[SOLVED] Math154 homework 6- stochastic convergence p0

MATH154
Homework 6
Stochastic Convergence
Problem 6.1: Consider the random variables Xn(x) = cos(nx) on [−π,π],B,dx/(2π).
a) By writing cos(nx) = Re(einx) and using a geometric series, verify that
.
This is Dn(x) − 1, where Dn(x) is called the Dirichlet kernel.
b) Verify that
c) First recollect from class why the assumptions of the weak law aresatisfied and restate the conclusion of that theorem about Sn/n. This should verify that Sn/n → 0 in L1 and so in probability.
d) Given a continuous even function f with E[f] = 0, the expectation an
= E[fXn] is called the n’th Fourier coefficient and g(x) = Pn anXn(x) is the cos-
Fourier series of n. The formula is called
Parceval’s identity. What geometric condition does assure it and what famous geometric theorem does it generalize?
Problem 6.2: a) Give an example of a sequence of random variables Xn → X for which we have convergence in probability but not complete convergence.
b) Give an example of a sequence of random variables Xn, where Xn → X in probability but where Xn → X in L1 does not happen.
c) Give an example of a sequence of random variables where Xn → X
in L1 but where the convergence is not in L2. Probability Theory
Problem 6.3: a) Is there for 1 ≤ p < ∞ a relation between Lp convergence and convergence almost everywhere?
b) Is there a relation between L∞ convergence and convergence almost everywhere?
c) Is there a relation between complete convergence and Lp convergence for p < ∞?
d) Is there a relation between complete convergence and L∞ convergence?
Law of Large numbers
Problem 6.4: The n’th Chebyshev polynomial is defined as Xn = Tn(x) = cos(narccos(x)). We have Tn(cos(t) = cos(nt).
a) Verify that Tn(x) is a polynomial of degree n and write down Tn(x) for n = 1,2,3,4.
b) We look at Tn(x) as a random variable on the probability space (Ω
= [−1,1],B,P = ). Check that the later indeed is a probability space.
c) Demonstrate (by showing all conditions) that you can use the weak lawof large numbers to establish that (1/n)Sn converges in probability to
0.
Problem 6.5: Let Xn(ω) be the n’th binary digit of ω ∈
[0,1]. a) Investigate the convergencein probability.
b) Verify that Sn has the Binomial distribution
.
c) Show directly and then use the weak law to see that Sn/n → 0 in probability. √ 2.

d) Verify that Sn/ n does not go to zero in√ L
e) Verify also that Sn/ n does not converge to 0 in distribution.