SS 9657 - Announcements


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Last updated November 16, 2017, 11:21

1) The first 43 pages of the course notes have been updated. The changes are mainly cosmetic except Theorem 93 and Lemma 105, where more details were added.

2) By accident I have uploaded an older version of Problem Set 3, which included an error in  problem (vi). Please download the modern day version. Unfortunately it contains 3 more problems but they should not be difficult.

  1. 3) This is a strategy for Problem (vi) on Problem set 3.

  1. a)Show that if \{A_n\} is a sequence of events such that P(A_n) -> 0 and \sum_{n=1}^\infty P(A_n^c \cap A_{n+1}) < \infty, then P(limsup A_n) = 0.

  1. b)Let B_n and C_n are the events from the hint in class. As explained in class show that P(B_n^c \cup C_n^c) = (1-1/n^{1-\epsilon})^n + 1 - (1-1/n^{1+\epsilon})^n.

If it happened that \sum_{n=1}^\infty P(B_n^c \cup C_n^c) < \infty, then...we would have P(liminf (B_n \cap C_n)) = 1.

But unfortunately that is not the case.

So, let A_n := B_n^c \cup C_n^c and show that the events \{A_n\} satisfy the condition in part a) above.

In particular show that for all n large P(A_n^c \cap A_{n+1}) \le (1-1/(n+1)^{1-\epsilon})^{n+1} + 1/(n+1)^{1+\epsilon}.

  1. 4)About Problems (v) on Problem set 3. The sets limsup \{X_n / log(n) \le 1+\epsilon \} and

\{ \limsup_{n \rightarrow \infty}  X_n / log(n) \le 1+\epsilon \} are different!!! Well, the first contains the second, but the second is the one you need.  This first is too big to be useful.

About Problem (vi) on Problem set 3: Let M_n:= \max_{1\le k \le n} X_n / long(n).

Show and use that  liminf  \{ 1-\epsilon  \le M_n  \le 1+\epsilon \} is contained in

\{ 1-\epsilon  \le \liminf_{n \rightarrow \infty} M_n \le \limsup_{n \rightarrow \infty} M_n  \le 1+\epsilon \}

5) Pages 44 to 58 of the course notes have been updated. The changes are mainly cosmetic.

7) About Problem Set 5 (v).

For part (a): show that since F_Y is continuous F_Y(F_Y^{-1}(\omega)) = \omega for all \omega \in (0,1)...

For part (b): show and use that {F(Y) < F(y)} \subset {Y <= y} \subset {F(Y) <= F(y)}...

  1. 6)About Problem Set 6 (vii). When E|X|^q = \infty, then the inequality holds trivially, but  what happens

when E|X|^p = \infty?

8) The course notes have been updated. The major changes are Section 2.1.2 and Exercise 41, but they are optional for you. I will experience a great joy if you read the new material and tell me that you did so. Still, the small changes are numerous, so I suggest you print a fresh copy of the notes. I will not be issuing another update this semester.