Rabi N. Bhattacharya

Rabi Bhattacharya has made signal contributions to central limit theory and normal approximation, particularly for sums of independent random vectors. His monograph in the area (Bhattacharya and Ranga Rao 1976) has become a classic, its importance being so great that it has had significant influence on mathematical statistics as well as probability. The methods developed in that monograph led to Bhattacharya and Ghosh’s (1978) seminal account of general Edgeworth expansions under the smooth function model, as it is now commonly called. That article had a profound influence on the development of bootstrap methods, not least because it provided a foundation for influential research that enabled different bootstrap methods to be compared. At a vital time in the evolution of bootstrap methods, it led to an authoritative and enduring assessment of many of the bootstrap’s important contributions.

The quasi-likelihood estimator and the Bayesian type estimator of the volatility parameter are in general asymptotically mixed normal. In case the limit is normal, the asymptotic expansion was derived in Yoshida 1997 as an application of the martingale expansion. The expansion for the asymptotically mixed normal distribution is then indispensable to develop the higher-order approximation and inference for the volatility. The classical approaches in limit theorems, where the limit is a process with independent increments or a simple mixture, do not work. We present asymptotic expansion of a martingale with asymptotically mixed normal distribution. The expansion formula is expressed by the adjoint of a random symbol with coefficients described by the Malliavin calculus, differently from the standard invariance principle. Applications to a quadratic form of a diffusion process (“realized volatility”) is discussed.

Normal approximation or, more generally the asymptotic theory, plays a fundamental role in the developments of modern probability and statistics. The one-dimensional central limit theorem and the Edgeworth expansion for independent real-valued random variables are well studied. We refer to the classical book by Petrov (1995). In the context of the multi-dimensional central limit theorem, Rabi Bhattacharya has made fundamental contributions to asymptotic expansions. The book by Bhattacharya and Ranga Rao (1976) is a standard reference. In this note I shall focus on two of his seminal papers (1975, 1977) on asymptotic expansions. Recent developments on normal approximation by Stein’s method and strong Gaussian approximation will also be discussed.

Berry-Essen bounds for the multi-dimensional central limit theorem. Bull. Amer. Math. Society. 74 (1968), 285–287. © 1968 American Mathematical Society. Rates of weak convergence and asymptotic expansions for classical central limit theorems. The Annals of Mathematical Statistics. 42 (1971), 241–259. ©1971 Institute of Mathematical Statistics. On errors of normal approximation. The Annals of Probability. 3 (1975), 815–828. ©1975 Institute of Mathematical Statistics. Refinements of the multidimensional central limit theorem and applications. The Annals of Probability. 5 (1977), 1–27. ©1977 Institute of Mathematical Statistics. On the validity of the formal Edgeworth expansion. The Annals of Statistics. 6 (1978), 434–451. ©1978 Institute of Mathematical Statistics.

As the name suggests, central limit theorem or CLT does play a central role in probability theory. Early masters like De Moivre, Laplace, Gauss, Lindeberg, Lévy, Kolmogorov, Lyapunov, and Bernstein studied the case of sums of independent random variables. Their results were then extended to sums of dependent random variables by various authors. For sums of the form ∑ _i V (X _i ) where X _i is a stationary Markov chain, the CLT was proved by Markov himself. The proof consisted of leaving large gaps to create enough independence but not large enough to make a difference in the sum. It is a bit delicate to balance the two and requires assumptions on the mixing properties of the stationary process. If the summands form a stationary sequence and the partial sums is a martingale relative to the natural filtration, it was observed by Paul Lévy that the CLT was valid under virtually no additional conditions. An early version of this result can be found in Doob’s book [5] on Stochastic Processes and a more modern version in Billingsley [3].

Criteria for recurrence and existence of invariant measures for multidimensional diffusions. The Annals of Probability. 6 (1978), 541–553. © 1978 Institute of Mathematical Statistics. On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete. 60 (1982), 185–201. © 1982 Springer-Verlag. A central limit theorem for diffusions with periodic coefficients. The Annals of Probability. 13 (1985), 385–396. © 1985 Institute of Mathematical Statistics. On the central limit theorems for diffusions with almost periodic coefficients. Sankhyā. The Indian Journal of Statistics. Series A 50 (1988), 9–25. © 1988 Indian Statistical Institute (with S. Ramasubramanian). Stability in distribution for a class of singular diffusions. The Annals of Probability. 20 (1992), 312–321. © 1992 Institute of Mathematical Statistics (with G. Basak). Speed of convergence to equilibrium and to normality for diffusions with multiple periodic scales, Stochastic Processes and their Applications. 80, pages 55–86. © 1999 Elsevier Science B.V. (with M. Denker & A. Goswami).

