Homework 8 Math280A Fall 2020
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Homework 8 Math280A Fall 2020
Due midnight Thursday December 1. This is the last homework assignment of the quarter, on law of large numbers and convergence of random series. Relevant sections in Durrett’s textbook 2.4, 2.5. Justify all your answers.
1. Let X1 , X2 , ... be i.i.d. with distribution
P ╱Xi = (-1)k k、= , k 2 2,
where c is the normalization constant so that the sum of probabilities is 1 over k e {2, 3, . . .}.
Let Sn = X1 + ... + Xn . Show that 匝 [IXi I] = -, but there is a finite constant µ so that Sn /n → µ in probability.
2. Let Xn be a sequence of i.i.d. random variables with 匝 [IXi I] < -, and 匝 [Xi] 0. Show that
→ 0 a.s. when n → -,
where Sn = X1 + ... + Xn .
3. (Betting on favorable game). Suppose you start with $1. On each bet, independently win the amount of your bet with probability + q and lose with probability - q , q e (0, ). Assume we always bet proportion a e (0, 1] of our current fortune. What is the optimal choice of a as a function of q?
4. Suppose {Xn , n 2 1} are independent random variables with 匝[Xn] = 0 for all n. If
匝 ╱Xn(2)己{|X石 |s1} + IXn I己{|X石 |>1}、< -,
n
then n Xn converges a.s.
5. Suppose {Xn , n 2 1} are independent with distributions
P(Xn = n-α ) = P(Xn = -n-α ) = .
Use the Kolmogorov convergence criterion to verify that if α > 1/2, then n Xn converges a.s. Use the three series theorem to verify that α > 1/2 is necessary for convergence.
6. Let X1 , X2 , . . . be i.i.d. and not constant 0. Let T (ω) be the radius of convergence of the power series n21 Xn (ω)zn , that is
T (ω) = sup{s e R2〇 : IXn (ω)Isn < -}.
Show that T (ω) = 1 a.s. or T (ω) = 0 a.s., according to 匝(log+ IX1 I) < - or = -. Here log+ 北 = max{0, log 北}.
2022-12-02