18 problems

Let $\{x(n)\}$ be a multivariate stationary process with spectral density $W$ on the unit circle $\mathbb{T}$. A matrix function $\Phi$ is Hermitian when $\Phi=\Phi^{*}$, and…

Let $rho ameq{}(0,1)$ be the stationary Bernoulli density, let $mathbb{P}rho$ be the corresponding TASEP path measure, and let $ht(j)$ be the height function. For $winmathbb{R}$, d…

Wiener's conjecture. Under certain conditions, a stationary process can be expressed as a one-sided function of a sequence of i.i.d. random variables.

Let $\mathsf{t_{mix}}$ denote the mixing-time parameter of the stationary $\alpha$-mixing stochastic process, and let the estimator be the one introduced in the paper for estimatin…

Conjecture on the absence of a good ARMA approximation. The process $X_t$ does not have a good ARMA model of the displayed form that is better than the simple $M+N$-term truncation…

Continuity conjecture. For every stationary ergodic process $Z$,

Let $H_0,H_1\subset\mathcal E$ be hypotheses for stationary ergodic distributions. A test is uniformly consistent if both errors converge to zero uniformly over the respective hypo…

Let $H_0$ and $H_1$ be hypotheses consisting of stationary ergodic distributions. A test is weakly asymmetrically consistent (respectively, weakly asymptotically consistent) when t…

Let $F$ be a degree distribution on the nonnegative integers, and assign directions to the stubs independently for each stub. Consider any rule for connecting right-pointing arrows…

AMP-UD large-system conjecture. AMP-UD can achieve the MMSE for stationary ergodic inputs in the limit of large linear systems where the matrix has i.i.d. random entries.

Universal denoising conjecture. Under some technical conditions, the modified universal denoiser achieves the MMSE asymptotically for unbounded stationary ergodic signals.

Let $G$ be a compact group, let $\widehat{G}$ denote its unitary dual, and let $Y$ be an ${\cal A}$-adapted stationary random field with Wold decomposition … where $Y^{(1)}$ is…

Consider a linear inverse problem in which an input signal ${\bf x}$, generated by a stationary ergodic source, is estimated from noisy measurements ${\bf y}$ and measurement matri…

Let the input signal be a stationary ergodic signal, and let the proposed universal denoiser use context quantization together with Gaussian-mixture model learning, without requiri…

Let $\widehat{f}_t$ denote the first dynamic principal component (DPC) of order $k$ constructed from a stationary vector process $\mathbf{z}_t$. For the first DPC of order $k$, the…

A linear process is understood in the setting above, with coefficient operators whose appropriate summability is assumed, and with finite moments as specified below. Four-moment su…

Let $\mathcal S$ be the set of all stationary processes, and let $\mathcal M$ be the set of all functions of stationary ergodic countable-state Markov chains. A solution to Problem…

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables, and define … … where $q=\max\{j:0<\omega_j<\pi\}$. Write $\log^+x=\max\{\log x,0\}$ and let $G$ ha…