19 research outputs found
A Note on Global Suprema of Band-Limited Spherical Random Functions
In this note, we investigate the behaviour of suprema for band-limited
spherical random fields. We prove upper and lower bound for the expected values
of these suprema, by means of metric entropy arguments and discrete
approximations; we then exploit the Borell-TIS inequality to establish almost
sure upper and lower bounds for their fluctuations. Band limited functions can
be viewed as restrictions on the sphere of random polynomials with increasing
degrees, and our results show that fluctuations scale as the square root of the
logarithm of these degrees
Central limit theorem for exponentially quasi-local statistics of spin models on Cayley graphs
Central limit theorems for linear statistics of lattice random fields
(including spin models) are usually proven under suitable mixing conditions or
quasi-associativity. Many interesting examples of spin models do not satisfy
mixing conditions, and on the other hand, it does not seem easy to show central
limit theorem for local statistics via quasi-associativity. In this work, we
prove general central limit theorems for local statistics and exponentially
quasi-local statistics of spin models on discrete Cayley graphs with polynomial
growth. Further, we supplement these results by proving similar central limit
theorems for random fields on discrete Cayley graphs and taking values in a
countable space but under the stronger assumptions of {\alpha}-mixing (for
local statistics) and exponential {\alpha}-mixing (for exponentially
quasi-local statistics). All our central limit theorems assume a suitable
variance lower bound like many others in the literature. We illustrate our
general central limit theorem with specific examples of lattice spin models and
statistics arising in computational topology, statistical physics and random
networks. Examples of clustering spin models include quasi-associated spin
models with fast decaying covariances like the off-critical Ising model, level
sets of Gaussian random fields with fast decaying covariances like the massive
Gaussian free field and determinantal point processes with fast decaying
kernels. Examples of local statistics include intrinsic volumes, face counts,
component counts of random cubical complexes while exponentially quasi-local
statistics include nearest neighbour distances in spin models and Betti numbers
of sub-critical random cubical complexes.Comment: Minor changes incorporated based on suggestions by referee
Random fields and the geometry of Wiener space
In this work we consider infinite dimensional extensions of some finite
dimensional Gaussian geometric functionals called the Gaussian Minkowski
functionals. These functionals appear as coefficients in the probability
content of a tube around a convex set under the standard
Gaussian law . Using these infinite dimensional extensions,
we consider geometric properties of some smooth random fields in the spirit of
[Random Fields and Geometry (2007) Springer] that can be expressed in terms of
reasonably smooth Wiener functionals.Comment: Published in at http://dx.doi.org/10.1214/11-AOP730 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Spatio-temporal Patterns of Indian Monsoon Rainfall
The primary objective of this paper is to analyze a set of canonical spatial
patterns that approximate the daily rainfall across the Indian region, as
identified in the companion paper where we developed a discrete representation
of the Indian summer monsoon rainfall using state variables with
spatio-temporal coherence maintained using a Markov Random Field prior. In
particular, we use these spatio-temporal patterns to study the variation of
rainfall during the monsoon season. Firstly, the ten patterns are divided into
three families of patterns distinguished by their total rainfall amount and
geographic spread. These families are then used to establish `active' and
`break' spells of the Indian monsoon at the all-India level. Subsequently, we
characterize the behavior of these patterns in time by estimating probabilities
of transition from one pattern to another across days in a season. Patterns
tend to be `sticky': the self-transition is the most common. We also identify
most commonly occurring sequences of patterns. This leads to a simple seasonal
evolution model for the summer monsoon rainfall. The discrete representation
introduced in the companion paper also identifies typical temporal rainfall
patterns for individual locations. This enables us to determine wet and dry
spells at local and regional scales. Lastly, we specify sets of locations that
tend to have such spells simultaneously, and thus come up with a new
regionalization of the landmass