GATE Convergence of Random Variables: Modes of Convergence – Previous Year Questions

10 GATE previous year questions on Modes of Convergence (Convergence of Random Variables, Statistics) with answers and explanations, from every paper.

1. GATE 2021 ST Q28 (1 mark, Numerical answer) – For each α>0,independent Bernoulli variables X n(α) have P(X n(α)=1)=1/n(2α). Let S=α>0:X n(α)→0 almost surely. Find inf S,to two decimals.
2. GATE 2022 ST Q45 (2 marks, Multiple select) – X n are independent exponential variables of mean 1/2. “i.o.” means infinitely often. Which hold?
3. GATE 2022 ST Q46 (2 marks, Multiple select) – X n has probabilities n/(n+1) at 0 and 1/(n+1) at n. Let X=0 almost surely. Which hold?
4. GATE 2023 ST Q21 (1 mark, Multiple choice) – All variables are defined on the same probability space. Which statement is true?
5. GATE 2023 ST Q32 (1 mark, Numerical answer) – X n are iid with mean 4 and variance 9. For Y n=(X₁+⋯+X n)/n,find lim as n→∞ of E[((Y n−4)/√n)²],as an integer.
6. GATE 2024 ST Q21 (1 mark, Multiple choice) – F n is the CDF of uniform(−n,n):0 below −n,(x+n)/(2n) on [−n,n),1 at and above n. Which holds?
7. GATE 2024 ST Q52 (2 marks, Multiple select) – Independent X n have exponential density e(−x/λ n)/λ n,x≥0, with λ n=10−Σᵢ₌₁ⁿ5/2(i−1). Which hold?
8. GATE 2025 ST Q50 (2 marks, Multiple select) – Independent X n converge almost surely to zero. Which are necessarily true?
9. GATE 2025 ST Q60 (2 marks, Numerical answer) – Independent X n take values ±1/2ⁿ, each with probability 1/2. If Σᵢ₌₁ⁿX i converges in distribution to U, find 6P(U≤2/3) (integer).
10. GATE 2026 ST Q53 (2 marks, Multiple select) – For n≥1, P(X n=x)=[1/(5n)](1−1/(5n))x,x=0,1,…. Let Z n=X n/n converge in distribution to V. Which hold?