From random partitions to self-similar processes
Abstract
Background: In probability theory, limit theorems characterize the scaling limits that arise from discrete stochastic models. Here, a discrete model refers to a collection of random variables over discrete temporal/spatial lattice, modeling for example the trajectory of a particle moving around in space, the fluctuations of stock prices and internet traffic rates, or the landscape of random surface representing temperatures and/or rainfall amounts in environmental studies. A ubiquitous phenomenon is that when the system under investigation becomes large, or equivalently the number of random variables becomes large, after appropriate scaling and normalization, different models may behave in very similar manners. Nowadays, it is well understood that such phenomena can be rigorously and elegantly described as limit theorems in probability theory, using stochastic processes to represent the limiting behavior of discrete models. For many important models and those to be considered in this project, the limiting processes, also known as the scaling limits, are self-similar processes. These are the processes, when looked at different time scales, that have the same law up to multiplicative constants depending on the scaling. Statement of objectives: The fundamental question that this project aims to address is the characterization of a new type of dependence that stems from certain random partitions. The two core random partitions playing the central role in this project are infinite urn schemes (essentially infinite exchangeable random partitions) and a non-exchangeable random-forest partition of integer lattice. Very recent results have revealed a new connection between these random partitions and fractional Brownian motions, in form of limit theorems. The project aims at following the recent breakthrough and exploring more the intriguing connection between random partitions and self-similar processes. Although random partitions have been extensively investigated as the fundamental object of stochastic combinatorial processes, a fast-developing area of probability, such limit theorems have been rarely seen in the literature, and new processes are expected to emerge. The main goals are (i) to extend the recently investigated one-dimensional models to random-field versions, (ii) to consider the heavy-tailed versions of the discrete models, and (iii) to investigate a specific non-exchangeable random partition and its limit. Methods of employments: Most results will be established in form of limit theorems. For this purpose, new mathematical tools will be developed, especially for the non-exchangeable random partition, combining techniques from probability, analysis, ergodic theory, and combinatorics. Significance: Now is the right moment to carry out intensive investigation on this topic, thanks to promising results in the past 3 years on models driven by random partitions scaling to fractional Brownian motions. These results open a new direction relating random combinatoric structures and limit theorems, with many exciting questions to be answered. Based on promising results by PI and collaborators, some scaling limits to be obtained in this project are expected to be new, and a rarely seen so-called scaling-transition phenomenon is expected to occur for a few random-field models. At the same time, the project also includes a few problems tailored to engage graduate students to start research in this new direction.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 14, 2019
- Source ID
- W911NF1710006
Entities
People
- Yizao Wang
Organizations
- Army Contracting Command
- United States Army
- University of Cincinnati