Bounds for the second largest eigenvalue of a transition matrix

The convergence rate of a Markov transition matrix is governed by the second largest eigenvalue, where the first largest eigenvalue is unity, under general regularity conditions. Garren and Smith (2000) constructed confidence intervals on this second largest eigenvalue, based on asymptotic normality theory, and performed simulations, which were somewhat limited in scope due to the reduced computing power of that time period. Herein we focus on simulating coverage intervals, using the advanced computing power of our current time period. Thus, we compare our simulated coverage intervals to the theoretical confidence intervals from Garren and Smith (2000).

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The eigenvalues of the empirical transition matrix of a Markov chain

Journal of Applied Probability ◽

10.1017/s0021900200112409 ◽

2004 ◽

Vol 41 (A) ◽

pp. 347-360

Author(s):

Geoffrey Pritchard ◽

David J. Scott

Keyword(s):

Markov Chain ◽

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Transition Matrix ◽

Largest Eigenvalue ◽

Independent Observations ◽

Second Largest Eigenvalue

This paper investigates the probabilistic behaviour of the eigenvalue of the empirical transition matrix of a Markov chain which is of largest modulus other than 1, loosely called the second-largest eigenvalue. A central limit theorem is obtained for nonmultiple eigenvalues of the empirical transition matrix. When the Markov chain is actually a sequence of independent observations the distribution of the second-largest eigenvalue is determined and a test for independence is developed. The independence case is considered in more detail when the Markov chain has only two states, and some applications are given.

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The eigenvalues of the empirical transition matrix of a Markov chain

Journal of Applied Probability ◽

10.1239/jap/1082552210 ◽

2004 ◽

Vol 41 (A) ◽

pp. 347-360 ◽

Cited By ~ 1

Author(s):

Geoffrey Pritchard ◽

David J. Scott

Keyword(s):

Markov Chain ◽

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Transition Matrix ◽

Largest Eigenvalue ◽

Independent Observations ◽

Second Largest Eigenvalue

This paper investigates the probabilistic behaviour of the eigenvalue of the empirical transition matrix of a Markov chain which is of largest modulus other than 1, loosely called the second-largest eigenvalue. A central limit theorem is obtained for nonmultiple eigenvalues of the empirical transition matrix. When the Markov chain is actually a sequence of independent observations the distribution of the second-largest eigenvalue is determined and a test for independence is developed. The independence case is considered in more detail when the Markov chain has only two states, and some applications are given.

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Graphs with the second signless Laplacian eigenvalue ≤ 4

Special Matrices ◽

10.1515/spma-2021-0152 ◽

2021 ◽

Vol 10 (1) ◽

pp. 131-152

Author(s):

Stephen Drury

Keyword(s):

General Solution ◽

Knapsack Problem ◽

Laplacian Eigenvalue ◽

Simple Graphs ◽

Largest Eigenvalue ◽

Signless Laplacian ◽

Second Largest Eigenvalue

Abstract We discuss the question of classifying the connected simple graphs H for which the second largest eigenvalue of the signless Laplacian Q(H) is ≤ 4. We discover that the question is inextricable linked to a knapsack problem with infinitely many allowed weights. We take the first few steps towards the general solution. We prove that this class of graphs is minor closed.

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