Guide Functional Equations in Probability Theory

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Econometrica 22 , — Aubin and H. Frankowska: Set-Valued Analysis.


Aumann: Integrals of set-valued functions. Cardinali, K. Nikodem and F. Papalini: Some results on stability and characterization of K-convexity of set-valued functions. Cascales and J. Rodrigeuz: Birkhoff integral for multi-valued functions. Castaing and M.

Valadier: Convex Analysis and Measurable Multifunctions. Notes in Math.

Cadariu and V. Pure Appl. ID 4 Radu: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math.

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Developments in Functional Equations and Related Topics

Radu: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. ID Debreu: Integration of correspondences.

II, Part I , — Diaz and B. Margolis: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Eshaghi Gordji, C. But look at the list of topics: dynamics, functional equations, infinite combinatorics and probability. How could I resist that? Of course, with a conference like this, the first question is: is the conference on the union of these topics, or their intersection?

The conference logo strongly suggests the latter. But as usual I was ready to plunge in with something connecting with more than one of the topics. I have thought about functional equations, but only in the power series ring or the ring of species ; but it seemed to me that sum-free sets would be the best fit: there is infinite combinatorics and probability; there is no dynamics, but some of the behaviour looks as if there should be!

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The organisers were Adam Ostaszewski and Nick Bingham. So the conference was centred on their common interests. I will say a bit about this in due course. I will try to give an overview of what I heard, but will not cover everything at the same level. Of course, a topologist, analyst or probabilist would give a very different account!

He talked about topological vector spaces carrying measures; in such a space, consider the set of vectors along which the measure is differentiable in one of several senses , one of which is apparently called the Cameron—Martin space. He had something called D C and explained that he originally named it after a Russian mathematician whose name began with S in transliterated form ; he had used the Cyrillic form, never imagining in those days that one day he would be lecturing about it in London!

David Applebaum talked about extending the notion of a Feller—Markov process from compact topological groups to symmetric spaces, and considering five questions that have been studied in the classical case, asking how they extend. People have known for a long time that these concepts are closely related, but he and his postdoc have a theorem to this effect.

In a compact 0-dimensional space like Cantor space, their theorem says that a map has the shadowing property if and only if it is conjugate to an inverse limit of shifts of finite type. Over arbitrary compact metric spaces they also have a necessary and sufficient condition, but it is more complicated, involving the notion of semi-conjugacy.

He explained that the aim of this kind of dynamics is to replace a complicated map on a simple space by a simple map a shift on a more complicated space finite type. I was the first speaker next morning. I arrived half an hour early, to find the coffee laid out but nobody else around. Soon Rebecca Lumb came along and logged in to the computer, so I could load my talk. I found that the clicker provided had the feature that the left button advances the slides, so I took it out and put in my own, which works the right way round.

Foundations of Information Theory. Fischer, P. Metrika 18 , — Fisher, I. The Making of Index Numbers. Kelley, New York. Forte, B. Hardy, G. University Press, Cambridge, pp. Hartley, R. Bell System Tech. Acta Math. Kannappan, P. Nath, P. Rao, C. MathSciNet Google Scholar. Redheffer, R. Probability Theory.

2019 IMO Problem 1 Solution: Functional Equation

Rota, G. Academic Press, New York, pp. Shanbhag, D.