Student Probability Seminar

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Limiting Distribution of Visits of Several Rotations to Shrinking Intervals

Speaker: Ilya Vinogradov

Location: Warren Weaver Hall 202

Date: Friday, March 25, 2011, 4:30 p.m.

Synopsis:

Let \(n\), \(N\), and \(d\) be positive integers. Let \(L(m, \alpha) = \sum_{i=1}^d m_i \alpha_i (\mod{1})\) be the trajectory of \(d\) rotations by \(\alpha_1, . . . , \alpha_d\) on the \(1\)-torus with \(1 \leq m_i \leq N\). Further let \(\xi_1, . . . , \xi_n \in [0, 1)\) and \(\sigma_1, . . . , \sigma_n > 0\) be fixed. We interprete the numbers of visits of \(L(m, \alpha)\) to \((\xi_1, \xi_1 + \sigma/N), . . . ,(\xi_n, \xi_n + \sigma_n/N)\) as a function of \(\alpha_1, . . . , \alpha_d\), and thus a random integer vector realized on the \(d\)-cube. We prove that as \(N\) tends to infinity, the distribution of this random vector has a limit that depends on the other parameters and has finitely many moments in each case. A subsequent limit transition can be made by letting \(d\) tend to infinity. In this setting we prove that the new limiting distribution is the multidimensional Poisson distribution unless an arithmetic obstruction arises. There results are a generalization of earlier work of J. Marklof.