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267
General state space Markov chains and MCMC algorithm
 PROBABILITY SURVEYS
, 2004
"... This paper surveys various results about Markov chains on general (noncountable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform e ..."
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Cited by 114 (27 self)
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This paper surveys various results about Markov chains on general (noncountable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform ergodicity are presented, along with quantitative bounds on the rate of convergence to stationarity. Many of these results are proved using direct coupling constructions based on minorisation and drift conditions. Necessary and sufficient conditions for Central Limit Theorems (CLTs) are also presented, in some cases proved via the Poisson Equation or direct regeneration constructions. Finally, optimal scaling and weak convergence results for MetropolisHastings algorithms are discussed. None of the results presented is new, though many of the proofs are. We also describe some Open Problems.
MaxWeight scheduling in a generalized switch: state space collapse and workload minimization in heavy traffic
 Annals of Applied Probability
, 2004
"... We consider a generalized switch model, which includes as special cases the model of multiuser data scheduling over a wireless medium, the inputqueued crossbar switch model and a discrete time version of a parallel server queueing system. Input flows n = 1,...,N are served in discrete time by a swi ..."
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Cited by 113 (9 self)
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We consider a generalized switch model, which includes as special cases the model of multiuser data scheduling over a wireless medium, the inputqueued crossbar switch model and a discrete time version of a parallel server queueing system. Input flows n = 1,...,N are served in discrete time by a switch. The switch state follows a finite state, discrete time Markov chain. In each state m, the switch chooses a scheduling decision k from a finite set K(m), which has the associated service rate vector (µ m 1 (k),..., µm N (k)). We consider a heavy traffic regime, and assume a Resource Pooling (RP) condition. Associated with this condition is a notion of workload X = � n ζnQn, whereζ = (ζ1,...,ζN) is some fixed nonzero vector with nonnegative components, and Q1,...,QN are the queue lengths. We study the MaxWeight discipline which always chooses a decision k maximizing n γn[Qn] β µ m n (k),thatis, k ∈ arg max γn[Qn] i n β µ m n (i), where β>0, γ1> 0,...,γN> 0 are arbitrary parameters. We prove that under MaxWeight scheduling and the RP condition, in the heavy traffic limit, the queue length process has the following properties: (a) The vector (γ1Q β 1,...,γNQ β N) is always proportional to ζ (this is “State Space Collapse”), (b) the workload process converges to a Reflected Brownian Motion, (c) MaxWeight minimizes the workload among all disciplines. As a corollary of these properties, MaxWeight asymptotically minimizes the holding cost rate n γnQ β+1
The Power of Two Random Choices: A Survey of Techniques and Results
 in Handbook of Randomized Computing
, 2000
"... ITo motivate this survey, we begin with a simple problem that demonstrates a powerful fundamental idea. Suppose that n balls are thrown into n bins, with each ball choosing a bin independently and uniformly at random. Then the maximum load, or the largest number of balls in any bin, is approximately ..."
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Cited by 99 (2 self)
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ITo motivate this survey, we begin with a simple problem that demonstrates a powerful fundamental idea. Suppose that n balls are thrown into n bins, with each ball choosing a bin independently and uniformly at random. Then the maximum load, or the largest number of balls in any bin, is approximately log n= log log n with high probability. Now suppose instead that the balls are placed sequentially, and each ball is placed in the least loaded of d 2 bins chosen independently and uniformly at random. Azar, Broder, Karlin, and Upfal showed that in this case, the maximum load is log log n= log d + (1) with high probability [ABKU99]. The important implication of this result is that even a small amount of choice can lead to drastically different results in load balancing. Indeed, having just two random choices (i.e.,...
How Useful Is Old Information
 IEEE Transactions on Parallel and Distributed Systems
, 2000
"... AbstractÐWe consider the problem of load balancing in dynamic distributed systems in cases where new incoming tasks can make use of old information. For example, consider a multiprocessor system where incoming tasks with exponentially distributed service requirements arrive as a Poisson process, the ..."
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Cited by 80 (10 self)
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AbstractÐWe consider the problem of load balancing in dynamic distributed systems in cases where new incoming tasks can make use of old information. For example, consider a multiprocessor system where incoming tasks with exponentially distributed service requirements arrive as a Poisson process, the tasks must choose a processor for service, and a task knows when making this choice the processor queue lengths from T seconds ago. What is a good strategy for choosing a processor in order for tasks to minimize their expected time in the system? Such models can also be used to describe settings where there is a transfer delay between the time a task enters a system and the time it reaches a processor for service. Our models are based on considering the behavior of limiting systems where the number of processors goes to infinity. The limiting systems can be shown to accurately describe the behavior of sufficiently large systems and simulations demonstrate that they are reasonably accurate even for systems with a small number of processors. Our studies of specific models demonstrate the importance of using randomness to break symmetry in these systems and yield important rules of thumb for system design. The most significant result is that only small amounts of queue length information can be extremely useful in these settings; for example, having incoming tasks choose the least loaded of two randomly chosen processors is extremely effective over a large range of possible system parameters. In contrast, using global information can actually degrade performance unless used carefully; for example, unlike most settings where the load information is current, having tasks go to the apparently least loaded server can significantly hurt performance. Index TermsÐLoad balancing, stale information, old information, queuing theory, large deviations. æ 1
Drawing inferences from statistics based on multiyear asset returns
 Journal of Financial Economics
, 1989
"... Researchers investigating the possibility of mean reversion in stock prices with statistics based on multiyear returns have noted difficulties in drawing inferences from these statistics because the approximating asymptotic distributions perform poorly. We develop an alternative asymptotic distribut ..."
