What’s a good (fairly advanced) book on probability, cumulants and CLT please?

Question

Can anyone recommend a book covering probability, cumulants and a proof of the CLT (ideally including a proof of Levy’s continuity theorem).

I studied Math undergrad and PhD many years ago, but don’t work in the field.

I’ve read Yuval Filmus‘s note on the CLT and the Wiki pages. Cumulants make intuitive sense, but e.g. I can’t see a proof of Levy’s continuity theorem online.

As a bonus, if there’s any version of the CLT for correlated random variable size , that would be interesting.

Thanks

Answer 1

I would recommend Introduction to mathematical statistics by Robert V. Hogg, Joseph Mckean, Allen T. Craig 7th Edition. (I'm not sponsored)

The first chapter gives you pretty detailed information about the probability and distributions. I also recommend you to look at the second and the third chapter which talks about multivariate distribution and some special distributions.

If you go to the Index page of the book, you can look for the CLT. The book that I currently own indicates that page 170,220,225,249,313,317,323-324, 340, 342, 352,370, 448, 456, 463, 541, 552, 570, 605, and 626 have information related to CLT.

I was struggling a lot with statistics, not coming from the math or stat background and this book really helped me a lot.

Thanks.

Answer 2

Thanks all. In the end I went for Probability with Martingales by David Williams. Seems like tough going but hopefully will be worthwhile in the end.