Thursday, 24 October 2013

Expectations of random sets in Banach spaces

P. Terán (2014). Journal of Convex Analysis 21(4), to appear.

A random set is a random element of a space of sets. Since the latter are not linear (in general, you cannot subtract a set from another), it is not possible to define a notion of expectation for random sets with all the properties of the usual expectation of random variables or vectors. Thus there exist many definitions of the expectation, with various properties.

Two very interesting definitions were proposed by Aumann and Herer. Aumann's expectation is defined by putting together the expectations of all selections of the random set (i.e. if a random variable/vector is taken by selecting one point of the random set, its expectation should be an element of the expectation of the random set). Thus it is analytical in that it relies on calculating integrals in the underlying linear space.

Herer's expectation is not analytical but geometrical, as it only uses the metric structure of the space. It is defined as the locus of all points which are closer to each point x than x is, in average, to the farthest point of the random set. In other words, call R(x) the radius of a ball centered in x that covers the random set, then the Herer expectation is the intersection of all balls with center any x and radius the expected value of R(x).

The aim of the paper is to study the relationships between those two notions. Since the Herer expectation is an intersection of balls, the simplest possibility is when it is either equal to Aumann's, or the intersection of all balls covering it. The first case I had already studied, though there was a gap in the proof of the non-compact case which is corrected here.

The main types of results are:
1. Sufficient conditions on the norm for the equality between the Herer expectation and the ball hull of the Aumann expectation.
2. Sufficient conditions on the kind of sets the random set takes on as values.
3. Inclusions valid without restricting either the norm or the possible values of the random set.

The paper is a very nice amalgam of random sets, bornological differentials, and Banach space geometry. I think it makes a convincing case of how all those ingredients fit together.

Up the line:
·(2008).
·Intersections of balls and the ball hull mapping (2010).

Down the line:
A paper on limit theorems is in the making.