20.21. 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] 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-X,>2-k]<2-k for m, nzn. Analyze P[IX-Xn>2-k]. (b) Show that X₁ →p X.

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Q: 24. Let (an) and (bn) be two sequences with general terms defined by 2n 1 2n 1 Σ and bn = an = k=n+1…

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Q: 1 a) Sn+1 = 3 – Sn 25, -35m b) Sn+1 Sn-1 9Sn c) Sn+1 S+1

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Q: 20.21. Suppose that the sequence (X) is fundamental in probability in the sense that for e positive…

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Q: 3. Let {an be a sequence in R. (a) Show that it has a monotonic subsequence. (b) Prove that a…

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Q: (b) Prove that a bounded sequence in R has a convergent subsequence.

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Q: 26. (Barndorff-Nielsen) Suppose {En} is a sequence of events such that lim_ P(En) = 0, ΣP(EnE+1) n.…

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Q: 5. Consider the sequence {a} defined recursively as : a = 0, a₁ = 1, an = an-1 + an-2. Prove by…

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Q: 3.5.4) Show directly from the definition that of (xn) and (yn) Cauchy sequences, then cxntyn) and…

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Q: Let {A,} be a non decreasing sequence of events of S, then lim P(A,) = P (lim A,) = P| U An %3D n =1

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Q: 15. Let {fn} be a sequence on C([a, b]). Assume that fn (x) 2 fn+1(x) for all x E [a, b]. Show that…

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Q: Consider the following recurrence T : N → N: T(0) = 0 T(n) = T(L) +T(L4J)+n_for n > 1 What can you…

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