Georgia Institute of Technology College of Sciences

Let $(S,\mathcal{A})$ be a measurable space and let $\mathcal{P}$ be a family of probability distributions on it. Given a Banach space $E$, a mapping $\theta:\mathcal{P}\to E$, and a smooth functional $f:E\to\mathbb{R}$, we consider the problem of estimating $f(\theta(P))$ from i.i.d. observations $X_1,\ldots,X_n\sim P$, where $P\in\mathcal{P}$. We write $f\in C^s(E)$ for a functional of Hölder smoothness $s=m+\rho$, where $m\ge 0$ is an integer and $\rho\in(0,1]$. Our aim is to construct estimators of $f(\theta(P))$ and to study their dependence on the sample size $n$, the smoothness $s$, and the dimension or complexity of the parameter $\theta(P)$.

When $\hat\theta_n$ is a $\sqrt{n}$-consistent base estimator of $\theta(P)$, the plug-in estimator $f(\hat\theta_n)$ is asymptotically efficient in classical low-dimensional models, but in high-dimensional and infinite-dimensional settings its bias is often too large to attain the rate $n^{-1/2}$. Existing bias reduction methods, based on iterated bootstrap or on linear aggregation of plug-in estimators, rely on concentration inequalities for $f(\hat\theta_n)$ that are available only for a limited class of models.

In this dissertation we study a class of estimators $T_f(X_1,\ldots,X_n)$ obtained from a Taylor expansion of $f$ about $\hat\theta_n$, of order determined by $s$, together with a sample split. Their analysis uses only bounds on the moments of the linear and higher order terms of $\hat\theta_n-\theta(P)$, rather than concentration inequalities. For functionals of smoothness $s\ge 1$, we derive upper bounds on the $L_p$-errors of $T_f$ whose dependence on $n$, on $s$, and on the dimension or complexity of the parameter matches the minimax lower bounds we obtain. We also give conditions under which these estimators are asymptotically normal and asymptotically efficient.

We develop these results in three settings: models composed of a large number of independent low-dimensional components, high-dimensional exponential families, and functionals of covariance operators in infinite-dimensional subgaussian models, where the complexity of the model is measured by the effective rank of the covariance operator.

In this dissertation, we use methods of probabilistic combinatorics to work towards a conjecture of Alon and Kim about coloring the edges of k-uniform t-simple hypergraphs. A hypergraph is k-uniform if every edge contains exactly k vertices, and t-simple if any two edges intersect in at most t vertices. The chromatic index of a hypergraph H is the smallest integer N such that one can properly color the edges of H with N colors.

In 1997, Alon and Kim conjectured that if H is a k-uniform t-simple hypergraph with maximum degree D sufficiently large, then the chromatic index \chi'(H) is upper bounded by (t-1+1/t+\epsilon)D. Using probabilistic techniques and a nibble coloring method, we prove a general coloring theorem stating that a k-uniform t-simple hypergraph H with large maximum degree D has chromatic index at most (b+\epsilon)kD, where b is a particular parameter derived from local structural information about H.

We use structural techniques to prove sharp upper bounds on b in the 3-uniform 2-simple, and 3-uniform 3-simple cases. In particular, we deduce as a corollary that for sufficiently large D, every 3-uniform 2-simple and 3-simple hypergraph of maximum degree at most D has chromatic index at most 2.358D and 2.679D, respectively. We also prove that for sufficiently large D, every 3-uniform 2-simple hypergraph has fractional chromatic index at most 2D.

We first investigate reproducing pairs in Hilbert spaces, with a focus on the discrete case. Reproducing pairs generalize frames and consist of two sequences $\Psi$ and $\Phi$, along with a bounded invertible operator $S_{\Psi,\Phi}$. The work examines sequences that are overcomplete by one element—that is, they become exact upon removal of a single element. A central result shows that if such a sequence admits a reproducing partner, the resulting exact subsequence must form a Schauder basis. This implies that systems like the Gaussian Gabor system at critical density, which lacks a Schauder basis, cannot have a reproducing partner. The result is further generalized to sequences overcomplete by finitely many elements.

Next, we introduce exponential reproducing pairs, where the sequences are weighted exponentials. The associated operator $S_{g\gamma}$ acts as a multiplication operator, and necessary and sufficient conditions are established for when a pair $(g, \gamma)$ forms an exponential reproducing pair.

Lastly, by extending a 2012 result of Heil and Yoon, we develop a two-dimensional theory for weighted exponential systems. It characterizes when weighted double exponential systems are minimal and complete, and provides necessary and sufficient conditions for exactness of arbitrary weighted systems.

Zoom Link: https://gatech.zoom.us/j/93221716846

In this thesis, we investigate three related problems at the intersection of analytic number theory and discrete harmonic analysis. Our primary goal is to understand discrete averaging operators over arithmetic sets—discrete analogues of classical continuous operators—and analyze their behavior using tools from harmonic analysis and additive combinatorics. The results deepen our understanding of how analytic and combinatorial techniques interact in the study of primes and other arithmetic structures.

The Zoom link to the meeting: https://gatech.zoom.us/j/99340322307