The Annals of Probability

Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption

Ion Grama, Ronan Lauvergnat, and Émile Le Page

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Consider a Markov chain $(X_{n})_{n\geq0}$ 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 $(y+S_{n})_{n\geq1}$ 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 $n\to+\infty$.

Article information

Source
Ann. Probab., Volume 46, Number 4 (2018), 1807-1877.

Dates
Received: July 2016
Revised: February 2017
First available in Project Euclid: 13 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1528876816

Digital Object Identifier
doi:10.1214/17-AOP1197

Mathematical Reviews number (MathSciNet)
MR3813980

Zentralblatt MATH identifier
06919013

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60F05: Central limit and other weak theorems 60J50: Boundary theory
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Keywords
Markov chain random walk exit time harmonic function limit theorem

Citation

Grama, Ion; Lauvergnat, Ronan; Le Page, Émile. Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption. Ann. Probab. 46 (2018), no. 4, 1807--1877. doi:10.1214/17-AOP1197. https://projecteuclid.org/euclid.aop/1528876816


Export citation

References

  • [1] Bertoin, J. and Doney, R. A. (1994). On conditioning a random walk to stay nonnegative. Ann. Probab. 22 2152–2167.
  • [2] Bolthausen, E. (1976). On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab. 4 480–485.
  • [3] Borovkov, A. A. (2004). On the asymptotic behavior of distributions of first-passage times, I. Math. Notes 75 23–37.
  • [4] Borovkov, A. A. (2004). On the asymptotics of distributions of first-passage times, II. Math. Notes 75 322–330.
  • [5] Buraczewski, D., Damek, E. and Guivarc’h, Y. (2010). Convergence to stable laws for a class of multidimensional stochastic recursions. Probab. Theory Related Fields 148 333–402.
  • [6] Caravenna, F. (2005). A local limit theorem for random walks conditioned to stay positive. Probab. Theory Related Fields 133 508–530.
  • [7] Dembo, A., Ding, J. and Gao, F. (2013). Persistence of iterated partial sums. Ann. Inst. Henri Poincaré Probab. Stat. 49 873–884.
  • [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.
  • [10] Denisov, D. and Wachtel, V. (2015). Exit times for integrated random walks. Ann. Inst. Henri Poincaré Probab. Stat. 51 167–193.
  • [11] Denisov, D. and Wachtel, V. (2015). Random walks in cones. Ann. Probab. 43 992–1044.
  • [12] Doney, R. A. (1989). On the asymptotic behaviour of first passage times for transient random walk. Probab. Theory Related Fields 81 239–246.
  • [13] Duraj, J. (2014). Random walks in cones: The case of nonzero drift. Stochastic Process. Appl. 124 1503–1518.
  • [14] Eichelsbacher, P. and König, W. (2008). Ordered random walks. Electron. J. Probab. 13 1307–1336.
  • [15] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. Wiley, New York.
  • [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.
  • [17] Gordin, M. I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 739–741.
  • [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.
  • [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.
  • [20] Grama, I., Le Page, É. and Peigné, M. (2017). Conditioned limit theorems for products of random matrices. Probab. Theory Related Fields 168 601–639.
  • [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.
  • [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. Systems 28 423–446.
  • [23] Iglehart, D. L. (1974). Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. 2 608–619.
  • [24] Iglehart, D. L. (1974). Random walks with negative drift conditioned to stay positive. J. Appl. Probab. 11 742–751.
  • [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.
  • [26] Kato, T. (1976). Perturbation Theory for Linear Operators. 2nd ed. Springer, Berlin.
  • [27] Lévy, P. (1937). Théorie de L’addition des Variables Aléatoires. Gauthier-Villars, Paris.
  • [28] Norman, M. F. (1972). Markov Processes and Learning Models. Academic Press, New York.
  • [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.
  • [32] Varopoulos, N. Th. (1999). Potential theory in conical domains. Math. Proc. Cambridge Philos. Soc. 125 335–384.
  • [33] Varopoulos, N. Th. (2000). Potential theory in conical domains. II. Math. Proc. Cambridge Philos. Soc. 129 301–319.
  • [34] Varopoulos, N. Th. (2001). Potential theory in Lipschitz domains. Canad. J. Math. 53 1057–1120.
  • [35] Vatutin, V. A. and Wachtel, V. (2009). Local probabilities for random walks conditioned to stay positive. Probab. Theory Related Fields 143 177–217.