Student Puzzle editor Anirban DasGupta poses these three puzzles, on three different topics. We encourage you to have a go at all of them! Send your solutions before September 21, 2026.

Puzzle 62.1:
Suppose $\{X_t, t = 1, 2, \cdots \}$ is a discrete time (weakly) stationary process and suppose that it has a bounded spectral density. Let $n \geq 2$. Prove that the eigenvalues of the autocovariance matrix of $(X_1, \cdots , X_n)$ are all bounded by some universal finite constant.

Puzzle 62.2
Suppose $X_1, X_2, \cdots $ are i.i.d. real valued random variables with a common distribution $P$ and that $P$ is absolutely continuous and has a compact support $C$. Given $x$, denote by $X_{n,NN}(x)$ a nearest neighbor of $x$ among $X_1, \cdots , X_n$.
Prove that $|x-X_{n,NN}(x)|$ converges uniformly to zero over $C$ with probability one.
Does this generalize to higher dimensions?
Does it also hold without the compact support assumption?

Puzzle 62.3
$X_1, \cdots , X_5$ are i.i.d. uniform on the set $\{-10, -9, \cdots , -1, 0, 1, \cdots , 9, 10\}$. Find the conditional probability $P(\sum_{i = 1}^{10}\, X_i^3 = 0\, |\sum_{i = 1}^{10}\, X_i = 0, \, \sum_{i \,\neq \, j = 1}^{10}\, X_i\,X_j = 0)$.

—

Solution to Puzzle

Hearty congratulations to Aniv Mazumder (Indian Statistical Institute, Delhi), whose solution Anirban DasGupta described as “a paragon of rigor and detail”! These puzzles (originally from the June/July issue) are viewable here.
Anirban explains:

Puzzle 61.1:
A straightforward calculation shows that the indicators of the events that the $i$th observation is a record are independent Bernoullis with parameter $1/i$. Therefore, $E(N_n) = \sum_{i = 1}^n\,1/i$ and $\mbox{Var}(N_n) = \sum_{i = 1}^n\,(i-1)/i^2$. Note that they are both asymptotic to $\log n$. For large $n$, the distribution of $N_n$ can be approximated by a Poisson with parameter $\log n$. Alternatively, the distribution of $\frac{N_n – \log n}{\sqrt{\log n}}$ can be approximated by a standard normal. The conditions of the Lindeberg CLT (also called the Lindeberg-Feller CLT) are verifiable without much difficulty. In particular, for $n = 10, E(N_n) = \frac{7381}{2520}, \mbox{Var}(N_n) = \frac{350339}{254016}$. Using the asymptotic approximations, for $n = 100, P(N_n > 8) \approx 0.0568295$.

Puzzle 61.2:
The Schwarz inequality gives that $\sup_{\bf{c}\, \in \, S}\,\bf{c}’\,\bf{Z} = ||Z||_2$, and by direct calculation, $E(||Z||_2) = \sqrt{2}\,\Gamma (\frac{p+1}{2})/\Gamma (\frac{p}{2})$. Now use Stirling to find the desired limit to be 1.

Puzzle 61.3:
This problem does require the loss function to be strictly convex in the second argument. For the given model, standard arguments show that a minimal sufficient statistic is $\sum_{i = 1}^n\,X_i^2$. The Rao–Blackwellizations of $\bar{X}$ and $s^2$ (both of which, by the way, are unbiased estimates of $\theta $) strictly dominate them pointwise in risk. This makes both $\bar{X}$ and $s^2$ inadmissible.