Problem Sets in Real Analysis and Probability Theory

Problem 1: Cauchy sequence and convergence

Problem 1 .Let (as n ‹E N/ be a real-valued sequence, defined by n   if n is even (—1) ?  if n is odd Is /a ,} Cauchy? Why or why not? Your answer should not depend on any results other than the definition of a Cauchy sequence and the definition of p . Does {andconverge as n —  ? Why or why not?

Problem 2 (Convergence for random variables). Let X be a uniform random variable on the unit interval [0, 1]. Let Yr —— go (K, where g is defined as follows. First, for any n a 1, let m be the largest integer such that 2 “ 1 /?. Then note that n can be written uniquely as n —— 2 “ + j, where 0 1 j < 2 “ . For such n, define where / denotes the indicator function.

Show that go ( converges in probability to zero, but for any value of A, there exists an infinite sequence of integers p where g„ ( is zero and an infinite sequence of integers nd where g„ (  iz 1. Does g ( converge to zero with probability one?

Problem 3. For each n * 1, let  n be a uniformly distributed random variable on the interval [0,1//?]. For x > 0, what is the limit of P (X n * x) as n

—  ? What is the limit of P (X  n 1 0) as n

—  ? If x < 0, what is the limit of P (X  1 x)?

Problem 4 (Adapted from Naiman: Delta-method, the Central Limit Theorem, and approximate distributions). Let X 1,    be an iid sample frog some distribution having mean  and variance cr.Recall that the Central Limit Theorem says that  n(X — p)  has an approximate N (0, ²) distribution as n tends to infini.

Let n(g(be a smooth function with g  ) 6—— 0. The purpose of this exercise is to justify the claim— g )) has distribution that is approximately N (0, g  (p) ² 6).

(a) Write down a first order Taylor's expansion with remainder for g about y.  (Here the remainder term should be quadratic in x.)

(b) Substitute X for x in the expansion in (a) and use this to express   n(g(  — g(p)) as a sum of two terms, one involving (X — y) (the linear term) and the other involving (X — y)  ² (the quadratic term).

(c) Explain why the linear term should have a distribution that is approximately N (0, g  (y) ²o²) as /? tends to infinity.

(d) Slutsky's Theorem, which we will  cover in lecture, concerns how convergence in probability and in distribution is impacted by sums and products.

Explain why the quadratic term should tend to zero in probability as n tends to infinity, and then use Slutsky's Theorem to show the distributional result you are required to prove. You can use, without proof, the fact that if h(x) is a continuous function on a closed, bounded interval [a, b}, then there exists /U such that IN(x)| ñ M for all xE [a, b}.