Suppose that the sequence (X) is fundamental in probability in the sense that for e positive...

 

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Suppose that the sequence (X) is fundamental in probability in the sense that for e positive there exists an N such that P[IX-X,>e]<e for m, n > N. (a) Prove there is a subsequence (X) and a random variable X such that lim, XX with probability 1. Hint: Choose increasing n such that P[IX-X₁>2-k] <2-k for m, n ≥n. Analyze P[IX-Xn₂>2-k]. nk m (b) Show that X→p X. n Suppose that the sequence (X) is fundamental in probability in the sense that for e positive there exists an N such that P[IX-X,>e]<e for m, n > N. (a) Prove there is a subsequence (X) and a random variable X such that lim, XX with probability 1. Hint: Choose increasing n such that P[IX-X₁>2-k] <2-k for m, n ≥n. Analyze P[IX-Xn₂>2-k]. nk m (b) Show that X→p X. n