Probability and Stochastic Processes: A Volume in Honour of Rajeeva L. Karandikar (Indian Statistical Institute Series)

Preface
Contents
About the Editors
Rajeeva Laxman Karandikar: An Appreciation
	1 Introduction
	2 Stochastic Calculus
	3 White Noise Theory
	4 Martingale Problems
	5 Psephology
	6 Other Excursions
	References
Spectrum of High-Dimensional Sample Covariance and Related Matrices: A Selective Review
	1 Introduction
	2 Spectral Distribution and Linear Spectral Statistic
	3 Non-commutative Probability
	4 Useful Probability Laws and Non-commutative Variables
	5 Matrices of Our Interest
	6 LSD and Joint Algebraic Convergence Results
		6.1 Wigner Matrix
		6.2 Sample Covariance Matrix
		6.3 Kendall's upper T Subscript pTp and Spearman's upper R Subscript pRp Matrices
		6.4 Separable Sample Covariance Matrix
		6.5 Cross-Covariance Matrix
		6.6 Autocovariance Matrix of a Real Time Series
		6.7 Generalized Covariance Matrices
		6.8 High-Dimensional Autocovariance Matrices
	7 Linear Spectral Statistics: Gaussian Limit
	8 Extreme Eigenvalues: Tracy–Widom Laws
	9 Spiked Covariance Matrices
	10 Applications
		10.1 Test of Independence
		10.2 Estimating the Number of Spikes
		10.3 Testing for High-Dimensional Time Series
		10.4 MA Order Determination
	References
Long-Time Behavior of Finite and Infinite Dimensional Reflected Brownian Motions
	1 Introduction
	2 Finite Dimensional RBM
		2.1 Ergodicity.
		2.2 Geometric Ergodicity
	3 Parameter and  Dimension Dependence on Rates of Convergence
		3.1 Main Result
		3.2 Outline of Approach
		3.3 Examples
	4 Dimension-Free Convergence Rates for Local Functionals of RBM
		4.1 A Weighted upper L Superscript 1L1-Distance Governing Dimension-Free Local Convergence
		4.2 Parameters and Assumptions
		4.3 Main Result
		4.4 Application to the Asymmetric Atlas Model
		4.5 Polynomial Local Convergence Rates for Perturbations from Stationarity of the Atlas Model
	5 Domains of Attraction of the Infinite Atlas Model
		5.1 Main Results
		5.2 Outline of Approach
	6 Extremal and Product Form invariant Distributions of Infinite Rank-Based Diffusions
		6.1 Main Results
		6.2 Connection with Prior Work in Interacting Particle Systems
		6.3 Proof Outline
	References
A Survey of Some Recent Developments in Measures of Association
	1 Introduction
	2 Why Does it Work?
	3 Asymptotic Distribution
	4 Power for Testing Independence
	5 Multivariate Extensions
	6 Measuring Conditional Dependence
	7 Application to Nonparametric Variable Selection
	8 A New Proposal: Generalization to Standard Borel Spaces
	9 R Packages
	References
Fractal Structure in the Directed Landscape
	1 Introduction
		1.1 Fractality in Brownian Motion
	2 The Directed Landscape
		2.1 Geodesics in script upper LmathcalL
	3 Geodesic Coalescence in script upper LmathcalL
		3.1 The Weight Difference Profile and Brownian Local Time
		3.2 Geodesic Local Time
		3.3 The Full Spatio-Temporal Weight Difference Profile
		3.4 Non-uniqueness of Semi-infinite Geodesics
	4 Conclusion
	References
Small Ball Probabilities for the Stochastic Heat Equation
	1 Discussion of the Proof When sigma left parenthesis t comma x comma u right parenthesis equals sigma left parenthesis t comma x right parenthesisσ(t,x,u)=σ(t,x)
	2 Final Comments
	References
Directed Spanning Forest for Ginibre Ensemble Is a Tree
	1 Introduction and Main Result
	2 The Ginibre Ensemble and Its Rigidity and Tolerance Properties
	3 Burton-Keane Argument for the Poisson DSF
		3.1 Common Part of the Argument
		3.2 Proof of Lemma 3.1
	4 Burton-Keane Argument for the Ginibre DSF and Proof of Theorem 1.1
	References
Convergence of Stochastic Approximation via Martingale and Converse Lyapunov Methods
	1 Introduction
	2 Statements of Main Results
		2.1 Preliminaries
		2.2 Theorem Statements
		2.3 Discussion of the Theorems
	3 Proof of Theorem1
	4 Proof of Theorem2
	5 Proof of Theorem3
	6 Proof of Theorem5
	7 Examples and Applications
	8 Conclusions and Future Work
	References