A Lepskiĭ-type stopping rule for the covariance estimation of multi-dimensional Lévy processes

We suppose that a Lévy process is observed at discrete time points. Starting from an asymptotically minimax family of estimators for the continuous part of the Lévy Khinchine characteristics, i.e., the covariance, we derive a data-driven parameter choice for the frequency of estimating the covariance. We investigate a Lepskiĭ-type stopping rule for the adaptive procedure. Consequently, we use a balancing principle for the best possible data-driven parameter. The adaptive estimator achieves almost the optimal rate. Numerical experiments with the proposed selection rule are also presented.

A Simple guide to complex world of overtone and combination bands: Theoretical simulation and interpretation of NIR spectra – summary of the workshop at NIR-2021 Beijing Conference

Between 18 and 21 October 2021, the 20th International Conference on NIR spectroscopy in Beijing took place. Despite this time being held as a virtual event, it was a highly successful symposium met with high interest from the wide audience – as evidenced by many excellent presentations, around which numerous vivid discussions developed. During the conference, four workshops were offered, focused at discussing few areas essential for NIR spectroscopy and its applications. Excellent workshops were provided by Professors Heinz Siesler, Hui Yan, Dolores Pérez-Marín and Tom Fearn, in which invaluable knowledge was shared with the participants of the conference. Among these renowned experts, I had the honour to offer my contribution with the workshop aimed at the physicochemical foundations of NIR spectroscopy, an area that seldom is exhaustively presented in the textbooks. The workshop aimed at shedding light on the complex world of overtone and combination bands and was met with a considerable interest from the participants. As many questions have been asked both during the dedicated Q&A session, as well as through other channels and private correspondence, I would like to provide a short recapitulation of the workshop in the form of this brief article. Some of the most essential ‘take home messages’, such as the origin of the intensity variation of the overtone bands and the famous ‘selection rule’ of the harmonic oscillator, among others will be briefly outlined here.

Graphs where Search Methods are Indistinguishable

Graph searching is one of the simplest and most widely used tools in graph algorithms. Every graph search method is defined using some partic-ular selection rule, and the analysis of the corre-sponding vertex orderings can aid greatly in de-vising algorithms, writing proofs of correctness, or recognition of various graph families. We study graphs where the sets of vertex order-ings produced by two di˙erent search methods coincide. We characterise such graph families for ten pairs from the best-known set of graph searches: Breadth First Search (BFS), Depth First Search (DFS), Lexicographic Breadth First Search (LexBFS) and Lexicographic Depth First Search (LexDFS), and Maximal Neighborhood Search (MNS).