Mathematics 741 - Advanced Probability Theory I - Fall 2008 - Zirbel Analysis warmup. These problems are representative of the analysis problems that arise in Mathematics 741. They are in no particular order and are of varying diﬃculty levels, so try them all rather than stopping when you ﬁnd a problem you cannot solve. Attempt all problems. I am interested in seeing partial solutions. 1. Let X : Ω → R . Set X + = max(0 ,X ) and X-= max(0 ,-X ). Show that X = X +-X-and that | X | = X + + X-. Do this by considering X ( ω ) for a given ω in Ω. 2. The squeeze law. Suppose that a n ≤ b n ≤ c n for all n = 1 , 2 ,... , and that ( a n ) and ( c n ) converge to the same ﬁnite limit L as n → ∞ . Show that ( b n ) converges to L as well. Use an ε-δ argument, not limsup. 3. If ( a n ) converges to a ﬁnite limit a as n → ∞ , show that lim N →∞ 1 N N X n =1 a n = a . 4.
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