Theoretical Statistics and Mathematics Unit Seminars

Some results in binary Recursive Sequences
Abstract: Beginning with the definition of binary recursive sequences and in paricular of Fibonacci sequences and Lucas sequences, we consider some features of these sequences. We shall also consider some diophantine equations in members of these sequences.

A statistical study of climate change: analysis of temperature records of Arctic seawater data
Abstract: We analyze a dataset on seawater pattern over the last few decades. For specificity, we restrict attention to temperature measures in the Arctic Ocean region for this talk. Our goal is to investigate whether there is a significant change of pattern in the Arctic Ocean seawater temperature, thus detecting climate change, after accounting for the systematic factors like location, depth, season, and the temporal and spatial dependence pattern of the observations. We do not explicitly model the spatio-temporal dependency pattern of the observations, but treat it as an extremely high dimensional nuisance parameter, and use techniques for estimation and inference that are insensitive to it. We use nonparametric curve fitting for weakly dependent observations to model different functions of seawater temperature, and then perform sequential tests to detect whether the function under consideration has changed its pattern from previous time-points. A complex resampling-based robustness study is used to decide whether the changes detected are chance aberrations. Finally, we separate the data in two parts based on observation-time, and use a block bootstrap based scheme to compare temperature patterns in the two regimes. The block-bootstrap based technique elicits probabilities that are the equivalents of size, power and $p$-value of our sequential testing procedure, under reasonable assumptions. The methodology used in this paper is applicable for any sequence of dependent observations on the climate, and unlike many other climate studies does not rely on computer simulated deterministic outputs, nor use of indirect historical data, nor rely on technical assumptions like linearity, Gaussian nature of random variables, specific dependency patterns, and so on. This work is jointly done with Qiqi Deng and Jie Xu.

Robust Estimation under Conditional Heteroscedasticity
Abstract: In this talk, we discuss estimation of parameters for heteroscedastic models. In particular, we propose classes of rank and M-estimators of the parameters associated with the conditional mean function of the heteroscedastic autoregressive models and the class of M-estimators of the parameters of the symmetric as well as asymmetric heteroscedasticity. We discuss theoretical as well as empirical properties of the proposed estimators. Applications to the analysis of the financial data are presented.

Cuntz-Li algebras and Inverse semigroups
Abstract: Let R be an integral domain such that every quotient R/mR is finite. Cuntz and Li associated a C*-algebra to R called the ring C*-algebra which is the C*-algebra on L^2(R) generated by the unitaries induced by the addition on R and the isometries induced by the multiplication on R. Cuntz and Li showed that this algebra is simple and purely infinite.
We will show how one can apply inverse semigroups to obtain these results. Some generalisation of these Cuntz-Li relations due to Quigg, Landstad and Kaliszewski will be discussed.

Algebraic values of meromorphic maps and transcendence
Abstract: A general theorem in transcendence theory, the so called " Schneider-Lang criterion", which roughly says that if a collection of meromorphic function satisfies a system of linear differential equations then there are only finitely many complex numbers at which the functions assumes algebraic values simultaneously and the cardinality of the set of complex points can be bounded by the growth of the functions and the degree of the number field. I will speak on this and on more recent progresses in this direction ( in particular the more challenging question of eliminating the dependence of the number field in the cardinality bound ).

Smoothness of Coalescence Hidden-variable Fractal Interpolation Surfaces
Abstract: Among the major recent developments in understanding the structures of objects found in nature, the term ``fractals'' occupies an important place. In this talk, beginning with the construction of Coalescence Hidden-variable Fractal Interpolation Surface(CHFIS) to simulate fractal surfaces whose graphs exhibit partly self-affine and partly non-self-affine nature, smoothness of a CHFIS will be described by its Lipschitz exponent.

Holiday

Group of rational points of algebraic groups
Abstract: Let G be an algebraic group (a group defined by polynomial equations) over a field K (not nec. algebraically closed, e.g. field of rationals, reals). If K is infinite, the group G(K) of K-points is dense in G, in most of situations. Hence group theoretic properties of G(K), e.g. its simplicity, its representation theory etc, are important in the study of G itself.
In this talk, we will discuss some celebrated problems concerning G(K) (e.g. the Kneser-Tits problem) and mention some recent research in the subject.

