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The two most important notions of fractal dimension are Hausdorff dimension, developed by Haus-dorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical… (More)

- Krishna B. Athreya, John M. Hitchcock, Jack H. Lutz, Elvira Mayordomo
- SIAM J. Comput.
- 2004

The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical… (More)

- Krishna B. Athreya, C. R. Pranesachar, Navin M. Singhi
- Eur. J. Comb.
- 1980

SUMMARY. The purpose of this paper is to study the problem of estimation of the stationary density and the transition density of a real-valued recurrent Markov chain. By using techniques of regenerative processes we are able to significantly reduce the strong hypotheses on the Markov chain such as Doeblin recurrence, stationarity, and mixing that were… (More)

The Markov chain simulation method has been successfully used in many problems, including some that arise in Bayesian statistics. We give a self-contained proof of the convergence of this method in general state spaces under conditions that are easy to verify.

- K B Athreya
- Journal of mathematical biology
- 1992

If qk is the extinction probability of a slightly supercritical branching process with offspring distribution [pkr:r = 0, 1, 2, ...], then it is shown that if supk sigma r r3pkr less than infinity, inf sigma 2k greater than 0, and mk----1, then 1-qk approximately 2(mk - 1)sigma -2k, where mk = sigma r rpkr, sigma 2k = k sigma r r2pkr - m2k. This provides a… (More)

- KRISHNA B. ATHREYA, ORJAN STENFLO
- 2001

For Markov chains that can be generated by iteration of i.i.d. random maps from the state space X into itself (this holds if X is Polish) it is shown that the Doeblin minorization condition is necessary and suucient for the method by Propp and Wilson for \perfect" sampling from the stationary distribution to be successful. Using only the transition… (More)

- Krishna B. Athreya
- Internet Mathematics
- 2007

Consider a network of sites growing over time such that at step n a newcomer chooses a vertex from the existing vertices with probability proportional to a function of the degree of that vertex, i.e., the number of other vertices that this vertex is connected to. This is called a preferential attachment random graph. The objects of interest are the growth… (More)

A sequence of random variables {X n } n≥0 is called regenerative if it can be broken up into iid components. The problem addressed in this paper is to determine under what conditions is a Markov chain regenerative. It is shown that an irreducible Markov chain with a countable state space is regenerative for any initial distribution if and only if it is… (More)

- Krishna B. Athreya, A. Weerasinghe
- Math. Oper. Res.
- 1992