A Theory of Probability Homework 5 Solutions
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چکیده
Exercise [2.1.5] Let > 0 and pick K = K( ) finite such that if k ≥ K then r(k) ≤ . Applying the Cauchy-Schwarz inequality for Xi −EXi and Xj −EXj we have that Cov(Xi, Xj) ≤ [Var(Xi)Var(Xj)] ≤ r(0) <∞ for all i, j. Thus, breaking the double sum in Var(Sn) = ∑n i,j=1 Cov(Xi, Xj) into {(i, j) : |i − j| < K} and {(i, j) : |i− j| ≥ K} gives the bound Var(Sn) ≤ 2Knr(0) + n . Dividing by n we see that lim supn Var(n Sn) ≤ . Since > 0 is arbitrary and ESn = nx, we have that nSn L → x (with convergence in probability as well).
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