Directory UMM :Data Elmu:jurnal:S:Stochastic Processes And Their Applications:Vol91.Issue2.2001:

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Stochastic Processes and their Applications 91 (2001) 337–338

www.elsevier.com/locate/spa

Corrigendum

Corrigendum to “Bounds on regeneration times and

convergence rates for Markov chains”

(

[Stochastic Processes and their Applications 80 (1999)

211–229]

G.O. Roberts

a;∗

, R.L. Tweedie

b

a_{Department of Mathematics and Statistics, University of Lancaster, Lancaster LA1 4YF, England, UK} b_{Division of Biostatistics, School of Public Health, A460 Mayo Building, Box 303 420 Delaware Street}

SE Minneapolis, MN 55455-0378, USA

Received 24 July 2000; accepted 24 July 2000

The authors would like to point out the following:

In Theorem 5.1(i) of Roberts and Tweedie (1999), we developed a bound on the cou-pling time of a Markov chain which is stated to hold for all n≥= log(V)=log−1_.

However, for the argument to be valid it is necessary that ˆ satisfying (51) should be

greater than 1. This imposes a further bound on n, namely

n ¿ +(1−)=:

If is small, this extra restriction can limit the applicability of the bound. This error became obvious in the calculations in Guglielmi and Tweedie (2000), where extremely

small values of occur leading to large values of the lower bound on n above. In

other examples, such as the convergence of slice samplers (Roberts and Rosenthal, 1999), this extra condition does not alter the bound on the actual convergence time (see Theorem 12 of that paper and the discussion following it).

We also note that the values of RT given in Table 1 for the rst three cases

(q= 0:51; b= 0:03) are incorrect. In these cases J ¡1 and so we should have used

case (a) of Theorem 2:3 rather than the case (b) used throughout Table 1. This gives,

in each case, RT = 1:0002. Hence, the improvement of the bounds in this parameter

region is rather less than claimed. We are grateful to John Kolassa who pointed this out to us.

References

Guglielmi, A., Tweedie, R.L., 2000. Mcmc estimation of the law of the mean of a Dirichlet process, unpublished.

(_{PII of original article: S0304-4149(98)00085-4. Work supported in part by NSF Grant DMS 9803682}

and EPSRC Grant GR /J19900.

∗_{Corresponding author.}

E-mail address:g.o.roberts@lancaster.ac.uk (G.O. Roberts).

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338 G.O. Roberts, R.L. Tweedie / Stochastic Processes and their Applications 91 (2001) 337–338

Roberts, G.O., Rosenthal, J.S., 1999. Convergence of slice sampler Markov chains. J. Roy. Statist. Soc. Ser. B 61, 643– 660.