Canonical representation defined

Question

Can anyone tell me what is meant exactly by a Canonical representation of a characteristic function, e.g. the Lévy-Khintchine canonical representation?

I have read several books and articles on the subject of characteristic function and found no definition of the distinction between something being canonical and something not being canonical.

Essentially all of these representations are mathematically identical, but I am wondering, if canonical refers to a certain form of representation? Or mayby something entirely different.

Thaks in advance!

Answer 1

Here is a definition with futher explanation of Levy-Khintchine canonical representation: https://www.encyclopediaofmath.org/index.php/L%C3%A9vy-Khinchin_canonical_representation

I quote the part that is relevant here:

To each infinitely-divisible distribution corresponds a unique set of characteristics $\gamma$ and $G$ in the Lévy–Khinchin canonical representation, and conversely, for any $\gamma$ and $G$ as above, the Lévy–Khinchin canonical representation determines the logarithm of the characteristic function of an infinitely-divisible distribution.

Generally speaking: a representation of a probability distribution in the sense used here is 'a way to write it'. As you remark: there are many ways to write the same probability distribution. What makes a way of writing canonical is that

1) its form or shape can be easily described using some dummy variables (e.g. in the example linked above $\gamma$ and $G$ will vary from distribution to distribution but for each distribution the LK-canonical representation looks like the formula given for some value of $\gamma$ and $G$)

and

2) that for each distribution there is a UNIQUE representation that has that form. So the distribution determines the value of the variables ($\gamma$ and $G$ in the example) appearing in the expression of what the canonical representation should look like.

A better known example of the same use of 'canonical' is the Jordan Canonical Form of a linear transformation. Yes, for every linear transformation there are many equivalent matrices representing it, but once you insist that the representation should have this particular form (with the Jordan blocks etc) it is (almost) uniquely determined.

This does not preclude the existence of other classes of representation that might be called canonical for the exactly the same reason. In fact this happens. Staying with the example of canonical forms of matrices Wikipedia lists several: https://en.wikipedia.org/wiki/Canonical_form.