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Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption. (English) Zbl 1430.60042
Summary: Consider a Markov chain \((X_{n})_{n\geq 0}\) with values in the state space \(\mathbb{X}\). Let \(f\) be a real function on \(\mathbb{X}\) and set \(S_{n}=\sum_{i=1}^{n}f(X_{i})\), \(n\geq1\). Let \(\mathbb{P}_{x}\) be the probability measure generated by the Markov chain starting at \(X_{0}=x\). For a starting point \(y\in\mathbb{R}\), denote by \(\tau_{y}\) the first moment when the Markov walk \((y+S_{n})_{n\geq 1}\) becomes nonpositive. Under the condition that \(S_{n}\) has zero drift, we find the asymptotics of the probability \(\mathbb{P}_{x}(\tau_{y}>n)\) and of the conditional law \(\mathbb{P}_{x}(y+S_{n}\leq \cdot\sqrt{n}\mid\tau_{y}>n)\) as \(n\rightarrow +\infty\).

MSC:
60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
60J50 Boundary theory for Markov processes
60J05 Discrete-time Markov processes on general state spaces
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60G40 Stopping times; optimal stopping problems; gambling theory
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References:
[1] Bertoin, J. and Doney, R. A. (1994). On conditioning a random walk to stay nonnegative. Ann. Probab.22 2152-2167. · Zbl 0834.60079
[2] Bolthausen, E. (1976). On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab.4 480-485. · Zbl 0336.60024
[3] Borovkov, A. A. (2004). On the asymptotic behavior of distributions of first-passage times, I. Math. Notes75 23-37. · Zbl 1108.60039
[4] Borovkov, A. A. (2004). On the asymptotics of distributions of first-passage times, II. Math. Notes75 322-330. · Zbl 1138.60035
[5] Buraczewski, D., Damek, E. and Guivarc’h, Y. (2010). Convergence to stable laws for a class of multidimensional stochastic recursions. Probab. Theory Related Fields148 333-402. · Zbl 1206.60025
[6] Caravenna, F. (2005). A local limit theorem for random walks conditioned to stay positive. Probab. Theory Related Fields133 508-530. · Zbl 1080.60045
[7] Dembo, A., Ding, J. and Gao, F. (2013). Persistence of iterated partial sums. Ann. Inst. Henri Poincaré Probab. Stat.49 873-884. · Zbl 1274.60144
[8] Denisov, D., Korshunov, D. and Wachtel, V. (2013). Harmonic functions and stationary distributions for asymptotically homogeneous transition kernels on \(\mathbb{z}^{+}\). Preprint. Available at arXiv:1312.2201[math].
[9] Denisov, D. and Wachtel, V. (2010). Conditional limit theorems for ordered random walks. Electron. J. Probab.15 292-322. · Zbl 1201.60040
[10] Denisov, D. and Wachtel, V. (2015). Exit times for integrated random walks. Ann. Inst. Henri Poincaré Probab. Stat.51 167-193. · Zbl 1310.60049
[11] Denisov, D. and Wachtel, V. (2015). Random walks in cones. Ann. Probab.43 992-1044. · Zbl 1332.60066
[12] Doney, R. A. (1989). On the asymptotic behaviour of first passage times for transient random walk. Probab. Theory Related Fields81 239-246. · Zbl 0643.60053
[13] Duraj, J. (2014). Random walks in cones: The case of nonzero drift. Stochastic Process. Appl.124 1503-1518. · Zbl 1319.60088
[14] Eichelsbacher, P. and König, W. (2008). Ordered random walks. Electron. J. Probab.13 1307-1336. · Zbl 1189.60092
[15] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. Wiley, New York. · Zbl 0219.60003
[16] Gao, Z., Guivarc’h, Y. and Le Page, É. (2015). Stable laws and spectral gap properties for affine random walks. Ann. Inst. Henri Poincaré Probab. Stat.51 319-348. · Zbl 1330.60016
[17] Gordin, M. I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR188 739-741. · Zbl 0212.50005
[18] Grama, I., Lauvergnat, R. and Le Page, É. (2018). Limit theorems for affine Markov walks conditioned to stay positive. Ann. Inst. Henri Poincaré Probab. Stat.54 529-568. · Zbl 1396.60080
[19] Grama, I., Le Page, É. and Peigné, M. (2014). On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains. Colloq. Math.134 1-55. · Zbl 1302.60057
[20] Grama, I., Le Page, É. and Peigné, M. (2017). Conditioned limit theorems for products of random matrices. Probab. Theory Related Fields168 601-639. · Zbl 1406.60015
[21] Guivarc’h, Y. and Hardy, J. (1988). Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. Henri Poincaré B, Probab. Stat.24 73-98. · Zbl 0649.60041
[22] Guivarc’h, Y. and Le Page, E. (2008). On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks. Ergodic Theory Dynam. Systems28 423-446. · Zbl 1154.37306
[23] Iglehart, D. L. (1974). Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab.2 608-619. · Zbl 0299.60053
[24] Iglehart, D. L. (1974). Random walks with negative drift conditioned to stay positive. J. Appl. Probab.11 742-751. · Zbl 0302.60038
[25] Ionescu Tulcea, C. T. and Marinescu, G. (1950). Théorie ergodique pour des classes d’opérations non complètement continues. Ann. of Math. (2) 52 140-147. · Zbl 0040.06502
[26] Kato, T. (1976). Perturbation Theory for Linear Operators. 2nd ed. Springer, Berlin. · Zbl 0342.47009
[27] Lévy, P. (1937). Théorie de L’addition des Variables Aléatoires. Gauthier-Villars, Paris. · Zbl 0016.17003
[28] Norman, M. F. (1972). Markov Processes and Learning Models. Academic Press, New York. · Zbl 0262.92003
[29] Presman, È. L. (1967). A boundary value problem for the sum of lattice random variables given on a finite regular Markov chain. Teor. Verojatnost. i Primenen.12 373-380.
[30] Presman, È. L. (1969). Factorization methods, and a boundary value problem for sums of random variables given on a Markov chain. Izv. Ross. Akad. Nauk Ser. Mat.33 861-900.
[31] Spitzer, F. (1976). Principles of Random Walk. 2nd ed. Springer, New York. · Zbl 0359.60003
[32] Varopoulos, N. Th. (1999). Potential theory in conical domains. Math. Proc. Cambridge Philos. Soc.125 335-384. · Zbl 0918.31008
[33] Varopoulos, N. Th. (2000). Potential theory in conical domains. II. Math. Proc. Cambridge Philos. Soc.129 301-319. · Zbl 0980.31007
[34] Varopoulos, N. Th. (2001). Potential theory in Lipschitz domains. Canad. J. Math.53 1057-1120. · Zbl 0983.60072
[35] Vatutin, V. A. and Wachtel, V. (2009). Local probabilities for random walks conditioned to stay positive. Probab. Theory Related Fields143 177-217. · Zbl 1158.60014
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