A class of Markov chains with no spectral gap

Authors:
Yevgeniy Kovchegov and Nicholas Michalowski

Journal:
Proc. Amer. Math. Soc. **141** (2013), 4317-4326

MSC (2010):
Primary 33C45, 60G05

Published electronically:
August 16, 2013

MathSciNet review:
3105873

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we extend the results of the research started by the first author in which Karlin-McGregor diagonalization of certain reversible Markov chains over countably infinite general state spaces by orthogonal polynomials was used to estimate the rate of convergence to a stationary distribution.

We use a method of Koornwinder to generate a large and interesting family of random walks which exhibits a lack of spectral gap, and a polynomial rate of convergence to the stationary distribution. For the Chebyshev type subfamily of Markov chains, we use asymptotic techniques to obtain an upper bound of order and a lower bound of order on the distance to the stationary distribution regardless of the initial state. Due to the lack of a spectral gap, these results lie outside the scope of geometric ergodicity theory.

**[1]**Milton Abramowitz and Irene A. Stegun (eds.),*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR**1225604****[2]**Mourad E. H. Ismail,*Classical and quantum orthogonal polynomials in one variable*, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR**2191786****[3]**Samuel Karlin and James McGregor,*Random walks*, Illinois J. Math.**3**(1959), 66–81. MR**0100927****[4]**Harry Kiesel and Jet Wimp,*A note on Koornwinder’s polynomials with weight function (1-𝑥)^{𝛼}(1+𝑥)^{𝛽}+𝑀𝛿(𝑥+1)+𝑁𝛿(𝑥-1)*, Numer. Algorithms**11**(1996), no. 1-4, 229–241. Orthogonal polynomials and numerical analysis (Luminy, 1994). MR**1383387**, 10.1007/BF02142499**[5]**Tom H. Koornwinder,*Orthogonal polynomials with weight function (1-𝑥)^{𝛼}(1+𝑥)^{𝛽}+𝑀𝛿(𝑥+1)+𝑁𝛿(𝑥-1)*, Canad. Math. Bull.**27**(1984), no. 2, 205–214. MR**740416**, 10.4153/CMB-1984-030-7**[6]**Yevgeniy Kovchegov,*Orthogonality and probability: beyond nearest neighbor transitions*, Electron. Commun. Probab.**14**(2009), 90–103. MR**2481669**, 10.1214/ECP.v14-1447**[7]**Yevgeniy Kovchegov,*Orthogonality and probability: mixing times*, Electron. Commun. Probab.**15**(2010), 59–67. MR**2595683**, 10.1214/ECP.v15-1525**[8]**Sean Meyn and Richard L. Tweedie,*Markov chains and stochastic stability*, 2nd ed., Cambridge University Press, Cambridge, 2009. With a prologue by Peter W. Glynn. MR**2509253****[9]**M. V. Menshikov and S. Yu. Popov,*Exact power estimates for countable Markov chains*, Markov Process. Related Fields**1**(1995), no. 1, 57–78. MR**1403077****[10]**Gábor Szegő,*Orthogonal polynomials*, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR**0372517****[11]**V. B. Uvarov,*The connection between systems of polynomials that are orthogonal with respect to different distribution functions*, Ž. Vyčisl. Mat. i Mat. Fiz.**9**(1969), 1253–1262 (Russian). MR**0262764**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
33C45,
60G05

Retrieve articles in all journals with MSC (2010): 33C45, 60G05

Additional Information

**Yevgeniy Kovchegov**

Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 97331

Email:
kovchegy@math.oregonstate.edu

**Nicholas Michalowski**

Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 97331

Email:
Nicholas.Michalowski@math.oregonstate.edu

DOI:
https://doi.org/10.1090/S0002-9939-2013-11697-7

Received by editor(s):
September 26, 2011

Received by editor(s) in revised form:
September 28, 2011, and February 9, 2012

Published electronically:
August 16, 2013

Communicated by:
Edward C. Waymire

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.