The many areas in mathematical statistics, probability theory, statistics processes, and mathematical analysis to which Professor Rabi Bhattacharya has made important and deep contributions include the topics in the title of this paper. In this paper we shall outline some interesting recent results in these topics.

The topic of random dynamical systems is extremely broad. However the focus of Rabi Bhattacharya’s work in this area is largely from the perspective of discrete parameter Markov processes on a general state space S, equipped with a suitable sigmafield \(\mathcal{S}\) of measurable subsets. Such Markov processes are either prescribed as evolutions defined by i.i.d. iterated random maps from S to S, or by such a representation theorem that holds for any discrete parameter Markov processes having stationary transition probabilities on a Borel subset S of a Polish space, with Borel sigmafield \(\mathcal{S}\). A theme of much of Rabi’s work is that of existence and uniqueness of invariant probabilities under conditions in which the Markov process may not be irreducible. These and corresponding problems concerning rates of convergence and various asymptotic limit theorems are representative of the research addressed here. Applications, particularly to geosciences and economics, are also a main theme of Rabi’s body of work in this area; however, these will be covered in separate essays and not treated here. The co-authored texts [6, 11] include a variety of such applications.

Asymptotics of a class of Markov processes which are not in general irreducible. The Annals of Probability. 16 (1988), 1333–1347. © 1988 Institute of Mathematical Statistics (with O. Lee). Random iterations of two quadratic maps. In: A Festschrift in honour of G. Kallianpur. Edited by Cambanis et al. Springer, New York, 1993, 13–22. © 1993 Springer-Verlag (with B.V. Rao). On a theorem of Dubins and Freedman. Journal of Theoretical Probability. 12 (1999), 1067–1087. © 1999 Springer-Verlag (with M. Majumdar). An approach to the existence of unique invariant probabilities for Markov processes. In: Limit Theorems in Probability and Statistics I. Balatonlelle 1999. Edited by I. Berkes, E. Csàki, M. Csörgő János Bolyai Mathematical Society, Budapest, 2002, 181–200. © 2002 János Bolyai Mathematical Society (with E.C. Waymire).

Random dynamical systems encountered in economics have certain distinctive characteristics that make them particularly well suited to analysis using the tools for studying Markov processes developed by Rabi N. Bhattacharya and his coauthors over the last few decades. In this essay we discuss the significance of these tools for both mathematicians and economists, provide some historical perspective, and review some recent related contributions.

The analysis of resource allocation over time under conditions of uncertainty occupies a central place in economics. A very widely used framework for the study of such dynamic resource allocation problems is one where stocks of “capital” or other assets accumulate over time through investment, and the accumulation process is subject to random “shocks.” Important economic problems studied in this framework include economic growth under productivity shocks, household accumulation of wealth with uncertain returns on savings, depletion of renewable and other natural resources whose natural growth is subject to environmental or climate related shocks, and the growth of pests and other invasive biological species whose expansion rates are affected by uncertainty. Though the economic models used to study these problems can be fairly elaborate, they often generate random dynamical systems where the intertemporal transition of capital or a related stock variable is determined by a “transition” function that depends on the previous period’s capital as well as the realizations of an exogenously specified stochastic process of “shocks.” The important problem for the economist is to then understand the asymptotic or limiting behavior of this dynamical system and how it depends on initial conditions, the nature of the transition function and the distribution of shocks. This naturally leads to the question of existence and (local or global) stability of a stochastic steady state i.e., an invariant distribution of the random dynamical system. Economists tend to view the limiting stochastic steady state of the dynamical system as a long run “equilibrium.” The rate of convergence to the limiting steady state is important for the strength of predictions made on the basis of this long run equilibrium. It is also important to be able to estimate the long run distribution of capital or resource stocks on the basis of finite data. Finally, whether or not similar economic systems that differ only in initial conditions may exhibit very different long run behavior depends on global stability of the steady state.

Dynamical systems subject to random shocks: an introduction. In: Symposium on dynamical systems subject to random shock. Economic Theory. 23 (2004), 13–38. © 2004 Springer-Verlag (with M. Majumdar). Random iterates of monotone maps. Revue of Economic Design. 14 (2010), 185–192. © 2010 Springer-Verlag (with M. Majumdar).

