Asymptotics of Gaussian processes and Markov chains
- Alex Zhai.
- [Stanford, California] : [Stanford University], 2018.
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- 1 online resource.
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- In this thesis, we present several results on the asymptotic behavior of Gaussian processes and Markov chains. In the first part, focused on Gaussian processes, we prove a central limit theorem for the sum of i.i.d. high-dimensional random vectors. Surprisingly, not much is known about the optimal dependence of the convergence rate on the dimension of the vectors. Our main contribution is to prove a convergence rate in quadratic transportation distance that is close to optimal in both the dimension and the number of summands. We next prove a result (based on joint work with Jian Ding and Ronen Eldan) about general Gaussian processes: we show that if the maximum of a Gaussian process is strongly concentrated around its expectation (called "superconcentration"), then with high probability the process has many near-maximal values with low pairwise correlations (called "multiple peaks"). Such phenomena naturally arise in the analysis of disordered systems in statistical physics, where the Gaussian process values correspond to energy levels. Our result adds to an overall picture of the behavior of superconcentrated Gaussian processes described by Chatterjee. The second part of the thesis contains results concerning asymptotic behavior of Markov chains. For random walk on a graph, we prove a sharpening of a relationship established by Ding, Lee, and Peres between the cover time and the Gaussian free field. In particular, our estimate implies that in families of graphs (of size growing to infinity) where the hitting time is asymptotically much smaller than the cover time, the cover time is exponentially concentrated around its expectation, and this expectation has a simple asymptotic formula in terms of the Gaussian free field. We also analyze the mixing time of a Markov chain, known as the product replacement walk, on n-tuples of elements of some finite group. One step of the walk involves randomly choosing two of the elements a and b and multiplying a by either b or the inverse of b, with equal probability. The product replacement walk has been extensively studied in the context of random generation of group elements and is part of a larger class of Markov chains that includes random walks on matrix groups over finite fields and certain interacting particle system models. Based on joint work with Yuval Peres and Ryokichi Tanaka, we prove that the product replacement walk exhibits a cutoff phenomenon as n goes to infinity: the chain rapidly transitions from being unmixed to mixed after around 3/2 n log(n) steps.
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- Submitted to the Department of Mathematics.
- Thesis Ph.D. Stanford University 2018.
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