Addition of runs to a supersaturated design
Abstract: The purpose of this presentation is to introduce a new class of extended $E(s^2)$-optimal two-level supersaturated designs obtained by adding runs to an existing $E(s^2)$-optimal two level supersaturated design. The extended design is a union of two optimal SSDs belonging to different classes. New lower bound to $E(s^2)$ has been obtained for the extended supersaturated designs.
This idea has been further extended to obtain multi-level supersaturated designs. Extended $E(\chi^2)$-optimal multi-level supersaturated design has been obtained by adding runs to an existing $E(\chi^2)$-optimal multi-level supersaturated design. The extended design is again a union of two $E(\chi^2)$-optimal multi-level supersaturated designs belonging to different classes. A lower bound to $E(\chi^2)$ has been obtained for the extended supersaturated designs. Some examples and a small catalogue of both $E(s^2)$-optimal and $E(\chi^2)$-optimal extended supersaturated designs has been prepared. These designs are available at www.iasri.res.in/design/.

Urn Models on One-dimensional Integer Lattice
Abstract: In this talk we will present a new urn model consisting of balls of infinite but countably many colors which we index by the integers. We will consider two special replacement matrices, one arriving from the right shift operator, and the other arriving from the simple symmetric random walk on the one dimensional integer lattice. We show using martingale techniques that in both the cases the expected proportion of colors converges to a standard normal distribution after an appropriate centering and scaling by $log n$. This shows that even though the associated Markov chain has different qualitative properties, namely one is transient and the other is null recurrent, the infinite color urn models have same asymptotic behavior. This is in sharp contrast to what is generally observed for finite color urn models.
(This is a joint work with Antar Bandyopadhyay, ISI, Delhi)

Extensions of $C^*$-algebra by compact operators
Abstract: We will show that the set of extensions of a unital, separable, nuclear $C^*$ algebra by compact operators modulo unitary equivalence forms an abelian group. As a consequence, we give a complete classification of essentially normal operators having same essential spectrum modulo an appropriate equivalence relation. These results are part of the celebrated Brown-Douglas-Fillmore theory.

On the conditional distributions of a partially observed Markov chain
Abstract: Let $X_n, n=0,1,2,...$ be an aperiodic positively recurrent Markov chain. Let $Y_n$ be an observation of $X_n$ at time $n$ obtained by an observation system which is not perfect. ($X_n$ is partially observed.)
Let $Z_n$ denote the conditional distribution of $X_n$ given all the observations up to time $n$. ($Z_n$ is thus a stochastic variable with values in the set of probability vectors on the state space of the Markov chain.)
Let $\mu_n$ denote the distribution of $Z_n$. Does it follow that there always exists a unique probability measure $\mu$ such that $\mu_n$ converges in distribution towards $\mu$ ? (Generalised version of Blackwell's conjecture from 1957.)

An Extreme point Characterization of Strategy-proof Probabilistic rules
Abstract: We consider collective decision problems with a finite number of agents who have single-peaked preferences on the real line. A probabilistic decision scheme assigns a probability distribution over the set of alternatives to every profile of reported preferences. The main result of the paper is a characterization of the class of unanimous and strategy-proof probabilistic schemes with the aid of deterministic rules as defined by H. Moulin (Public Choice 35 (1980), 437-455). Thereby, the work of Moulin (1980) is extended to the probabilistic framework. Here we give an extreme point characterization of the aforesaid class of probabilistic schemes with deterministic rules as extreme points. Thereby, we show that in single-peaked preference domains any strategy-proof probabilistic rule can be expressed as a convex combination of the strategy proof deterministic rules. This characterization helps in solving other problems such as finding the mechanism that maximizes ex-ante total expected utility of all agents. This is because, given this characterization, maximum ex-ante total utility can be obtained over the set of deterministic rules.
Co-authors: Hans Peters, Arunava Sen, and Ton Storcken.

