Problems for Thursday, August 29th. From the textbook:
Page 8, problems 1.3, 1.5, 1.6
Pages 13-14, problems 1.8, 1.9, 1.16, 1.18, 1.20
Problems for Thursday, September 5th. From the textbook:
Pages 20-22, problems 1.26, 1.28, 1.31, 1..32, 1.33, 1.34, 1.35, 1.37, 1.38, 1.42, 1.43, 1.44.
Page 44, exercise 2.2.
Problems for Thursday, September 12th. From the textbook:
Pages 46-48, problems 2.4, 2.7, 2.10, 2.11, 2.16, 2.17
Page 49, exercise 2.20
Page 53, problems 2.22, 2.30
Poblems for Thursday, September 19th. From the textbook:
Page 56, Exercise 2.37
Page 84, problems 3.2, 3.3, 3.4, 3.5, 3.6
Problems for Thursday, September 26th.
As there are few problems in the first sections of Chapter 3, we go back to Chapter 2 to pick up some:
Pages 64-65, problems 2.38, 2.40, 2.41, 2,42, 2.43, 2.44, 2.47, 2.50
Problems for Thursday, October 3rd:
Page 120, Ecercise 4.1 (There are misprints in part (a) and (b). The statements should be: \(\liminf_n\Lambda_n=\bigcup_{m=1}^{\infty}\bigcap_{j=m}^{\infty}\Lambda_j\) og \(\limsup_n\Lambda_n=\bigcap_{m=1}^{\infty}\bigcup_{j=m}^{\infty}\Lambda_j\))
Page 126, Exercise 4.3
Paghe 128, problems 4.4, 4.5, 4.7, 4.9
Problems for Thursday, October 10th:
Page 126-128, problems 4.11, 4.12, 4.13, 4.23, 4.26, 4.28
Problems for Thursday, October 17th:
Page 143, Exercise 5.6
Page 145, Exercise 5.7, 5.8
Page 147-148, Problems 5.9 (assume \(p>0\)), 5.10, 5.11, 5.12, 5.13
There is a confusing misprint in 5.11d) where \(X_k^2\) (in the formula), should be \(S_k^2\).
Otherwise, the degree of difficulty varies a lot this week, but much should be manageable. I do think, however, that Exercise 5.12 should have had a hint or a part a): Before you attempt the problem, it is a good idea to show that if \(\lim_{n\to\infty}\lim_{m\to\infty}P(\sup_{n<k\leq m}|S_n-S_k|\geq\delta)=0\) for all \(\delta>0\), then the sequence \(\{S_n\}\) converges a.s.. (Think Cauchy sequences.)
Problems for Thursday, October 24th:
Page 161-162: Problems 6.2, 6.3, 6.4, 6.5, 6.6, 6.8
There is a misprint in 6.4; the right hand side of the formula should be \(e^{\frac{z^2\sigma^2}{2}+z\mu}\) - plus instead of minus in the last term (and don't make this problem into a competition in contour integration if you don't want to; a hand waiving argument suffices). There is also a condition lacking in 6.6a): The \(X_n\)'s should be assumed to be independent (Extra problem: Show by example that the statement is false without this extra condition)
Otherwise the problems this week do not seem too unreasonable; just remember that if two distributions have the same characteristic function, then they are equal.
Problems for Thursday, October 31st:
Page 162: Problems 6.9, 6.11, 6.12
Page 175: Problems 6.21, 6.22, 6.24
Most of these problems should be reasonable, although there may be an advantage to have some experience of real analysis for some of them.
Problems for Thursday, November 7st:
Page 188: Problems 6.26, 6.29, 6.31, 6.34 (the last one assumes that you know what a metric is. If you don't, wait for the presentation!)
Page 199: Exercise 7.1
Page 211: Problems 7.14, 7.15, 7.16.
Problem 6.34 is very tough, especially showing that convergence in distribution implies convergence in the Lévy metric. It's a help first to find the following description of the Lévy metric:
\(d(F,G)=\inf\{\epsilon>0:\forall x[G(x-\epsilon)-\epsilon\leq F(x)\leq G(x+\epsilon)+\epsilon]\}\)
but even then the problem is tough.
Problems for Thursday, November 14th:
Page 274: Problems 8.1, 8.2, 8.3, 8.5, 8.6, 8.7
Comment: Problems 8.2, 8.3, 8.5, and 8.6 can be done quite quickly if you have the right approach. In 8.7 the way of attack should be clear, but you have to juggle the limits the right way. I haven't found a quick way of doing 8.1, and this seems to be the toughest problem this week. This problem also uses a notation I am not quite sure has been introduced: If \(A\) is an event, and \(\mathcal{F}\) is a \(\sigma\)-algebra, interpret \(P[A|\mathcal{F}]\) as \(E[\mathbb{1}_A|\mathcal{F}]\)
Problems for Thursday, November 21st:
Pages 288- 290: Problems 9.2, 9.3, 9.5, 9.7, 9.8, 9.9, 9.10 (for the notation \(U(0,1)\) in problem 9.10, see page 89)
November 28th:
I will not do problems on November 28th, but concentrate on reviewing the syllabus. If you want to do problems on your own, you may try 9.17-9.21.