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.
Tuesday, 13 November 2007
Wednesday, 31 October 2007
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.
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.
Friday, 26 October 2007
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.
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.
Thursday, 18 October 2007
A general law of large numbers, with applications
P. Terán, I. Molchanov (2006). In Soft methods for integrated uncertainty modelling (J.Lawry, E.Miranda, A.Bugarin, S.Li, M.A.Gil, P. Grzegorzewski, O. Hryniewicz, editors), 153-160. Springer, Berlin.
[Proceedings of the 3rd Intl. Conf. on Soft Methods in Statistics and Probability] [Invited session Probability of imprecisely valued random elements with applications]
We tried to draw some attention to our JTP paper among the fuzzy community, by showing that spaces of fuzzy sets are examples of the general`convex combination spaces' used there. A c.c.s. is much more general than a Banach space (e.g. convolution of probability measures, max-product, global NPC spaces).
Two applications are presented:
-a strong law of large numbers for fuzzy random variables in non-Banach spaces,
-a strong law of large numbers for level-2 fuzzy random variables.
These results cannot be obtained with the usual methods relying on Banach spaces.
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:
Nothing yet. Some of the compactness methods in the proof are reused in
On a uniform law of large numbers for random sets and subdifferentials of random functions.
To download, click on the title or here.
[Proceedings of the 3rd Intl. Conf. on Soft Methods in Statistics and Probability] [Invited session Probability of imprecisely valued random elements with applications]
We tried to draw some attention to our JTP paper among the fuzzy community, by showing that spaces of fuzzy sets are examples of the general`convex combination spaces' used there. A c.c.s. is much more general than a Banach space (e.g. convolution of probability measures, max-product, global NPC spaces).
Two applications are presented:
-a strong law of large numbers for fuzzy random variables in non-Banach spaces,
-a strong law of large numbers for level-2 fuzzy random variables.
These results cannot be obtained with the usual methods relying on Banach spaces.
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:
Nothing yet. Some of the compactness methods in the proof are reused in
On a uniform law of large numbers for random sets and subdifferentials of random functions.
To download, click on the title or here.
Monday, 15 October 2007
Hi
A fact about modern scientific research, and one researchers may want to ponder, is that no-one cares about you. Just as tons and tons of new papers are being printed and published right as I speak, you should know that Pallas Athena will visit no-one in dreams to tell them to read your paper.
In the old days, researchers went to the university library to browse through a small number of journals, looking for new, interesting papers. Today, a journal's output tends to become more and more heterogenous, so that only a small fraction of papers in a given journal are of interest to any given scientist. As a consequence, the only way to stay up-to-date is to check periodically the websites of at least ten to twenty journals.
In one of those mining sessions, one will read the titles of several hundreds of papers. For each of them, an effort is required to decide whether the title is promising enough to justify downloading and summarily checking the paper.
It is quite easier to draw people to your paper by using your name as a hook. That is, if you have a name. If you don't, and even more if you don't have what it takes to write catchy, slightly deluding titles, you are uphill to having your paper read by more than half a dozen people in the world, your mom included. So much for the advent of the information era.
This blog constitutes, thus, my uphill effort. Although most of my work is easily traceable or accessible using MathSciNet, Sciencedirect or whatever, why not devote a little time to care personally about the people who randomly get to read me? If they have found out something of value, they may google for more with no result.
So you will be able to find here:
·My new papers, as soon as a stable preprint version is available;
·Some of my old papers, whose uploading constitutes no copyright infringement;
·Comments on my papers;
·Conference communications (I typically do not rebuild this material into papers).
I hope some will be interesting to you. The blog structure makes it easy to filter the other material (hints: click on the labels, use the search box) and to interact with me.
In the old days, researchers went to the university library to browse through a small number of journals, looking for new, interesting papers. Today, a journal's output tends to become more and more heterogenous, so that only a small fraction of papers in a given journal are of interest to any given scientist. As a consequence, the only way to stay up-to-date is to check periodically the websites of at least ten to twenty journals.
In one of those mining sessions, one will read the titles of several hundreds of papers. For each of them, an effort is required to decide whether the title is promising enough to justify downloading and summarily checking the paper.
It is quite easier to draw people to your paper by using your name as a hook. That is, if you have a name. If you don't, and even more if you don't have what it takes to write catchy, slightly deluding titles, you are uphill to having your paper read by more than half a dozen people in the world, your mom included. So much for the advent of the information era.
This blog constitutes, thus, my uphill effort. Although most of my work is easily traceable or accessible using MathSciNet, Sciencedirect or whatever, why not devote a little time to care personally about the people who randomly get to read me? If they have found out something of value, they may google for more with no result.
So you will be able to find here:
·My new papers, as soon as a stable preprint version is available;
·Some of my old papers, whose uploading constitutes no copyright infringement;
·Comments on my papers;
·Conference communications (I typically do not rebuild this material into papers).
I hope some will be interesting to you. The blog structure makes it easy to filter the other material (hints: click on the labels, use the search box) and to interact with me.
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