Page 300 - Probability and Statistical Inference

5. Concepts of Stochastic Convergence 277

such that U and U are independent, U is N(0, 1 + 1/n), and U is N(0, 1), for
n
n
each n ≥ 1. Show that
(i) as n → ∞. {Hint: For x ∈ ℜ, check that P(U ≤ x)
n
→ Φ (x) as n → ∞.};

(ii) as n → ∞. {Hint: Since U and U are independent,
n
U – U is distributed as N(0, 2 + 1/n). Hence,
n
as n → ∞}.
5.3.15 Let X , ..., X be iid Uniform (0, 1). Let us denote a sequence of
1 n
random variables, .
t
(i) Show that the mgf, M (t) = 1/t(e – 1), t ≠ 0;
X1
a
(ii) Show that the mgf, M (t) = {1/2(e – e )/a} where a
n
–a
Un
(iii) Show that
= exp{t /24};
2
(iv) Use part (iii) to argue that as n → ∞.
5.3.16 Let X , ..., X be iid N(0, 1) where k = 2 , n ≥ 1. We denote
n
1 k

Find the limiting (as n → ∞) distribution of W = U /V . {Hint: Let ,
n n n
2 –1
j = 1, ..., n. Each random variable Y has the Cauchy pdf f(y) = π (1 + y ) ,
–1
j
–∞ < y < ∞. Also, 1/n U has the same Cauchy pdf. Is it possible to conclude
n
that as n → ∞ where W has the same Cauchy pdf? Would Slutsky’s
Theorem suffice?}
5.3.17 Let X , ..., X be iid N(0, 1), n ≥ 1. We denote:
1 n

Show that as n → ∞. {Hint: Use both CLT and Slutsky’s
Theorem.}
5.3.18 (Normal Approximation to the Binomial Distribution)
Suppose that X , ..., X are iid Bernoulli(p), 0 < p < 1, n ≥ 1. We know
1 n