Part CCSIR NET December 2024the-product-vanishes-but-its-nth-root-does-not

The product vanishes but its nth root does not

Let $U_{1}, U_{2}, \dots$ be a sequence of independent and identically distributed (i.i.d.) U(0, 1) random variables. Let $G_{n} = (\prod(i=1..n) U_{i})^(1/n)$ be the geometric mean of $U_{1}, U_{2}, \dots, U_{n}$ for $n \in \mathbb{N}$ . Let $X_{1}$ and $X_{2}$ be degenerate random variables such that $P(X_{1} = 0) = 1$ and $P(X_{2} = 1/e) = 1$ . Then, which of the following statements are true?

A. $G_{n}$ converges in r-th mean to $X_{1}$ as $n \to \infty$ , for any r > 0
B. $G_{n}$ converges in probability to $X_{1}$ as $n \to \infty$
C. $G_{n}$ converges in distribution to $X_{2}$ as $n \to \infty$ ✓ Correct
D. $G_{n}$ converges in r-th mean to $X_{2}$ as $n \to \infty$ , for any r > 0✓ 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 limit assumed to exist.

See pricing

50 are analysed free — try those first.

The trap it tests

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

Drill statements like this

The product vanishes but its nth root does not

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