On the equivalence of Aumann and Herer expectations of random sets

P. Terán (2008). Test 17, 505-514.


Trying to understand when the Herer expectation of a random set falls into the abstract definition of expectation in A law of large numbers in a metric space with a convex combination operation unexpectedly triggered a line of research on the Herer expectation itself.

This paper characterizes those separable Banach spaces where the Herer expectation is identical with the Aumann expectation, thus solving an open problem in Ilya Molchanov's book Theory of random sets as you can read here.

A few other results are presented: for instance, if the dual space is separable then the Aumann expectation is the intersection of the Herer expectations with respect to all equivalent norms. Nice!

It must be mentioned that the proof of the main theorem contains a gap and so applies only to Hausdorff approximable random sets. That does not compromise the solution of the problem as stated in Ilya's book. A different proof will be presented in forthcoming work.


Up the line:
·A law of large numbers in a metric space with a convex combination operation (w. Ilya Molchanov). Downloadable from Ilya's website.

Down the line:
·Intersections of balls and the ball hull mapping (2010).
·Expectations of random sets in Banach spaces (accepted 2013).

To download, click on the title or here. This preprint differs in minor details from the published version.

On convergence in necessity and its laws of large numbers

P. Terán (2008). In: Soft Methods for Handling Variability and Imprecision, 289--296. Springer, Berlin.

[Proceedings of the 4th Intl. Conf. on Soft Methods in Statistics and Probability]


An interesting question is what happens to random variables when the probability measure is replaced by a non-additive measure. That topic has been intermitently studied in the fuzzy literature for over 20 years, and has also received attention from economy theorists.

I've had a few ideas about that for some time. I took the opportunity to give an invited lecture at the University of Extremadura to put them in order and start writing a paper -which, however, won't be ready until the long awaited 36-hour-day regulations will be enforced.

This paper presents some convergence results, in an attempt to clarify the difference between LLNs for possibilistic variables and LLNs for their distributions identified with fuzzy sets.

It also shows that usual techniques, relying on shape assumptions related to the t-norm generators, can be effectively replaced by other techniques independent of the particularities of the t-norm modelling the interactivity between the variables.


Up the line:
Strong law of large numbers for t-normed arithmetics (2008)

Down the line:
An evolution of this paper with proofs and new results will be typed later this year. (I don't think that will happen. It's 2013 now and I doubt a journal would be excited to publish a full-length version of a 2008 conference paper. Why bother typing it then?- PT, Mar 18th 2013)


To download, click on the title or here.

Strong law of large numbers for t-normed arithmetics

P. Terán (2008). Fuzzy Sets and Systems 159, 343-360.


This paper was conceived and written in late 2001 and early 2002. For years, I stubbornly tried to have it published in a Statistics & Probability journal. A tale of this epic yet unfruitful quest was told in my personal blog (in Spanish).

I finally quit and submitted it to FSS, where I knew knowledgeable reviewers would be used.

Young people do all sorts of unrewarding things.


Up the line:
This started a new line.

Down the line:
A lot of subsequent material found its way to publication before the paper itself did.
·On limit theorems for t-normed sums of fuzzy random variables (2004).
·A law of large numbers in a metric space with a convex combination operation (2006, w. Ilya Molchanov). You may download it from Ilya's website.
·Probabilistic foundations for measurement modelling with fuzzy random variables (2007).
There is a further paper on the making, quite interesting.

To download, click on the title or here.

On a uniform law of large numbers for random sets and subdifferentials of random functions

P. Terán (2008). Statistics and Probability Letters 78, 42-49.


While spending a night in the (South) Tenerife Airport, and that means a lot of time to kill, I read a JMAA paper by Alexander Shapiro and Huifu Xu where they obtained a strong law of large numbers for subdifferentials of random functions as a tool for consistency analysis of stationary points in non-convex non-smooth stochastic optimization (math jargon rules, doesn't it?)

Their LLN, however sufficient for their purpose, depended on a sort of `blurring radius' parameter r>0 and so didn't cover the `exact' case r=0 unless the functions be additionally assumed to be continuous.

That looked like a perfect benchmark for the abstract LLN Ilya Molchanov and I had proved. It turned out that the strongest case r=0 held under upper semicontinuity (provided a separability condition on the range of the multifunction) or even weaker conditions, showing that continuity in fact played no role in the problem.


Up the line:
A law of large numbers in a metric space with a convex combination operation (w. Ilya Molchanov). You may download a (non-final) preprint copy from Ilya's website.

Down the line:
On consistency of stationary points of stochastic optimization problems in a Banach space (200x).

To download, click here. This is *the* good version; SPL has published their own version which I don't endorse in any way.

A continuity theorem for cores of random closed sets

P. Terán (2008). Proceedings of the American Mathematical Society. 136, 4417-4425.


The starting point is Zvi Artstein's 1983 paper on distributions of random sets. Among other things, he proved that, if a sequence of distributions of random compact sets converges weakly in the Hausdorff metric, their cores (or sets of distributions of selections) also converge in the Hausdorff metric defined in the space of compact sets of distributions. The proof is a quite laborious one, and I have never been able to read it through.

On a rainy summer afternoon I killed some time by proving it using the Skorokhod representation theorem. I worked a bit harder and adapted the new proof to get an extension to the unbounded case in locally compact separable Hausdorff spaces with the Fell topology. Then I checked Artstein again and saw that he already knew that (using the one-point compactification-- quite smarter than me). I realized that, to get something publishable, I would need a proof of the general unbounded case.

By Christmas, I had that general proof. It involves some results and notions from Hyperspace Topology which were developed in the nineties. It is a quite symphonic proof, with a large number of elements assembled together in a very nice way.


Up the line:
This starts a new line.

Down the line:
Nothing yet. I have some material I will finish and prepare for publication as soon as 36-hour days are available.

To download, click on the title or here.