In its broadest sense, understanding the dispersion of particles suspended in fluid media is a classic problem that has motivated a tremendous amount of laboratory and field experimentation as well as mathematical and physical theory for centuries. The theory traces back to celebrated work of such historically eminent scientists as Adolf Fick, Albert Einstein, Marian Smoluchowski, Jean Perrin, and Geoffrey I. Taylor, to name a few of the most prominent historic figures. The richness of the problem is reflected in the development of new mathematical, statistical, and computational tools that have resulted from continued explorations of this phenomena beyond the framework of advection-dispersion in a pure liquid. The work of Rabi Bhattacharya, in collaboration with hydrologist Vijay Gupta, stands out for the important theoretical insights provided to contemporary understanding of this phenomena in heterogeneous media over a range of space-time scales. This chapter is an attempt to provide some overview and context for the salient features of these contributions.

The basic equations governing the motion of a fluid are well understood. For simplicity, we shall refer to the case of an incompressible, constant density, viscous Newtonian fluid; the velocity vector field \(u\left (t,x\right )\) and pressure scalar field \(p\left (t,x\right )\) satisfy the classical Navier–Stokes equations (in dimension 3) with viscosity ν > 0 with appropriate initial and boundary conditions depending on the problem. For relatively simple fluid motions, these equations give us a very good tool for simulations and physical understanding. But there are complex fluid motions, those usually called turbulent, where special features are experimentally or numerically observed which do not have a clear explanation yet from the Navier–Stokes equations. In a sense, there is something at the foundation of fluid dynamics that is still unclear. For later reference, let us mention that this happens when a certain parameter R, called Reynolds number, is very large.

On a statistical theory of solute transport in porous media. SIAM Journal on Applied Mathematics 37 (1979), 485–498. © 1979 Society for Industrial and Applied Mathematics (with V.K. Gupta). On the Taylor-Aris theory of solute transport in a capillary. SIAM Journal on Applied Mathematics 44 (1984), 33–39. © 1984 Society for Industrial and Applied Mathematics (with V.K. Gupta). Asymptotics of solute dispersion in periodic porous media. SIAM Journal on Applied Mathematics 49 (1989), 86–98. © 1984 Society for Industrial and Applied Mathematics (with V.K. Gupta and H. F. Walker). Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media. The Annals of Applied Probability 9 (1999), 951–1020. © 1999 Institute of Mathematical Statistics. Majorizing kernel and stochastic cascades with application to incompressible Navier Stokes equations. Transactions of the American Mathematical Society 355 (2003), 5003–5040. © 2003 American Mathematical Society.

One of the many fundamental contributions that Rabi Bhattacharya, together with his coauthors, has made is the development of a general nonparametric theory of statistical inference on manifolds, in particular related to both intrinsic and extrinsic Fréchet means of probability distributions thereon (cf. Bhattacharya and Bhattacharya 2012, Bhattacharya, Patrangenaru 2013 and 2005). With the increasing importance of statistical analysis for non-Euclidean data in many applications, there is much scope for further advances related to this particular broad area of research. In the following, we concentrate on two particular important themes in data analysis on manifolds: nonparametric bootstrap methods and nonparametric curve fitting.

We review some aspects of the Bhattacharya-Patrangenaru asymptotic theory for intrinsic and extrinsic means on manifolds, some of the problems involved, many of which are still open, and survey some of its impacts on the community.

Large sample theory of intrinsic and extrinsic sample means on manifolds. I. The Annals of Statistics. 31 (2003), 1–29. © 2003 Institute of Mathematical Statistics (with V. Patrangenaru). Large sample theory of intrinsic and extrinsic sample means on manifolds. II. The Annals of Statistics. 33 (2005), 1225–1259. © 2005 Institute of Mathematical Statistics (with V. Patrangenaru). Statistics on Riemannian manifolds: asymptotic distribution and curvature. Proceedings of the American Mathematical Society. 136 (2008), 2959–2967. © 2008 American Mathematical Society (with A. Bhattacharya). Statistics on manifolds with applications to shape spaces. In: Perspectives in mathematical sciences. Statistical Science Interdisciplinary Research 7, World Scientific Publishing 2009, 41–70. © 2009 World Scientific Publishing Co. Pte. Ltd. (with A. Bhattacharya).