TY - JOUR

T1 - The scaling mean and a Law of Large Permanents

AU - Bochi, Jairo

AU - Iommi, Godofredo

AU - Ponce, Mario

N1 - Funding Information:
J.B., G.I. and M.P. were partially supported by the Center of Dynamical Systems and Related Fields code ACT1103 and by FONDECYT projects 1140202 , 1110040 , and 1140988 , respectively.
Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/4/9

Y1 - 2016/4/9

N2 - In this paper we study two types of means of the entries of a nonnegative matrix: the permanental mean, which is defined using permanents, and the scaling mean, which is defined in terms of an optimization problem. We explore relations between these two means, making use of important results by Egorychev and Falikman (the van der Waerden conjecture), Friedland, Sinkhorn, and others. We also define a scaling mean for functions in a much more general context. Our main result is a Law of Large Permanents, a pointwise ergodic theorem for permanental means of dynamically defined matrices that expresses the limit as a functional scaling mean. The concepts introduced in this paper are general enough so to include as particular cases certain classical types of means, as for example symmetric means and Muirhead means. As a corollary, we reobtain a formula of Halász and Székely for the limit of the symmetric means of a stationary random process.

AB - In this paper we study two types of means of the entries of a nonnegative matrix: the permanental mean, which is defined using permanents, and the scaling mean, which is defined in terms of an optimization problem. We explore relations between these two means, making use of important results by Egorychev and Falikman (the van der Waerden conjecture), Friedland, Sinkhorn, and others. We also define a scaling mean for functions in a much more general context. Our main result is a Law of Large Permanents, a pointwise ergodic theorem for permanental means of dynamically defined matrices that expresses the limit as a functional scaling mean. The concepts introduced in this paper are general enough so to include as particular cases certain classical types of means, as for example symmetric means and Muirhead means. As a corollary, we reobtain a formula of Halász and Székely for the limit of the symmetric means of a stationary random process.

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U2 - 10.1016/j.aim.2016.01.013

DO - 10.1016/j.aim.2016.01.013

M3 - Article

AN - SCOPUS:84957089069

SN - 0001-8708

VL - 292

SP - 374

EP - 409

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -