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Bangalore Probability Seminar 2012
Venue: Indian Statistical Institute (auditorium) and Indian Institute of Science (Mathematics dept)
Organizers: Siva Athreya and Manjunath Krishnapur
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23 Apr, ISI, 2:15 PM
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Probability Distributions over the Integers with Some Useful Independence Properties
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Navin Kashyap (Indian Institute of Science)
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For an integer-valued random variable Z, let [Z] denote the binary-valued rv Z mod 2. Suppose that X, Y are iid integer-valued rvs, each having probability mass function p. It is easy to check that for non-degenerate p, the sum X+Y cannot be independent of X or Y. On the other hand, if we take any p such that Pr[[X] = 0] = 1/2, then [X+Y] is independent of each of [X] and [Y]. The question we ask is: can p be chosen such that X+Y is independent of each of [X] and [Y]? We require that Pr[[X]=0] = 1/2; the question is trivially answered otherwise. This question is motivated by a practical application: secure physical-layer function computation. We will give a general construction of p that answers our question in the affirmative. We will extend this idea to the setting of integer lattices (generalizing Z) and finite Abelian groups (generalizing [Z]). The tools used are compactly-supported characteristic functions and the Poisson summation formula. We will also describe a result of Ehm, Gneiting and Richards (Trans. AMS, 2004) that gives a complete and precise solution to the extremal problem of determining the probability distribution in R^n with characteristic function supported within the unit ball, having the least variance among such distributions.
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23 Apr, ISI, 3:30 PM
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Nodal length of random eigenfunctions of the Laplacian on the 2-d torus
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Manjunath Krishnapur (Indian Institute of Science)
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We consider Gaussian linear combinations of eigenfunctions of the Laplacian on the 2-dimensional torus for a given eigenvalue. The mean of the length of the nodal set is easy to compute. As the dimension of the eigenspace goes to infinity, we find asymptotics of the variance of the nodal length. The proof has two main steps. First, the Kac-Rice formulas from probability give a formula for the variance in terms of the covariance kernel of the Gaussian random function. The covariance kernel is of an arithmetic nature and is analyzed using tools from additive combinatorics. This is joint work with Igor Wigman and Pär Kurlberg.
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