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1. Give an example of two sequences of nonnegative numbers tex2html_wrap_inline37 and tex2html_wrap_inline39 such that

displaymath41

and, for any tex2html_wrap_inline43 , we have tex2html_wrap_inline45 .

2. Explain why the function defined by tex2html_wrap_inline47 for tex2html_wrap_inline49 and 0 for tex2html_wrap_inline53 is not a characteristic function.

3. Give an example of iid tex2html_wrap_inline55 such that tex2html_wrap_inline57 converges in distribution but not in probability. (Hint: upon assuming that tex2html_wrap_inline57 tends in probability to a random variable tex2html_wrap_inline63 , prove that tex2html_wrap_inline63 is constant (a.s.) due to Kolmogorov's 0-1 law.)

4. Give an example of a sequence of moment generating functions tex2html_wrap_inline59 such that tex2html_wrap_inline61 exists for any tex2html_wrap_inline63 , but this limit is not a continuous function of tex2html_wrap_inline65 .

5. Let (X,Y,Z) be a 3 dimensional normal variable. Prove that X,Y,Z are independent iff they are pairwise uncorrelated.

6. Let tex2html_wrap_inline71 be a Wiener process, tex2html_wrap_inline73 . We know that tex2html_wrap_inline75 . In particular, tex2html_wrap_inline77 where tex2html_wrap_inline79 and tex2html_wrap_inline81 is the first time tex2html_wrap_inline71 hits either tex2html_wrap_inline85 or tex2html_wrap_inline87 . By using that tex2html_wrap_inline89 is a martingale, prove that

displaymath91

7. For any constant tex2html_wrap_inline155 and tex2html_wrap_inline162 define

displaymath161

We know that tex2html_wrap_inline163

displaymath165

By using change of variables show that tex2html_wrap_inline163

displaymath169

By letting tex2html_wrap_inline159 conclude that tex2html_wrap_inline161 equals 0 if tex2html_wrap_inline168 and tex2html_wrap_inline165 if tex2html_wrap_inline172 . (Warning: the case tex2html_wrap_inline174 requires a little bit extra attention.)



Nicolai V. Krylov
Mon Jun 1 17:08:42 CDT 1998