## Large deviation bounds for Markov chains (0)

Citations: | 9 - 0 self |

### BibTeX

@TECHREPORT{Kahale_largedeviation,

author = {Nabil Kahale},

title = {Large deviation bounds for Markov chains},

institution = {},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study the fraction of time that a Markov chain spends in a given subset of states. We give an exponential bound on the probability that it exceeds its expectation by a constant factor. Our bound depends on the mixing properties of the chain, and is asymptotically optimal for a certain class of Markov chains. It beats the best previously known results in this direction. We present an application to the leader election problem. 1 Introduction Let (Xm ); m 1, be an irreducible Markov chain on a finite state space V with transition matrix P and stationary distribution ß. We assume that P is reversible, that is ß(u)P (u; v) = ß(v)P (v; u); for all u; v 2 V: (1) Let A be a proper subset of V . Denote by ß(A) = P u2A ß(u) the stationary probability of the set A, and by S l = ØA (X 1 ) + ØA (X 2 ) + \Delta \Delta \Delta + ØA (X l ) the number of steps the Markov chain is inside A. It is known [2] that, for any initial distribution, the fraction S l =l converges almost surely to ß(A) ...

### Citations

376 | Reversible Markov chains and random walks on graphs
- Aldous, Fill
(Show Context)
Citation Context ... by ��(A) = P u2A ��(u) the stationary probability of the set A, and by S l = ��A (X 1 ) + ��A (X 2 ) + \Delta \Delta \Delta + ��A (X l ) the number of steps the Markov chain is in=-=side A. It is known [2] that, f-=-or any initial distribution, the fraction S l =l converges almost surely to ��(A) as l goes to infinity. This lead Aldous [3] to propose the following sampling technique: ��(A) can be estimate... |

317 |
Non-negative Matrices and Markov Chains
- Seneta
- 1981
(Show Context)
Citation Context ... ? �� ) basis (e 0 ; e 1 ; : : : ; e n\Gamma1 ), where n = jV j. The first element e 0 of the basis can be chosen to be the all ones column vector 1, since P1 = 1. The theory of non-negative matri=-=ces [16]-=- shows that 1 is the largest eigenvalue of P is absolute value. Denote bysi the eigenvalue corresponding to e i , for 1sisn \Gamma 1, and assume thats1s2s\Delta \Delta \Deltasn\Gamma1 . Since the chai... |

299 | Approximating the permanent
- Jerrum, Sinclair
- 1989
(Show Context)
Citation Context ...on 4, the decay rate in our bound is better than the one in [7] by at least a constant factor. When ��(A) is small, which is the case in several applications such as approximating the dense perman=-=ent [12], it-=- beats it by a factor of \Theta(��(A) \Gamma1 ). As in [4, 7, 10], the proof of our main result relies on computing an upper bound on the generating function of S l . The main new idea in our pape... |

260 |
Approximate counting, uniform generation and rapidly mixing Markov chains
- Sinclair, Jerrum
- 1989
(Show Context)
Citation Context ...are equal. This case can be further reduced to the case where the state space consists of two elements. Random walks have been used in many areas of Computer Science, such as approximation algorithms =-=[12, 17]-=- (See also [14], and references therein), complexity and cryptography [1, 8, 11], and distributed computing [15]. The rest of the paper is organized as follows. Section 2 contains basic results and de... |

203 |
A guided tour of Chernoff bounds
- Hagerup, Rüb
- 1990
(Show Context)
Citation Context ...= (1 \Gamma ��(A))=(1 \Gamma fi��(A)) and OE(�� 0 ) = i 1\Gamma��(A) 1\Gammafi��(A) j 1\Gammafi��(A) fi \Gammafi ��(A) . In this case, the bound in Theorem 3.1 coincides wi=-=th the usual Chernoff bound [9], up to a multiplicative-=- constant. An easy continuity argument shows that if fi and ��(A) are fixed, then OE(�� 0 ) goes to i 1\Gamma��(A) 1\Gammafi��(A) j 1\Gammafi��(A) fi \Gammafi ��(A) ass1 goes t... |

184 | How to recycle random bits
- Impagliazzo, Zuckerman
- 1989
(Show Context)
Citation Context ...xceeds ��(A) by a given amount. A bound on the variance of S l in terms of the mixing properties of the chain was established in [3, 14]. An exponential bound on the tail of S l =l was established=-= in [5, 11]-=- in a special case, and in a more general setting in [7]. In this paper, we establish a bound on the tail of the distribution of S l =l that beats the previously known bounds. As the previous bounds, ... |

123 |
Random walks in a convex body and an improved volume algorithm
- Lovász, Simonovits
- 1993
(Show Context)
Citation Context ...erefore important to establish a bound on the probability that S l =l exceeds ��(A) by a given amount. A bound on the variance of S l in terms of the mixing properties of the chain was established=-= in [3, 14]-=-. An exponential bound on the tail of S l =l was established in [5, 11] in a special case, and in a more general setting in [7]. In this paper, we establish a bound on the tail of the distribution of ... |