Congruences between modular forms and eigenvarieties
Abstract: Congruences between modular forms have been discovered by Ramanujan around 1910; their systematic study started only in 1970 with the work of Serre and Swinnerton-Dyer. A new phenomenon, congruences between cusp forms, has been discovered numerically by the japanese mathematician Doi in 1975. These congruences have been systematized by Hida and Coleman in the 80's in what is called $p$-adic families of cusp forms; they are now best seen through the geometry of a curve (or even a higher dimensional variety) called the Eigenvariety (or the Hecke Variety). I'll summarize these progresses with examples.

Congruences between modular forms
Abstract: I will describe sketchily how the theory of modular forms modulo $p$ as developed by Serre, Swinnerton-Dyer and Katz in the 70's can explain some of the celebrated Ramanujan congruences.

Dependence comparisons and their applications
Abstract: Suppose we want to compare the degree of dependence between the components of two bivariate random vectors. Rather than using a single summary statistic such as coefficient of correlation, it is more useful and informative to compare the whole distributions or some aspects of them. For this purpose, various partial orders have been introduced in the literature assuming that their marginal distributions are identical. But in many problems of practical interest, this is not the case. It will be discussed as how to modify the existing partial orders to over come this difficulty. The modified partial orders for dependence are copula based. Applications of these orders to compare the degree of dependence between pairs of order statistics, record values and generalized order statistics will be discussed. We also consider the case when the random variables are independent but not identically distributed. Explicit expressions are obtained for the population value of Kendall's coefficient of concordance between any two order statistics as well as record values of a random sample.

Adaptive Lasso estimators and the Bootstrap
Abstract: We will describe some consistency results for bootstrapped Adaptive Lasso estimators in fixed and increasing dimensions. It will be shown that the bootstrap provides higher order accuracy in comparison to the normal (oracle-based) approximation.

Description of basis vectors of the joint kernel for a class of Hilbert modules
Abstract: We will be describing the basis of the joint kernel at origin of module of the form [I]. This problem is motivated from the computation of invariants for such modules by blow up technique. There it is necessary to obtain a description of the basis vectors in terms of reproducing kernel. The proof is based on the basic properties of the characteristic space introduced by Chen-Guo and realizing certain coefficient matrix as sum of Grammian of some polynomials with respect to Fock inner product. This is a joint work with Gadadhar Misra from Indian Institute of Science.

On Asymptotically Harmonic Spaces
Abstract: First I will present my result: If $M$ is an asymptotically harmonic manifold of constant $h > 0$, then $M$ is a hyperbolic manifold of constant sectional curvature $\lbrace -h^2 \rbrace \ \lbrace 4 \rbrace$. Then I will talk about my two recent results viz: (1) The above result can be extended to the case $h = 0$. (2) Non-compact, flat manifolds of dimension $3$ exhibit a rigidity property.

Martingale Solutions for Stochastic Navier-Stokes Equations with Itō-Lévy Noise
Abstract: In this talk, we discuss the solvability of martingale problem for the stochastic Navier-Stokes equations with Itō-Lévy noise under appropriate conditions in bounded and unbounded domains in $R^d, d = 2, 3$. The tightness criteria for the laws of a sequence of semimartingales is obtained from a theorem of Rebolledo as formulated by Metivier for the Lusin space valued processes. The existence of martingale solutions (in the sense of Stroock and Varadhan) relies on a Minty stochastic lemma which is essentially obtained from a local monotonicity of the drift term.

SRB-measure leaks
Abstract: In this talk, we shall study about the escaping rate of the Sinai-Ruelle-Bowen (SRB) measure through holes of positive measure constructed in the Julia set of hyperbolic rational maps (open dynamics). The dependence of this rate on the size and position of the hole shall be explained. For an easier and better understanding, the simple quadratic polynomial restricted on the unit circle will be analysed thoroughly.

Some results on generalized Stein's identity and its applications
Abstract: Stein's identity and its role in inference procedures has been widely discussed in the literature. In this talk, we discuss generalization of this identity applicable to a general class of distributions. We derive several existing results as special cases of our general identity and discuss some applications.

Some results on model identification ofnon-linear time series
Abstract: Information theoretic approach has significant role in analyzing the serial dependence of non-linear non-Gaussian time series models. In this talk, we discuss the properties of autoinformation and partial autoinformation functions in the context of model identification. We compare the autoinformation and partial autoinformation functions with autocorrelation and partial autocorrelation functions for some well known models.