGoogle Scholar
Lamperti, J.. Maximal branching processes and ‘long-range percolation’. J. Appl. Probab., 7:89–98, 1970.CrossRefGoogle Scholar
Lamperti, J.. Remarks on maximal branching processes. Theory Probab. Appl., XVII:46–54, 1972.Google Scholar
Levin, D. and Peres, Y.. Markov Chains and Mixing Times, 2nd Ed. American Mathematical Society, 2017.CrossRefGoogle Scholar
Li, B., Ni, W., and Constantinescu, C.. Risk models with premiums adjusted to claims number. Insurance Math. Econom., 65:94–102, 2015.CrossRefGoogle Scholar
Maejima, M.. On local limit theorems and Blackwell’s renewal theorem for independent random variables. Ann. Inst. Stat. Math., 27:507–520, 1975.CrossRefGoogle Scholar
Malyshev, V. A. and Menshikov, M. V.. Ergodicity, continuity and analyticity of countable Markov chains. Trans. Moscow Math. Soc., 1:1–48, 1982.Google Scholar
Marciniak, E. and Palmowski, Z.. On the optimal dividend problem for insurance risk models with surplus-dependent premiums. J. Optim. Theory Appl., 168: 723–742, 2016.CrossRefGoogle Scholar
Menshikov, M., Popov, S., and Wade, A.. Non-Homogeneous Random Walks. Lyapunov Function Methods for Near-Critical Stochastic Systems. Cambridge University Press, Cambridge, 2017.Google Scholar
Menshikov, M. V. and Popov, S. Y.. Exact power estimates for countable Markov chains. Markov Proc. Relat. Fields, 1:57–78, 1995.Google Scholar
Menshikov, M. V. and Wade, A. R.. Rate of escape and central limit theorem for the supercritical Lamperti problem. Stochastic Process. Appl., 120:2078–2099, 2010.CrossRefGoogle Scholar
Menshikov, M. V., Asymont, I. M., and Yasnogorodskii, R.. Markov processes with asymptotically zero drifts. Probl. Inform. Trans., 31:248–261, 1995.Google Scholar
Menshikov, M. V., Vachkovskaia, M., and Wade, A. R.. Asymptotic behaviour of randomly reflecting billiards in unbounded domains. J. Stat. Phys., 132: 1097–1133, 2008.CrossRefGoogle Scholar
Meyn, S. and Tweedie, R.. Markov Chains and Stochastic Stability. Wiley, 1993.CrossRefGoogle Scholar
Moustafa, M. D.. Input–output Markov processes. Proc. Koninkl. Ned. Akad. Wetensch. Ser. A, 60:112–118, 1957.Google Scholar
Nagaev, A. V.. Renewal theorems in . Theory Probab. Appl., 24:572–581, 1980.CrossRefGoogle Scholar
Nagaev, S. V.. Large deviations of sums of independent random variables. Ann. Probab., 7:745–789, 1979.CrossRefGoogle Scholar
Nagaev, S. V. and Khan, L. V.. Limit theorems for a critical Galton–Watson branching process with migration. Theory Probab. Appl., 24:514–525, 1980.Google Scholar
Nagaev, S. V. and Wachtel, V. I.. Probability inequalities for a critical Galton–Watson process. Theory Probab. Appl., 50:225–247, 2006.CrossRefGoogle Scholar
Petrov, V. V.. Sums of Independent Random Variables. Springer, Berlin, 1975.Google Scholar
Pitman, J.. One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. Appl. Probab., 7:511–526, 1975.CrossRefGoogle Scholar
Popov, S.. Two-Dimensional Random Walk – from Path Counting to Random Interlacements. Cambridge University Press, Cambridge, 2021.Google Scholar
Revuz, D. and Yor, M.. Continuous Martingales and Brownian Motion, 3rd Ed. Springer, Berlin, 1999.CrossRefGoogle Scholar
Rogozin, B.. Asymptotics of renewal functions. Theory Probab. Appl., 21: 669–686, 1977.CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J.. Stochastic Processes for Insurance and Finance. Wiley, Chichester, 1998.Google Scholar
Rosenkrantz, W. A.. A local limit theorem for a certain class of random walks. Ann. Math. Statist., 37:855–859, 1966.CrossRefGoogle Scholar
Sandrić, N.. Recurrence and transience property for a class of Markov chains. Bernoulli, 19:2167–2199, 2013.CrossRefGoogle Scholar
Seneta, E.. An explicit-limit theorem for the critical Galton–Watson process with immigration. J. Royal Statist. Soc. Ser. B, 32:149–152, 1970.CrossRefGoogle Scholar
Shurenkov, V.. On Markov renewal theory. Theory Probab. Appl., 29:247–265, 1984.CrossRefGoogle Scholar
Smith, W. L.. On some general renewal theorems for nonidentically distributed variables. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. 2: Contributions to Probability Theory, 467–514, 1961.Google Scholar
Spitzer, F.. Principles of Random Walk. D. Van Nostrand, Princeton, N.J.– Toronto–London, 1964.CrossRefGoogle Scholar
Vatutin, V. A.. The asymptotic probability of the first degeneration for branching processes with immigration. Theory Probab. Appl., 19:25–34, 1974.CrossRefGoogle Scholar
Vatutin, V. A.. A critical Galton–Watson branching process with emigration. Theory Probab. Appl., 22:465–481, 1977.CrossRefGoogle Scholar
Vatutin, V. A.. A conditional limit theorem for a critical branching process with immigration. Math. Notes, 21:405–411, 1977.CrossRefGoogle Scholar
Vervaat, W.. On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Probab., 11:750–783, 1979.CrossRefGoogle Scholar
Voit, M.. Strong laws of large numbers for random walks associated with a class of one-dimensional convolution structures. Monatsh. Math., 113:59–74, 1992.CrossRefGoogle Scholar
Voit, M.. Central limit theorems for Markov processes associated with Laguerre polynomials. J. Math. Anal. Appl., 182:731–741, 1994.CrossRefGoogle Scholar
Vysotsky, V.. Limit theorems for random walks that avoid bounded sets, with applications to the largest gap problem. Stochastic Process. Appl., 125: 1886–1910, 2015.CrossRefGoogle Scholar
Williamson, J. A.. Some renewal theorems for non-negative independent random variables. Trans. Amer. Math. Soc., 114:417–445, 1965.CrossRefGoogle Scholar
Woess, W.. Denumerable Markov Chains. European Mathematical Society, Zurich, 2009.CrossRefGoogle Scholar
Zubkov, A. M.. The life spans of a branching process with immigration. Theory Probab. Appl., 17:174–183, 1972.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Denis Denisov, University of Manchester, Dmitry Korshunov, Lancaster University, Vitali Wachtel, Universität Bielefeld, Germany
  • Book: Markov Chains with Asymptotically Zero Drift
  • Online publication: 24 April 2025
  • Chapter DOI: https://doi.org/10.1017/9781009554237.013
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Denis Denisov, University of Manchester, Dmitry Korshunov, Lancaster University, Vitali Wachtel, Universität Bielefeld, Germany
  • Book: Markov Chains with Asymptotically Zero Drift
  • Online publication: 24 April 2025
  • Chapter DOI: https://doi.org/10.1017/9781009554237.013
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Denis Denisov, University of Manchester, Dmitry Korshunov, Lancaster University, Vitali Wachtel, Universität Bielefeld, Germany
  • Book: Markov Chains with Asymptotically Zero Drift
  • Online publication: 24 April 2025
  • Chapter DOI: https://doi.org/10.1017/9781009554237.013
Available formats
×