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GATE 2024 ST – Question 21

Convergence of Random Variables · Modes of Convergence · 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?

  1. Pointwise convergence everywhere to a CDF
  2. Pointwise convergence everywhere, but not to a CDF
  3. No convergence at any real x
  4. Convergence at some x but not others

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Show answer and explanation

Correct answer: (B) Pointwise convergence everywhere, but not to a CDF

Explanation

For any fixed finite x, eventually |x|<n and F_n(x)=1/2+x/(2n)→1/2. The constant limit has neither the required zero limit at −∞ nor one limit at +∞, so is not a CDF. **B**.

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