Part CCSIR NET December 2024only-a-constant-limit-reverses-the-arrow

Only a constant limit reverses the arrow

Suppose that a sequence of random variables { $X_{n}$ } $_{n} \geq_{1}$ and the random variable X are defined on the same probability space. Then which of the following statements are true?

A. $X_{n}$ converges to X almost surely as $n \to \infty$ implies that $X_{n}$ converges to X in probability as $n \to \infty$ .✓
B. $X_{n}$ converges to X in probability as $n \to \infty$ implies that $X_{n}$ converges to X almost surely as $n \to \infty$ .
C.If $\sum (n = 1..\infty)$ ℙ[| $X_{n} - X$ | $> δ] < \infty$ for all $δ > 0$ , then $X_{n}$ converges to X almost surely as $n \to \infty$ .✓
D.If $X_{n}$ converges to X in distribution as $n \to \infty$ , and X is a constant with probability 1, then $X_{n}$ converges to X in probability as $n \to \infty$ .✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Only a constant limit reverses the arrow

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