a Use Tchebychevs inequality to show that if E(X n ) n and var(X n ) 2 n , then (X n n ) O p ( n ) b Suppose that Y 1 , , Y n are independent with E(Y i ) and var(Y i ) 2 for i 1, ,n Let Y n ( i Y i ) n Apply part (a) to show that Y n O p (n 1 2 )

Question: a. Use Tchebychevs inequality to show that if E(X n ) = n and var(X n ) = 2 n < , then

a. Use Tchebychev’s inequality to show that if E(X_n) = µ_n and var(X_n) = σ²_n < ∞, then (X_n – µ_n) = O_p(σ_n).

b. Suppose that Y₁,..., Y_n are independent with E(Y_i) = µ and var(Y_i) = σ² for i = 1,...,n. Let Y̅_n = (∑_i Y_i)/n. Apply part (a) to show that Y̅_n – µ = O_p(n^–1/2).

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