Monday, 23 September 2013

Jensen's inequality for random elements in metric spaces and some applications

P. Terán (2014). Journal of Mathematical Analysis and Applications 414, 756-766.

(By the way, this is my thirtieth paper.)

This paper has a funny story. I was invited to give a talk a couple of years ago. From the background of the people issuing the invitation, it looked clear that they had found my second paper with Ilya Molchanov. The situation was awkward, because it simply makes no sense to fly somebody from abroad to give you a two-hour lecture on a topic he has only written one paper about. So I expected that they would soon realize I was not fit for what they wanted and the invitation would be withdrawn (and that's what happened).

But in the meantime I grew increasingly concerned. What if they did know what they were doing? What if they just had an insane lot of money to spend? If I waited and the invitation never got withdrawn, I 'd have to show up and speak for two hours, and what was I going to say?

The topic of the original paper was the Law of Large Numbers for random elements of metric spaces. Under some axiomatic conditions on the way averages are constructed (maybe not via algebraic operations), we constructed an expectation operator and proved the LLN for it. It seemed to me that the first thing those people were going to ask me was: What are the properties of that expectation? Does it enjoy some of the nice properties of the expectation defined in less general spaces using Lebesgue or Bochner integrals? Unfortunately the paper, being a paper, had paid no attention to any properties unnecessary to acheive the paper's aim (proving the Law of Large Numbers).

I thought: I will prove Jensen's inequality for that expectation. That way they will realize that it is well-behaved and plausibly has more nice properties, even if I can't claim that it has.

Once it became clarified that the talk would not happen, I worked for some time on applications and called it a paper. It's fun because the paper's path is quite unusual: we prove Jensen's inequality from the Law of Large Numbers; then we prove a Dominated Convergence Theorem from Jensen's inequality; and then we prove a Monotone Convergence Theorem from the Dominated Convergence Theorem.

Abstract: Jensen's inequality is extended to metric spaces endowed with a convex combination operation. Applications include a dominated convergence theorem for both random elements and random sets, a monotone convergence theorem for random sets, and other results on set-valued expectations in metric spaces and on random probability measures. Some of the applications are valid for random sets as well as random elements, extending results known for Banach spaces to more general metric spaces.

Up the line:
·A law of large numbers in a metric space with a convex combination operation (2006, w. Ilya Molchanov). Downloadable from Ilya's website.

Down the line:
Nothing being prepared.

To download, click on the title or here.

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