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Shuffling cards and stopping times
 In Proceedings of the 43rd IEEE Conference on Decision and Control
, 1986
"... 1. Introduction. How many times must a deck of cards be shuffled until it is close to random? There is an elementary technique which often yields sharp estimates in such problems. The method is best understood through a simple example. EXAMPLE1. Top in at random shuffle. Consider the following metho ..."
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Cited by 130 (16 self)
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1. Introduction. How many times must a deck of cards be shuffled until it is close to random? There is an elementary technique which often yields sharp estimates in such problems. The method is best understood through a simple example. EXAMPLE1. Top in at random shuffle. Consider the following method of mixing a deck of cards: the top card is removed and inserted into the deck at a random position. This procedure is
Rates of Convergence for Gibbs Sampling for Variance Component Models
 Ann. Stat
, 1991
"... This paper analyzes the Gibbs sampler applied to a standard variance component model, and considers the question of how many iterations are required for convergence. It is proved that for K location parameters, with J observations each, the number of iterations required for convergence (for large K ..."
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Cited by 40 (10 self)
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This paper analyzes the Gibbs sampler applied to a standard variance component model, and considers the question of how many iterations are required for convergence. It is proved that for K location parameters, with J observations each, the number of iterations required for convergence (for large K and J) is a constant times
Approximate pvalues for local sequence alignments
 Ann. Statist
, 2000
"... Siegmund and Yakir (2000) have given an approximate pvalue when two independent, identically distributed sequences from a nite alphabet are optimally aligned based on a scoring system that rewards similarities according to a general scoring matrix and penalizes gaps (insertions and deletions). The ..."
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Cited by 39 (1 self)
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Siegmund and Yakir (2000) have given an approximate pvalue when two independent, identically distributed sequences from a nite alphabet are optimally aligned based on a scoring system that rewards similarities according to a general scoring matrix and penalizes gaps (insertions and deletions). The approximation involves an innite sequence of difculttocompute parameters. In this paper, it is shown by numerical studies that these reduce to essentially two numerically distinct parameters, which can be computed as onedimensional numerical integrals. For an arbitrary scoring matrix and afne gap penalty, this modied approximation is easily evaluated. Comparison with published numerical results show that it is reasonably accurate. Key words: local alignment, afne gap penalty, pvalue, Markov renewal theory. 1.
Rates of Convergence for Data Augmentation on Finite Sample Spaces
 Ann. Appl. Prob
, 1993
"... this paper, we examine this rate of convergence more carefully. We restrict our attention to the case where ..."
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Cited by 24 (11 self)
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this paper, we examine this rate of convergence more carefully. We restrict our attention to the case where
Onedimensional linear recursions with Markovdependent coefficients
, 2004
"... For a class of stationary Markovdependent sequences (ξn,ρn) ∈ R 2, we consider the random linear recursion Sn = ξn + ρnSn−1, n ∈ Z, and show that the distribution tail of its stationary solution has a power law decay. An application to random walks in random environments is discussed. MSC2000: pri ..."
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Cited by 16 (0 self)
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For a class of stationary Markovdependent sequences (ξn,ρn) ∈ R 2, we consider the random linear recursion Sn = ξn + ρnSn−1, n ∈ Z, and show that the distribution tail of its stationary solution has a power law decay. An application to random walks in random environments is discussed. MSC2000: primary 60K15; secondary 60K20, 60K37.
Information ranking and power laws on trees
 ADVANCES IN APPLIED PROBABILITY, 42(4), 2010
, 2010
"... We consider the stochastic analysis of information ranking algorithms of large interconnected data sets, e.g. Google’s PageRank algorithm for ranking pages on the World Wide Web. The stochastic formulation of the problem results in an equation of the form R D N∑ = Q + CiRi, where N, Q, {Ri}i≥1, {C, ..."
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Cited by 15 (11 self)
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We consider the stochastic analysis of information ranking algorithms of large interconnected data sets, e.g. Google’s PageRank algorithm for ranking pages on the World Wide Web. The stochastic formulation of the problem results in an equation of the form R D N∑ = Q + CiRi, where N, Q, {Ri}i≥1, {C, Ci}i≥1 are independent nonnegative random variables, {C, Ci}i≥1 are identically distributed, and {Ri}i≥1 are independent copies of R; D = stands for equality in distribution. We study the asymptotic properties of the distribution of R that, in the context of PageRank, represents the frequencies of highly ranked pages. The preceding equation is interesting in its own right since it belongs to a more general class of weighted branching processes that have been found useful in the analysis of many other algorithms. Our first main result shows that if ENE[C α] = 1, α> 0 and Q, N satisfy additional moment conditions, then R has a power law distribution of index α. This result is obtained using a new approach based on an extension of Goldie’s (1991) implicit renewal theorem. Furthermore, when N is regularly varying of index α> 1, ENE[C α] < 1 and Q, C have higher moments than α, then the distributions of R and N are tail equivalent. The latter result is derived via a novel sample path large deviation method for recursive random sums. Similarly, we characterize the situation when the distribution of R is determined by the tail of Q. The preceding approaches may be of independent interest, as they can be used for analyzing other functionals on trees. We also briefly discuss the engineering implications of our results.
IMPLICIT RENEWAL THEORY AND POWER TAILS ON TREES
 APPLIED PROBABILITY TRUST
, 2012
"... We extend Goldie’s (1991) Implicit Renewal Theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power tail asymptotics of the distributions of the solutions R to R D N∑ ..."
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Cited by 12 (7 self)
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We extend Goldie’s (1991) Implicit Renewal Theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power tail asymptotics of the distributions of the solutions R to R D N∑
RENEWAL THEORY FOR FUNCTIONALS OF A MARKOV CHAIN WITH COMPACT STATE SPACE
, 2003
"... Motivated by multivariate random recurrence equations we prove a new analogue of the Key Renewal Theorem for functionals of a Markov chain with compact state space in the spirit of Kesten [Ann. Probab. 2 (1974) 355–386]. Compactness of the state space and a certain continuity condition allows us to ..."
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Cited by 10 (4 self)
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Motivated by multivariate random recurrence equations we prove a new analogue of the Key Renewal Theorem for functionals of a Markov chain with compact state space in the spirit of Kesten [Ann. Probab. 2 (1974) 355–386]. Compactness of the state space and a certain continuity condition allows us to simplify Kesten’s proof considerably.
Renewal theory in analysis of tries and strings
, 2009
"... We give a survey of a number of simple applications of renewal theory to problems on random strings and tries: insertion depth, size, insertion mode and imbalance of tries; variations for btries and Patricia tries; Khodak and Tunstall codes. ..."
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Cited by 4 (1 self)
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We give a survey of a number of simple applications of renewal theory to problems on random strings and tries: insertion depth, size, insertion mode and imbalance of tries; variations for btries and Patricia tries; Khodak and Tunstall codes.