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On the equivalence between Stein and de Bruijn identities [article]

Sangwoo Park, Erchin Serpedin, Khalid Qaraqe
2012 arXiv   pre-print
This paper focuses on proving the equivalence between Stein's identity and de Bruijn's identity. Given some conditions, we prove that Stein's identity is equivalent to de Bruijn's identity.  ...  For arbitrary but fixed input and noise distributions, there exist relations between the first derivative of the differential entropy and the posterior mean.  ...  Also, the equivalence between De Bruijn's identity and I-MMSE identity was shown in [11] .  ... 
arXiv:1202.0015v4 fatcat:wu7dfp47gfcmrpijnqszug3ncq

IT Formulae for Gamma Target: Mutual Information and Relative Entropy

Benjamin Arras, Yvik Swan
2018 IEEE Transactions on Information Theory  
The proof relies on a rescaled version of De Bruijn identity for gamma target distribution together with a stochastic representation for the gamma-specific Fisher information.  ...  This relation is reminiscent of the well-known link between the derivative of the input-output mutual information of additive Gaussian noise channel with respect to the signal-to-noise ratio and the minimum  ...  ACKNOWLEDGMENT We thank two anonymous referees and the Associate Editor for their pertinent remarks which have led to substantial improvement of the paper.  ... 
doi:10.1109/tit.2017.2759279 fatcat:pm7wkrljrbaedj7elouvsnsi4i

IT formulae for gamma target: mutual information and relative entropy [article]

Benjamin Arras, Yvik Swan
2016 arXiv   pre-print
The proof relies on a rescaled version of De Bruijn identity for gamma target distribution together with a stochastic representation for the gamma specific Fisher information.  ...  This relation is reminiscent of the well-known link between the derivative of the input-output mutual information of additive Gaussian noise channel with respect to the signal-to-noise ratio and the minimum  ...  Acknowledgements The research of BA is funded by a Welcome Grant from Liege University.  ... 
arXiv:1609.02342v1 fatcat:v75kgd5opfhj3daiz5ctoxtbjy

A stroll along the gamma [article]

Benjamin Arras, Yvik Swan
2016 arXiv   pre-print
We use these results to prove a differential and an integrated De Bruijn identity which hold under minimal conditions, hereby extending the classical formulae which follow from Bakry, Emery and Ledoux's  ...  entropy, Stein discrepancy and standardized Fisher information for the gamma law with α≥ 1/2.  ...  Acknowledgements The authors thank Michel Ledoux, Ivan Nourdin and Giovanni Peccati for fruitful discussions on a preliminary version of this paper.  ... 
arXiv:1511.04923v3 fatcat:iayq3oaukva3leddnu4pwn4ck4

A stroll along the gamma

Benjamin Arras, Yvik Swan
2017 Stochastic Processes and their Applications  
We use these results to prove a differential and an integrated De Bruijn identity which hold under minimal conditions, hereby extending the classical formulae which follow from Bakry, Emery and Ledoux's  ...  entropy, Stein discrepancy and standardized Fisher information for the gamma law with α ≥ 1/2.  ...  Acknowledgements The authors thank Michel Ledoux, Ivan Nourdin and Giovanni Peccati for fruitful discussions on a preliminary version of this paper.  ... 
doi:10.1016/j.spa.2017.03.012 fatcat:kqvohxglzbdczifud646ljlmpy

Page 743 of Mathematical Reviews Vol. 39, Issue 4 [page]

1970 Mathematical Reviews  
These equivalence relations have the property that if the interval [z, y] is equivalent to the interval [z, w), then there is a one-to-one correspondence between their sub-intervals such that corresponding  ...  Monthly 72 (1965), 135-138; MR 30 #3863], the Cohen- Landau identity, the Anderson-Apostol- -Hélder identity and 33 corollaries.  ... 

q-gram analysis and urn models

Pierre Nicodème
2003 Discrete Mathematics & Theoretical Computer Science  
In the exact and dependent model, when omitting border effects, a $q$-gramin a random sequence depends on the $q-1$ preceding $q$-grams.  ...  In an approximate and independent model, we draw randomly a $q$-gram at each position, independently of the others positions. Using ball and urn models, we analyze the independent model.  ...  Acknowledgement We thank Philippe Jacquet and Bruno Salvy for helpful discussions.  ... 
doi:10.46298/dmtcs.3322 fatcat:jmrucvvw2va6tlmqk6dcin3dde

Geometrical, Algebraic, Functional and Correlation Inequalities Applied in Support of James-Stein Estimator for Multidimensional Projections

