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GATE 2025 ST – Question 50

Convergence of Random Variables · Modes of Convergence · 2 marks · Multiple select

Independent X_n converge almost surely to zero. Which are necessarily true?

  1. E(X_n³)→0
  2. X_n⁷→0 in probability
  3. For every ε>0,Σₙ₌₁∞P(|X_n|≥ε)<∞
  4. X_n²+X_n+5→5 almost surely

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Show answer and explanation

Correct answer: (B) X_n⁷→0 in probability; (C) For every ε>0,Σₙ₌₁∞P(|X_n|≥ε)<∞; (D) X_n²+X_n+5→5 almost surely

Explanation

Continuous mapping gives X_n⁷→0 almost surely and hence in probability, and gives D directly. For fixed ε the events {|X_n|≥ε} are independent and occur only finitely often almost surely. The second Borel–Cantelli lemma therefore forces the probability sum to be finite. Moments need not converge: independent X_n=n with probability 1/n² and zero otherwise converge a.s. to zero, but E X_n³=n. **B,C,D**.

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