Correct answer: (A) √nY_n converges in distribution to N(0,1).; (B) Y_n converges in second mean to zero.; (C) Y_n+1/n converges in probability to zero.
Explanation
The disjoint pair products are iid with mean 0 and variance E X₁²E X₂²=1. The CLT proves A; E Y_n²=1/n proves B and thus probability convergence. Adding deterministic 1/n proves C. D fails,for example,for iid fair ±1 variables. **A,B,C**.
More Laws of Large Numbers and Central Limit Theorem questions
GATE 2023 ST Q61– X n are iid Exp(rate 1). Let Y n= X (2n)−X (2n−1) and Ȳ n=(Y₁+⋯+Y n)/n. If √n[exp(−Ȳ…
GATE 2022 ST Q31– X i are iid with density x(−5/6)e(−x/8)/[√2 Γ(1/6)] on x>0. Let X̄ n be their mean. If…
GATE 2021 ST Q29– X n are iid Uniform(0,2). Set Z n=−ln[(∏ (i=1)n(2−X i))(1/n)]. Find its almost-sure…
GATE 2021 ST Q59– X n are iid Exp(rate 1),and X (n)=maxX₁,…,X n. If X (n)−ln n converges in distribution to…
GATE 2026 ST Q54– Independent X (2k−1)~Bin(1,θ),X (2k)~Bin(1,1−θ),0<θ<1. The Y k are iid Poisson(λ),λ>0. Put…
GATE 2025 ST Q16– The X n are iid with density 1/[π(1+x²)] on ℝ. Define Y n=1/2+arctan(X n)/π. Which…