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Patricia Garmirian
Norbert Wiener Assistant Professor
Patricia Garmirian
Contact Info:
Tufts University
Department of Mathematics
503 Boston Avenue
Medford, MA 02155

Email @tufts.edu:

Probability and stochastic processes

My first area of research is the Central Limit Theorem (CLT). My dissertation advisor Vladimir Dobric and I gave a new, direct and elementary proof of the general CLT. Two of our important stepping-stones are, first, a new, similarly direct and elementary proof of the CLT for Rademacher random variables defined on \([0,1]\). The second important stepping-stone is a new result for Bernstein polynomials of continuous functions. Bernstein polynomials are a fundamental object of mathematical analysis. It is well known that Bernstein polynomials of a continuous function on intervals \([0,b_{n}]\) when \(n\) tends to infinity return the value of the function for an appropriate rate of \(b_{n}\), but uniform convergence is sacrificed. Nothing was known for the symmetric interval \([-b_{n},b_{n}]\). We have proven that for these intervals the limit does not recover the function but rather its integral with respect to Gaussian measure. The extension to our direct proof of the of the general CLT involves a new and surprising connection between the CLT and the Haar basis on \([0, 1]\): the i.i.d. sequence of random variable is transformed to a sequence defined on \([0,1]\) and the random variables in the transformed sequence are then expanded with respect to the Haar basis. We are continuing this work by studying random variables which take values in other spaces, such as \(\mathbb{R}^{n}\) or \(S^{n}\).

My second area of research is fractional Brownian motion. My work with Vladimir Dobric on the estimation of the concentration of measure for fractional Brownian motion requires finding the intersections of ellipsoidal and spherical shells for Gaussian measure in \(\mathbb{R}^{n}\). Gaussian measure is concentrated on a small shell of a sphere of radius \(\sqrt{n}\). We want to determine how large this shell must be to include the majority of the Gaussian measure. This result determines the rate of convergence of averages of squares for fractional Brownian increments. It also requires understanding the spectrum of the covariance operator as a function of dimension \(n\) and the Hurst index. To help understand the spectrum, we have computed the exact rate of the largest eigenvalue of this operator. We are continuing to study the distribution of the eigenvalues for this operator to obtain a better estimation.