Description

Department of Statistics
STAT 210A: Introduction to Mathematical Statistics
Problem Set 11
Fall 2014

Problem 11.1
(Sub-Gaussian bounds and means/variances). Consider a random variable X such that

for all λ ∈ R.
(a) Show that E[X] = µ.
(b) Show that Var(X) ≤ σ2.
(b) Suppose that the smallest possible σ satisfying the inequality above is chosen. Is it then true that Var(X) = σ2? Prove or disprove.
Problem 11.2
(Gaussian maxima). Let be an i.i.d. sequence of N(0,σ2) random variables, and consider the random variable Z = maxk=1,…,n |Xk|.
(a) Prove that
for all n ≥ 2.
Gaussian normal variate.
(b) Prove that
E[Z] ≥ (1 − 1/e)p2σ2 logn for all n ≥ 5.
(c) Prove that .
Problem 11.3
(Bernstein and expectations). Consider a nonnegative random variable that satisfies a concentration inequality of the form

for positive constants (ν,b) and C ≥ 1.
1
√ √
(a) Show that E[Z] ≤ 2ν( π + logC) + 4B(1 + logC).
(b) Let {Xk}nk=1 be an i.i.d. sequence of zero-mean variables satisfying the Bernstein condition (2.16). Use part (a) to show that
.
Problem 11.4
(Sub-Gaussian random matrices). Consider the random matrix Q = gB, where g ∈ R is a zero-mean sub-Gaussian variable with parameter σ.
(a) Assume that g has a distribution symmetric around zero, and B ∈ Sd×d is a deterministic matrix. Show that Q is sub-Gaussian with matrix parameter V = c2σ2B2, for some universal constant c.
(b) Now assume that B ∈ Sd×d is random and independent of g, with the operator norm of B upper bounded by a constant b almost surely. Now show that Q is sub-Gaussian with matrix parameter V = c2b2σ2Id×d.
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