Mathematics · Probability Theory
Suppose that a sequence of random variables {Xn}n≥1 and the random variable X are defined on the same probability space. Which of the following statements is NOT necessarily true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1If Xn converges to X in distribution as n → ∞, and X is a constant with probability 1, then Xn converges to X in probability as n → ∞.
- 2If Σn=1∞ P(|Xn - X| > δ) 0, then Xn converges to X almost surely as n → ∞.
- 3Xn converges to X in probability as n → ∞ implies that Xn converges to X almost surely as n → ∞.✓
- 4Xn converges to X almost surely as n → ∞ implies that Xn converges to X in probability as n → ∞.
Correct answer
Xn converges to X in probability as n → ∞ implies that Xn converges to X almost surely as n → ∞.
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