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MATH 6040 (Fall '18)

There will be a

It is mostly out of the textbook and will be posted below.

A big part of your grade will come from homework.

Make sure you work on it in a timely fashion, otherwise you WILL fall behind.

There will be

1.1, 1.3, 1.9, 1.11, 1.12, 1.13, 1.16, 1.22, 1.24, 1.28

2.1, 2.3, 2.4, 2.5, 2.7

Prove Lemmas 3.4, 3.9, and 3.11 (one of the two claims needs finiteness, which one?!)

Prove that the Borel sigma-algebra on (0,1] is the same as the sigma-algebra generated by intervals of the form (a,b].

3.2, 3.9, 3.10, 3.11, 3.13, 3.14, 3.17(this one shows that existence of nonmeasurable sets relies on the axiom of choice!)

4.1, 4.6, Construct a counter example to Fatou's lemma, 4.16, 4.18, 4.28, 4.29

4.9-4.11, 4.17, 4.21, 4.22, 4.23, 4.24, 4.27

5.2, 5.3, 5.4, 5.7, 5.13, 5.17

5.5, 5.8, 5.14, 5.18

6.2, 6.4, 6.5, 6.7, 6.9, 6.10, 6.14, 6.19, 6.27, 6.28, 6.29, 6.30, 6.37

6.3, 6.6, 6.16, 6.17, 6.20, 6.23, 6.25, 6.26, 6.32

7.2, 7.7, 7.10, 7.16, 7.18 (can use 7.14(3)), 7.19, 7.20, 7.25, 7.26, 7.30, 7.32, 7.40

7.3, 7.6, 7.8, 7.14(3), 7.15, 7.21, 7.42, 7.44

8.3, 8.4, 8.8, 8.14, 8.15, 8.25, 8.31, 8.34, 8.35, 8.39, 8.40, 8.51

8.14 continued: prove that $S_n$ is recurrent when $p=1/2$ and transient when $p\not=1/2$, i.e. $P(S_n\not=0\ \forall n\ge1)=0$ for $p=1/2$ and $>0$ for $p\not=1/2$.

Prove also that when $p=1/2$, $S_n$ is in fact null recurrent, i.e. that $E[\text{return time to }0]=\infty$. (When this is finite, the process is called positive recurrent)

8.16, 8.20, 8.26, 8.27, 8.30, 8.36, 8.38, 8.43, 8.47, 8.48, 8.56