Michael Fundator
2018 Applied and Computational Mathematics  
of James-Stein estimator against classical least squares as applied to Cumulant Analysis, Associate Random Variables, and Time Series Analysis.  ...  Isoperimetric, Milman reverse, Hilbert, Widder, Fan-Taussky-Todd, Landau, and Fortuin-Kasteleyn-Ginibre (FKG) inequalities in n dimensions in investigations of multidimensional estimators support the use  ...  Therefore, the process X (t) with above properties can be represented by composition of Brownian motion processes in finite-dimensional time model. ≤ and N. G. de Bruijn and H. S.  ... 
doi:10.11648/j.acm.20180703.14 fatcat:wl4wlm2g7zcktedq3bcozxiwre

Products of Universal Cycles [chapter]

Persi Diaconis, Ron Graham
2008 A Lifetime of Puzzles  
Introduction A de Bruijn cycle is a sequence of zeros and ones such that each window of width k running along the sequence shows a different binary k-tuple.  ...  A friendly introduction to de Bruijn sequences is in Stein [15]; a more comprehensive survey is in Fredericksen [4]; see Knuth [13] for an extensive discussion.  ...  The y-sequence uses zeros and ones, such as 0011, for the usual de Bruijn sequence for window width k = 2. Here, RS = 12.  ... 
doi:10.1201/b10573-6 fatcat:3mndykydcbe6jgzai3c7q7u6pa

The Monotonicity of Information in the Central Limit Theorem and Entropy Power Inequalities

Mokshay Madiman, Andrew Barron
2006 2006 IEEE International Symposium on Information Theory  
Extensions to the more general case of independent, not identically distributed summands are also presented. New families of Fisher information and entropy power inequalities are discussed.  ...  We provide a simple proof of the monotonicity of information in the Central Limit Theorem for i.i.d. summands.  ...  Proof: Recall the integral form of the de Bruijn identity, which is now a standard method to "lift" results from Fisher divergence to relative entropy.  ... 
doi:10.1109/isit.2006.261882 dblp:conf/isit/MadimanB06 fatcat:oaghpoohsvctrfiggqxofypcuy

Generalized Entropy Power Inequalities and Monotonicity Properties of Information

M. Madiman, A. Barron
2007 IEEE Transactions on Information Theory  
As a consequence, a simple proof of the monotonicity of information in central limit theorems is obtained, both in the setting of independent and identically distributed (i.i.d.) summands as well as in  ...  New families of Fisher information and entropy power inequalities for sums of independent random variables are presented.  ...  ACKNOWLEDGMENT We are indebted to Prasad Tetali for several interesting conversations and for the beneficial influence of Prasad's joint work with the first author on entropy inequalities.  ... 
doi:10.1109/tit.2007.899484 fatcat:ljw7fozq7ralzjreich662dfna

Maximal correlation and the rate of Fisher information convergence in the Central Limit Theorem [article]

Oliver Johnson
2019 arXiv   pre-print
We consider the behaviour of the Fisher information of scaled sums of independent and identically distributed random variables in the Central Limit Theorem regime.  ...  We prove that assuming this eigenvalue satisfies a strict inequality, an O(1/n) rate of convergence and a strengthened form of monotonicity hold.  ...  The author would like to thank the Associate Editor, and three anonymous referees for their close reading of this paper and extremely helpful suggestions.  ... 
arXiv:1905.11913v2 fatcat:6w6ht3o22ve33c2wqkcamr2xhe

Random Feedback Shift Registers, and the Limit Distribution for Largest Cycle Lengths [article]

Richard Arratia and E. Rodney Canfield and Alfred W. Hales
2021 arXiv   pre-print
Such behavior was conjectured by Golomb, Welch, and Goldstein in 1959.  ...  in distribution to the same Poisson-Dirichlet limit as holds for random permutations in 𝒮_N, with all N!  ...  to respect the de Bruijn property.  ... 
arXiv:1903.09183v3 fatcat:cpqdguhjmfda3j6s5psaep4iwy

Hessian transport gradient flows [article]

Wuchen Li, Lexing Ying
2019 arXiv   pre-print
The new gradient flow is a generalized Fokker-Planck equation and is associated with a stochastic differential equation that depends on the reference measure.  ...  Several examples of Hessian transport gradient flows and the associated stochastic differential equations are presented, including the ones for the reverse Kullback--Leibler divergence, alpha-divergence  ...  Our study is the first step to bridge Hessian geometry, Wasserstein geometry, and divergence functions. Several fundamental questions arise.  ... 
arXiv:1905.04556v2 fatcat:74mxpl4pwzarfcexgq7ehhyaeq

Counting Even and Odd Partitions

Martin Klazar
2003 The American mathematical monthly  
I would like to thank an anonymous referee for interesting and useful comments.  ...  I was supported by the project LN00A056 of the Ministry of Education of the Czech Republic. I am grateful to M. Noy for his comments.  ...  See de Bruijn [2, p. 108] or Lovász [7, Problem 1.9b] for more precise asymptotics. We show that S ± n remains superexponential. Theorem 3.  ... 
doi:10.2307/3647908 fatcat:dqksopapvbdp7obra6r2ozkcri
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