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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 {XnX_{n}}n≥1_{n}\ge_{1} and the random variable X are defined on the same probability space. Then which of the following statements are true?

  1. A.XnX_{n} converges to X almost surely as n→∞n \to \infty implies that XnX_{n} converges to X in probability as n→∞n \to \infty.✓ Correct
  2. B.XnX_{n} converges to X in probability as n→∞n \to \infty implies that XnX_{n} converges to X almost surely as n→∞n \to \infty.
  3. C.If ∑(n=1..∞)\sum(n=1..\infty) ℙ[|Xn−XX_{n} - X| >δ]<∞> \delta] < \infty for all δ>0\delta > 0, then XnX_{n} converges to X almost surely as n→∞n \to \infty.✓ Correct
  4. D.If XnX_{n} converges to X in distribution as n→∞n \to \infty, and X is a constant with probability 1, then XnX_{n} converges to X in probability as n→∞n \to \infty.✓ Correct

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, and why this one tests converse assumed.

See pricing

50 are analysed free — try those first.

The trap it tests

Converse assumed

The theorem runs one way. You used it in the other.

Drill statements like this

Related counterexample: Convergence in probability implies almost sure convergence

More on this topic

The chapter behind this: Modes of convergence and the limit theorems — free to read

From Limit Theorems and Markov Chains › Modes of convergence, WLLN, SLLN, CLT

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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