People
Patricia Garmirian
Norbert Wiener Assistant Professor
Contact Info:
Tufts University
Department of Mathematics
503 Boston Avenue
BromfieldPearson
Medford, MA 02155
Email @tufts.edu:
patricia.garmirian
Expertise:
Probability and stochastic processes
Research:
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
steppingstones are, first, a new, similarly direct and elementary
proof of the CLT for Rademacher random variables defined on
\([0,1]\). The second important steppingstone 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.
