Anish Sarkar

Bio: Anish Sarkar is an academic researcher from Indian Statistical Institute. The author has contributed to research in topics: Vertex (geometry) & Almost surely. The author has an hindex of 13, co-authored 37 publications receiving 410 citations. Previous affiliations of Anish Sarkar include Indian Statistical Institute, Delhi Centre.

Random oriented trees: A model of drainage networks

Abstract: Consider the d-dimensional lattice Ζ d where each vertex is "open" or "closed" with probability p or 1-p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w satisfy w(d)=v(d)-1. In case of nonuniqueness of such a vertex w, we choose any one of the closest vertices with equal probability and independently of the other random mechanisms. It is shown that this random graph is a tree almost surely for d=2 and 3 and it is an infinite collection of distinct trees for d≥4. In addition, for any dimension, we show that there is no bi-infinite path in the tree and we also obtain central limit theorems of (a) the number of vertices of a fixed degree ν and (b) the number of edges of a fixed length l.

Nonuniversality and continuity of the critical covered volume fraction in continuum percolation

Abstract: We establish, using mathematically rigorous methods, that the critical covered volume fraction (CVF) for a continuum percolation model with overlapping balls of random sizes is not a universal constant independent of the distribution of the size of the balls. In addition, we show that the critical CVF is a continuous function of the distribution of the radius random variable, in the sense that if a sequence of random variables converges weakly to some random variable, then the critical CVF based on these random variables converges to the critical CVF of the limiting random variable.

On the coverage of space by random sets

Abstract: Let ˘ 1 ;˘ 2 ;::: be a Poisson point process of density  on (0;1) d ; d 1 and letˆ;ˆ 1 ;ˆ 2 ;::: be i.i.d. positive random variables independent of the point process. LetC := [ i1 f˘ i + [0;ˆ i ] d g. If, for some t>0, (t;1) d C;then we say that (0;1) d iseventually covered. We show that the eventual coverage of (0;1) d depends on the be-haviour of xP(ˆ>x) as x!1as well as on whether d= 1 or d2. These resultsare quite dissimilar to those known for complete coverage of R d by such Poisson Booleanmodels (Hall [3]).In addition, we consider the region C := [ fi1:X i =1g [i;i+ ˆ i ], where X 1 ;X 2 ;:::isa f0;1gvalued Markov chain and ˆ;ˆ 1 ;ˆ 2 ;:::are i.i.d. positive integer valued randomvariables independent of the Markov chain. We study the eventual coverage properties ofthis random set C. 1 Introduction In this paper we address two issues. One of these arises from genome analysis, while the othercomplements the results on complete coverage in stochastic geometry.In genomics, contig analysis is the method employed in sequencing or identifying the nucleotidesof a DNA sequence. This method involves cloning to obtain many identical copies of thesequence. Each such copy is then fragmented (by bio-chemical means) into many contigs orrandom segments, with each contig being of random length and starting from some randompoint of the sequence. After sequencing each of the random segments obtained from all the

Brownian web in the scaling limit of supercritical oriented percolation in dimension 1 + 1

Abstract: We prove that, after centering and diffusively rescaling space and time, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice Z^2_{even}:={(x,i) in Z^2: x+i is even} converges in distribution to the Brownian web. This proves a conjecture of Wu and Zhang. Our key observation is that each rightmost infinite open path can be approximated by a percolation exploration cluster, and different exploration clusters evolve independently before they intersect.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Applied Probability and Queues

Abstract: (1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.