Suppose U sim U(0, 1). Then the sequence Xn = begincases sqrtn & if 0<U<1/n^2, n ge 1 0, & otherwise endcases converges

(A) both with probability one and in mean square

Q32Section I (Objective (Q1–Q80))32 of 80

Probability — modes of convergence for a degenerating sequence

Suppose $U \sim U(0, 1)$ . Then the sequence

$X_n = \begin{cases} \sqrt{n} & \text{if } 0<U<\dfrac{1}{n^2}, n \ge 1 \ 0, & \text{otherwise} \end{cases}$

converges

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