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Let S be a stochastic sequence of bounded ræ Which of the following assertions is equivalent

Let S be a stochastic sequence of bounded ræ Which of the following assertions is equivalent

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let S be a stochastic sequence of bounded random vectors 1, X2, ... such that X, → x in mean-square as t → ∞ for some random variable X. Which of the following assertions is equivalent to the definition of convergence in probability? There is only 1 correct answer. (1) When t is sufficiently large, the probability that X, → ĩ is less than some arbitrarily chosen positive number e converges to one. (2) The probability that x, X as t→ o is some sufficiently small strictly positive number e. (3) The probability that 1, X2, ... does not converge to K as t → o is exactly equal to zero. (4) The variance of X, converges to zero as t → ∞. (5) The variance of x, – ĩ converges to zero as t → ∞. (6) As t → ∞, the probability distribution of x, converges to the probability distribution of ,41.
Transcribed Image Text:Let S be a stochastic sequence of bounded random vectors 1, X2, ... such that X, → x in mean-square as t → ∞ for some random variable X. Which of the following assertions is equivalent to the definition of convergence in probability? There is only 1 correct answer. (1) When t is sufficiently large, the probability that X, → ĩ is less than some arbitrarily chosen positive number e converges to one. (2) The probability that x, X as t→ o is some sufficiently small strictly positive number e. (3) The probability that 1, X2, ... does not converge to K as t → o is exactly equal to zero. (4) The variance of X, converges to zero as t → ∞. (5) The variance of x, – ĩ converges to zero as t → ∞. (6) As t → ∞, the probability distribution of x, converges to the probability distribution of ,41.
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