Lefschetz Center for Dynamical Systems Seminar
Large Deviations and Small Ball Estimates for Fractional Brownian Motions | |
Abstract:
Intermittency is one of the important phenomena in turbulence. Simply
put intermittency is the fact that
the probability distribution functions (PDF's) for quantities transported
by a turbulent flow are asymptotically
broad - wider than a Gaussian distribution.
We present some work (with R.M. McLaughlin (UNC)) on a model of passive scalar intermittency originally due to Majda: \[T_t = \gamma(t) x \frac{\partial T}{\partial y} + \Delta T \] where $\gamma(t)$ is a random process, and $T$ is a passive scalar (for instance a dye) which is advected by the random (shear) flow. Majda was able to explicitly calculate moments of the distribution of the scalar $T$. McLaughlin and B. were able to calculate the large $N$ asymptotics of the moments of the distribution and, by a large deviations/Tauberian type argument calculate the distribution of the quantity $T$.
I will also talk about some recent work on a generalization of this model. A similar calculation can be done for this generalized model, which involves calculating the asymptotics of a certain compact eigenvalue problem. As a by-product of this calculation one finds the (previously unknown) optimal constants in a certain probabilistic "small ball" estimate for the probability that a fractional Brownian motion stays in a small ball in $L_2$.
Brown University Center for Statistical Sciences Seminar
Department of Biostatistics, Harvard School of Public Health | |
4th Floor Conference Room Questions - Contact Jason Roy at 863-9559, e-mail jroy@stat.brown.edu or Ilana Gareen at 863-1758, e-mail igareen@stat.brown.edu |
Abstract: In this seminar we consider methods for analyzing discrete longitudinal data when there are dropouts. In order to correct for the bias that can arise when there is nonignorable dropout, two broad classes of joint models for the multivariate outcomes and dropout indicators have been proposed: selection models and pattern-mixture models. In this seminar we focus on methods based on pattern-mixture models and highlight some of the distinct advantages and disadvantages of the pattern-mixture modeling approach. Finally, an example using data from a clinical trial of contracepting women is used to illustrate the methods.
Stochastic Systems Seminar
Abstract: Precise limit theorems are developed for Markov processes on general state spaces, under natural (and in some cases minimal) assumptions. We consider the partial sums of a real-valued functional of an ergodic Markov process. Assuming only that the process is geometrically ergodic and the functional is bounded, we obtain:
(1). Edgeworth expansions for the distribution function of the normalized partial sums.
(2). The partial sums are shown to satisfy a large deviations principle (LDP) in a neighborhood of the mean. This result, proved under geometric ergodicity alone, cannot in general be strengthened to a full LDP.
(3). Exact large deviations expansions, refining the above LDP.
The main analytical tool used in the proofs is a multiplicative ergodic theorem. This, in turn, is deduced from the regularity properties of the solutions of an associated eigenvalue problem, the multiplicative Poisson equation.
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