Conditional limit theorems under Markov conditioning

Csiszár, Imre; Cover, Thomas M.; Choi, ByoungSeon

doi:10.1109/tit.1987.1057385

Cited by 96 publications

(57 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The classical (conditional) Kullback-Leibler information (informational divergence or relative entropy) is denoted by D and entropy by H. 10,11,22 Specifically, for a probability distribution Q on X 2 , transition (or conditional) probabilities P (v|u), u, v ∈ X , and a probability distribution p on X , we define…”

Section: Main Results For Simple Casementioning

confidence: 99%

“…8, which employed the method of types 10,11,22,33 . In the present case, second-order (Markov) types rather than the usual types are used.…”

Section: B the Methods Of Typesmentioning

confidence: 99%

“…. , z |X | ) = max i |z i |, 22 the set of all probability distributions is compact, and D(Q) = D(Q||P ) is continuous in its effective domain {Q | D(Q) < ∞} (cf., the proof of Lemma 2 in Csiszár et al 22 ). This completes the proof.…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 2 more Smart Citations

A lower bound on the quantum capacity of channels with correlated errors

Hamada¹

2002

Journal of Mathematical Physics

View full text Add to dashboard Cite

The highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by 1 − exp[−nE(R) + o(n)] for some function E(R) on noisy quantum channels that are subject to not necessarily independent errors. The E(R) is positive below some threshold R 0 , which implies R 0 is a lower bound on the quantum capacity. This work is an extension of the author's previous works [M. Hamada, Phys. Rev. A, 65, 052305 (2002), e-Print quant-ph/0109114, LANL, 2001, and M. Hamada, e-Print quant-ph/0112103, LANL, 2001], which presented the bound for channels subject to independent errors, or channels modeled as tensor products of copies of a completely positive linear map. The relation of the channel class treated in this paper to those in the previous works are similar to that of Markov chains to sequences of independent identically distributed random variables.

show abstract

Section: Main Results For Simple Casementioning

confidence: 99%

“…8, which employed the method of types 10,11,22,33 . In the present case, second-order (Markov) types rather than the usual types are used.…”

Section: B the Methods Of Typesmentioning

confidence: 99%

See 1 more Smart Citation

A lower bound on the quantum capacity of channels with correlated errors

Hamada¹

2002

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…Boza [14], Davisson, Longo, Sgarro [15], Natarajan [16], Csiszár, Cover, Choi [17], and Csiszár [18]. (Throughout this paper, the base of the logarithm is |Σ|.…”

Section: B Markov Typementioning

confidence: 99%

On the VC-dimension of binary codes

Weinberger

Shayevitz

2017

2017 IEEE International Symposium on Information Theory (ISIT)

View full text Add to dashboard Cite

Abstract-We investigate the asymptotic rates of length-n binary codes with VC-dimension at most dn and minimum distance at least δn. Two upper bounds are obtained, one as a simple corollary of a result by Haussler and the other via a shortening approach combining Sauer-Shelah lemma and the linear programming bound. Two lower bounds are given using Gilbert-Varshamov type arguments over constant-weight and Markov-type sets.

show abstract

“…sequence [7], [8], [6]. But in the case of Markov chains they have been much less successful, their applicability limited essentially only to the more elementary case of finite state-space Markov chains; see [9] and the references therein. This difficulty reflects, to some extent, the intrinsic limitations of the informationtheoretic methods, but it is also due to the much greater complexity of the field of large deviations for general Markov chains.…”

Section: Introductionmentioning

confidence: 99%

Exponential bounds and stopping rules for MCMC and general Markov chains

Kontoyiannis

Lastras-Montaño

Meyn

2006

Proceedings of the 1st International Conference on Performance Evaluation Methodolgies and Tools - Valuetools '06

View full text Add to dashboard Cite

Abstract-We develop explicit, general bounds for the probability that the empirical sample averages of a function of a Markov chain on a general alphabet will exceed the steady-state mean of that function by a given amount. Our bounds combine simple information-theoretic ideas together with techniques from optimization and some fairly elementary tools from analysis. In one direction, motivated by central problems in simulation, we develop bounds for the general class of "geometrically ergodic" Markov chains. These bounds take a form that is particularly suited to simulation problems, and they naturally lead to a new class of sampling criteria. These are illustrated by several examples. In another direction, we obtain a new bound for the important special class of Doeblin chains; this bound is optimal, in the sense that in the special case of independent and identically distributed random variables it essentially reduces to the classical Hoeffding bound.

show abstract

Conditional limit theorems under Markov conditioning

Cited by 96 publications

References 15 publications

A lower bound on the quantum capacity of channels with correlated errors

A lower bound on the quantum capacity of channels with correlated errors

On the VC-dimension of binary codes

Exponential bounds and stopping rules for MCMC and general Markov chains

Contact Info

Product

Resources

About