Reference no: EM132224055

Problem 1: Sequential convergence implies Cesaro convergence, i.e. if a real sequence (x_n) converges to λ then so does the sequence of arithmetic means (x‾_n) where x‾_n = (x₁+ ....+ x_n)/n. (The converse is not true in general; let x_n = (-1)ⁿ.) This extends to almost sure convergence; if X_n →^a,s A then X‾_n →^a,s A.

Try to construct an example showing this need not be the case for convergence in probability. Suppose the X_n's are independent and P(X_n = c_n) = P_n and P(X_n = 0) = 1- p_n, where the c_n's are positive and bounded away from zero, and pn 0 as it oo. The 'right' choice of the p_n allows you to find a simple explicit expression for P(X‾_n ≤ ε) if ε < 1.

Problem 2:
(i) Show that the sequence a_n defined by (1.1.7) is also given by the single formula

a_n = 1 + (-1)ⁿ.1/n

(ii) If a_n is the nth member of the sequence (1.1.9), find a formula for a_n analogous to that of (i).

Problem 3: (i) Show that the sequence (1.1.9) tends to 0 as n → ∞ .
(ii) Make a table of a_n given by (1.1.9) for n = 75, 76, . . . , 100.

Problem 4: In Example 2.2.1, if X_i = ±3^i-1 with probability 1/2 each, show that X¯ → ∞ with probability 1/2 and X¯ → -∞ with probability 1/2.

[Hint: Evaluate the smallest value that (X₁ + ... + X_n)/n can take on when X_n = 3^n-1.]

Problem 5: If X₁, . . . , X_m and Y₁, . . . , Y_n are independent with common mean ξ and variances σ² and τ², respectively, use (2.2.1) to show that the grand mean (X₁ + ... + X_m + Y₁ + ... + Y_n)/ (m + n) is a consistent estimator of ξ provided m + n → ∞.

Problem 6: In the standard two-way random effects model, it is assumed that

X_ij = A_i + U_ij (j = 1, . . . , m; i = 1, . . . , s)

with the A's and U's all independently, normally distributed with

E (A_i) = ξ, Var (A_i) = σ_A² , E (U_ij) = 0, Var (U_ij) = σ².

Show that X¯ = ∑∑X_ij/sm is
(i) not consistent for ξ if m → ∞ and s remains fixed,
(ii) consistent for ξ if s → ∞ and m remains fixed.