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Last week (10. Feb to 16. Feb) we talked about pi-systems and d-systems. We proved that a d-system that is also at pi-system is a sigma-algebra. Then we proved Dynkin's lemma, and the corollary saying that two finite measures that agree on a pi-system and agree and are equal when evaluated \Omega, agree on the sigma-algebra generated by that pi-system. On Thursday we proved a lemma about conditional expectation, then we derived the inversion formula, and proved that c.f.'s are unique. Finally, we derived an inversion expression for the density of random variables whose c.f. are integrable. 

Exercises: 2.17, 2.18, 2.20, 2.27, 2.28, 2.31, 2.38

This week we have finished up the proof of the Portmanteau theorem. Then we proved a lemma showing that for weak convergence to take place, it is enough that the probability measure involved converge for all sets in a pi-system that are such that all open sets can be written as finite unions of sets in this pi-system. That weak convergence is equivalent to c.d.f. convergence is a corollary of this lemma. Next we proved various lemmata combining convergence in distribution and convergence in probability, ending with the Cram{\'e}r--Slutsky rules.Then we got to tightness. We have proved Helly's theorem in the real line, and Prokhorov's theorem. Lastly, we introduced characteristic functions (c.f.), and found the c.f. of a standard normal random variable. 

Exercises: 2.22--2.26, as well as A.36, A.39, and A.40

This week we have proved a lemma relating convergence in probability and convergence almost surely, and used this to prove the continuous mapping theorem for convergence in probability. We have proved the Portmanteau theorem, and (almost) proved that any of the statements of that theorem are equivalent with the c.d.f. definition of convergence in distribution. See Ex. 2.19(e) in the errata to the book. We have also proved the continuous mapping theorem for convergence in distribution. Next week we'll get to the Cramer--Slutsky rules, and then introduce the notion of tightness. 

This week you may try Exercises~2.9, 2.11, 2.12, 2.13, 2.15, 2.16, and 2.19(e). For 2.19(e), see the errata in stk4090errata.pdf

We've started the course! Emil gave a general introduction on Tue Jan 21, and pointed to themes of the main curriculum-to-be: decently big chunks of Chs 2, 5, 9, somewhat smaller chunks of Chs 6, 7, and a dozen or so Statistical Stories. Emil captains the first weeks, then Nils does half of Ch 5, etc. 

Exercises, for Thu Jan 23 and next week: (a) so much as you can of Exercises 2.1 - 2.7; (b) skim through the pages of Ch1, to check that you know the basics (if not all details). 

Note: the preliminary versions PartOne and PartTwo will be shortened, after more polishing & processing. Also: all comments on the book-to-be are welcomed. And Emil promised you could all email him even in the middle of the night. 

Emil & Nils 

We've placed PartOne (lots of exercises) and PartTwo (lots of Statistical Stories) on the site; take copies for your own computers (as the files may be taken away from the course site later). These are from a preliminary January 2025 version of the book-to-be, "Statistical Inference: 777 Exercises, 77 Stories, and Solutions", by NL Hjort and EAa Stoltenberg. 

The course material will essentially be: big chunks of Chs 2, 5, 9, plus the Appendix; smaller chunks of Chs 6, 7; and a dozen or so of the Statistical Stories. You are also expected to know the core material covered in Chs 1, 3, 4, rom STK 4011/9011 or related courses elsewhere, covering statistical models, parameters, confidence, testing.

A more detailed list for the curriculum will come later.

Nils and Emil

The Statistical Large-Sample Theory course stk 4090 (with stk 9090 for the PhD subset) will be given by Nils Lid Hjort and Emil Stoltenberg, with teaching taking place Tuesdays and Thursdays 10:15 to 12:00. We start Tuesday January 21.

The course material will be based on preliminary versions of chapters from the forthcoming book "Statistical Inference: 777 Exercises, 77 Stories, and Solutions to All", by Hjort and Stoltenberg (Cambridge University Press, 2025); a pdf will be given to the course participants in early January. Specifically, the chapters involved are 1, 2, 3, 5, 7, 9, 10, and a proper subset of the Statistical Stories will also be included.

You may skim through the 95 pages of course notes, by Hjort, on the course site for 2020.