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Part CCSIR NET June 2023clt-at-the-boundary

Clt at the boundary

Suppose X1,X2,…X_{1}, X_{2}, \dots are independent and identically distributed N(0,1) random variables and Yn=X14+X24+⋯+Xn4Y_{n} = X_{1}^{4} + X_{2}^{4} + \cdots + X_{n}^{4}. Which of the following probabilities converge to 1/2 as n→∞n \to \infty?

  1. A.P{Yn∈[0,2n]Y_{n} \in [0, 2n]}
  2. B.P{Yn∈[n,3n]Y_{n} \in [n, 3n]}✓ Correct
  3. C.P{Yn∈[2n,4n]Y_{n} \in [2n, 4n]}
  4. D.P{Yn∈[3n,5n]Y_{n} \in [3n, 5n]}✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: Convergence in probability implies almost sure convergence

More on this topic

The chapter behind this: Modes of convergence and the limit theorems — free to read

From Limit Theorems and Markov Chains › Modes of convergence, WLLN, SLLN, CLT

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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