Talk:Transition kernel

Product/composition interchanged

Hello I had some trouble with contradicting definitions in the literature for this article. The two books i am referring to use the product/composition of a kernel interchanged, so a product in Klenke is a composition in Kallenberg, a composition in Klenke is a product in Kallenberg. I chose to stick with Klenke for the following reasons:

• Like this, the composition of kernels corresponds to the composition of the corresponding Operator
• The product measure can be interpreted as a special case of the product of kernels (set $S=\{s\}$ and $\kappa ^{2}\equiv \nu$.)

It would be great if someone has got additional sources on this topic that might help to solve this or at least give a clue about which version is more common. --NikelsenH (talk) 10:19, 6 January 2018 (UTC)

I made a similar observation and I have the same preference as you for the same reasons. I further point out that Kallenberg is not completely consistent in his assignment of words "product" and "composition". I suppose, you were referring to a recent edition of Kallenberg's "Foundations of Modern Probability" or "Random Measures, Theory and Applications", where $\kappa \otimes \nu$ and $\kappa \nu$ are called "composition" and "product", respectively. However, in Kallenberg's 1997 edition of "Foundations of Modern Probability", he calls $\kappa \nu$ a "composition" on page 20, and $\kappa \otimes \nu$ a "product" on page 92. This early edition does not yet have a separate chapter dedicated to a thorough introduction of kernels and related definitions, so the words "product" and "composition" in relation to kernels do not seem to ever appear on the same page in this earlier book. Whether Prof. Kallenberg's change of mind was a thought-through decision or an editorial artefact, remains an open question to me. Maybe someone familiar with research in this niche could comment on the issue. Fortunately, the symbols $\kappa \otimes \nu$ and $\kappa \nu$ seem to be used consistently (if we consider Klenke's $\kappa \cdot \nu$ synonymous to Kallenberg's $\kappa \nu$ when both objects are kernels). AVM2019 (talk) 21:27, 26 December 2022 (UTC)

Motivation

Why are transition kernels crafted this way? What benefits do they offer? 77.164.35.146 (talk) 07:44, 20 August 2023 (UTC)