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Cited by 72 (3 self)
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Researchers investigating the possibility of mean reversion in stock prices with statistics based on multiyear returns have noted difficulties in drawing inferences from these statistics because the approximating asymptotic distributions perform poorly. We develop an alternative asymptotic distribution theory for statistics involving multiyear returns. These distributions diPier markedly from those implied by the conventional theory. The alternative theory provides substantially better approximations to the relevant finitesample distributions. It also leads to empirical inferences much less at odds with the hypothesis of no mean reversion.
Optimal scaling of discrete approximations to Langevin diffusions
 J. R. Statist. Soc. B
, 1997
"... . We consider the optimal scaling problem for proposal distributions in HastingsMetropolis algorithms derived from Langevin diffusions. We prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterised by its overall acceptance rate, independ ..."
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Cited by 66 (22 self)
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. We consider the optimal scaling problem for proposal distributions in HastingsMetropolis algorithms derived from Langevin diffusions. We prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterised by its overall acceptance rate, independently of the target distribution. The asymptotically optimal acceptance rate is 0:574. We show that as a function of dimension n, the complexity of the algorithm is O(n 1=3 ), which compares favourably with the O(n) complexity of randomwalk Metropolis algorithms. We illustrate this comparison with a number of example simulations. Keywords. Langevin algorithm, HastingsMetropolis, Markov chain Monte Carlo, weak convergence. * Statistical Laboratory, University of Cambridge, Cambridge CB2 1SB, U.K. Internet: G.O.Roberts@statslab.cam.ac.uk. ** Department of Statistics, University of Toronto, Toronto, Ontario, Canada M5S 1A1. Internet: jeff@utstat.toronto.edu. Supported in part by NSERC o...
Linear Multiuser Receivers in Random Environments
 IEEE Trans. Inform. Theory
, 2000
"... We study the signaltointerference (SIR) performance of linear multiuser receivers in random environments, where signals from the users arrive in "random directions." Such random environment may arise in a DSCDMA system with random signature sequences, or in a system with antenna diversity where t ..."
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Cited by 55 (2 self)
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We study the signaltointerference (SIR) performance of linear multiuser receivers in random environments, where signals from the users arrive in "random directions." Such random environment may arise in a DSCDMA system with random signature sequences, or in a system with antenna diversity where the randomness is due to channel fading. Assuming that such random directions can be tracked by the receiver, the resulting SIR performance is a function of the directions and therefore also random. We study the asymptotic distribution of this random performance in the regime where both the number of users and the number of degrees of freedom in the system are large, but keeping their ratio fixed. Our results show that for both the decorrelator and the minimum meansquare error (MMSE) receiver, the variance of the SIR distribution decreases like 1 , and the SIR distribution is asymptotically Gaussian. We compute closedform expressions for the asymptotic means and variances for both receivers. Simulation results are presented to verify the accuracy of the asymptotic results for finitesized systems.
Quenched invariance principle for simple random walk on percolation clusters
, 2005
"... We consider the simple random walk on a twodimensional supercritical infinite percolation cluster. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to a nondegenerate Brownian motion. ..."
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Cited by 42 (3 self)
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We consider the simple random walk on a twodimensional supercritical infinite percolation cluster. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to a nondegenerate Brownian motion.
Smoluchowski’s coagulation equation: uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent
 Ann. Appl. Probab
, 1999
"... Abstract. Sufficient conditions are given for existence and uniqueness in Smoluchowski’s coagulation equation, for a wide class of coagulation kernels and initial mass distributions. An example of nonuniqueness is constructed. The stochastic coalescent is shown to converge weakly to the solution of ..."
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Cited by 34 (2 self)
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Abstract. Sufficient conditions are given for existence and uniqueness in Smoluchowski’s coagulation equation, for a wide class of coagulation kernels and initial mass distributions. An example of nonuniqueness is constructed. The stochastic coalescent is shown to converge weakly to the solution of Smoluchowski’s equation. 1.
Structure of large random hypergraphs
 Ann. Appl. Probab
, 2005
"... The theme of this paper is the derivation of analytic formulae for certain large combinatorial structures. The formulae are obtained via fluid limits of pure jump type Markov processes, established under simple conditions on the Laplace transforms of their Lévy kernels. Furthermore, a related Gaussi ..."
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Cited by 28 (3 self)
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The theme of this paper is the derivation of analytic formulae for certain large combinatorial structures. The formulae are obtained via fluid limits of pure jump type Markov processes, established under simple conditions on the Laplace transforms of their Lévy kernels. Furthermore, a related Gaussian approximation allows us to describe the randomness which may persist in the limit when certain parameters take critical values. Our method is quite general, but is applied here to vertex identifiability in random hypergraphs. A vertex v is identifiable in n steps if there is a hyperedge containing v all of whose other vertices are identifiable in fewer steps. We say that a hyperedge is identifiable if every one of its vertices is identifiable. Our analytic formulae describe the asymptotics of the number of identifiable vertices and the number of identifiable hyperedges for a Poisson(β) random hypergraph Λ on a set V of N vertices, in the limit as N → ∞. Here β is a formal power series with nonnegative coefficients β0,β1,..., and (Λ(A))A⊆V are independent Poisson random variables such that Λ(A), the number of hyperedges on A, has mean Nβj / ( N) j whenever A  = j.