98 |
Deterministic simulation in LOGSPACE
- Ajtai, Komlós, et al.
- 1987
(Show Context)
Citation Context ...consists of two elements. Random walks have been used in many areas of Computer Science, such as approximation algorithms [12, 17] (See also [14], and references therein), complexity and cryptography =-=[1, 8, 11]-=-, and distributed computing [15]. The rest of the paper is organized as follows. Section 2 contains basic results and definitions. Section 3 contains the proof of our main result. We establish more ex... |

80 | A Chernoff Bound for Random Walks on Expander Graphs
- Gillman
- 1998
(Show Context)
Citation Context ... l in terms of the mixing properties of the chain was established in [3, 14]. An exponential bound on the tail of S l =l was established in [5, 11] in a special case, and in a more general setting in =-=[7]-=-. In this paper, we establish a bound on the tail of the distribution of S l =l that beats the previously known bounds. As the previous bounds, ours depends on the second largest eigenvalue of P . We ... |

77 |
On the Markov Chain Simulation Method for Uniform Combinatorial Distributions and Simulated Annealing
- Aldous
- 1987
(Show Context)
Citation Context ... l ) the number of steps the Markov chain is inside A. It is known [2] that, for any initial distribution, the fraction S l =l converges almost surely to ��(A) as l goes to infinity. This lead Ald=-=ous [3] to -=-propose the following sampling technique: ��(A) can be estimated by simulating the Markov chain for l steps and computing the fraction S l =l of steps it spends in A. Typically, the size of A is e... |

53 | Security Preserving Amplification of Hardness
- Goldreich, Impagliazzo, et al.
- 1990
(Show Context)
Citation Context ...consists of two elements. Random walks have been used in many areas of Computer Science, such as approximation algorithms [12, 17] (See also [14], and references therein), complexity and cryptography =-=[1, 8, 11]-=-, and distributed computing [15]. The rest of the paper is organized as follows. Section 2 contains basic results and definitions. Section 3 contains the proof of our main result. We establish more ex... |

37 |
deterministic amplification, and weak random sources
- Dispersers
- 1989
(Show Context)
Citation Context ...xceeds ��(A) by a given amount. A bound on the variance of S l in terms of the mixing properties of the chain was established in [3, 14]. An exponential bound on the tail of S l =l was established=-= in [5, 11]-=- in a special case, and in a more general setting in [7]. In this paper, we establish a bound on the tail of the distribution of S l =l that beats the previously known bounds. As the previous bounds, ... |

20 |
Better expansion for Ramanujan graphs
- Kahale
- 1992
(Show Context)
Citation Context ...aker bounds in Section 4, and present an application to the leader election problem. We also give a bound on the probability that the Markov chain stays inside A for l steps, generalizing a result in =-=[13]. In Section 5, -=-we show the tightness of our main result. 2 Preliminaries Let L 2 (��) be the set of real valued functions on V , with the scalar product !; ? �� : ! f; g ? �� = X x2V ��(x)f(x)g(x): F... |

17 | Simple and efficient leader election in the full information model
- Ostrovsky, Rajagopalan, et al.
- 1994
(Show Context)
Citation Context ... have been used in many areas of Computer Science, such as approximation algorithms [12, 17] (See also [14], and references therein), complexity and cryptography [1, 8, 11], and distributed computing =-=[15]-=-. The rest of the paper is organized as follows. Section 2 contains basic results and definitions. Section 3 contains the proof of our main result. We establish more explicit but weaker bounds in Sect... |

4 |
A measure for asymptotic efficiency of a hypothesis based on the sum of observations
- Chernoff
- 1952
(Show Context)
Citation Context ... by at least a constant factor. When ��(A) is small, which is the case in several applications such as approximating the dense permanent [12], it beats it by a factor of \Theta(��(A) \Gamma1 )=-=. As in [4, 7, 10]-=-, the proof of our main result relies on computing an upper bound on the generating function of S l . The main new idea in our paper is to reduce the analysis to the case where all the eigenvalues of ... |

3 |
Central limit theorems and statistical inference for finite Markov chains. Z. Wahrscheinlichkeitstheorie und Verw
- Höglund
- 1974
(Show Context)
Citation Context ... by at least a constant factor. When ��(A) is small, which is the case in several applications such as approximating the dense permanent [12], it beats it by a factor of \Theta(��(A) \Gamma1 )=-=. As in [4, 7, 10]-=-, the proof of our main result relies on computing an upper bound on the generating function of S l . The main new idea in our paper is to reduce the analysis to the case where all the eigenvalues of ... |