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S. Sethuraman and S.R.S. Varadhan, "Large deviations for the current and tagged particle in 1D nearest-neighbor symmetric simple exclusion", Annals of Probability 41, no. 3A, 1461-1512 (2013) New York University

2 S. R. S. VARADHAN 1. Introduction The theory of large deviations deals with rates at which probabilities of certain events decay as a natural parameter in the problem varies. It is best to think of a speci c example to clarify the idea. Let us suppose that we have nindependent and identically distributed random variables fX

Y. Kifer and S.R.S. Varadhan, "Nonconventional limit theorems in discrete and continuous time via martingales", Annals of Probability 42, no. 2, 649-688 (2014)

Sections available now on .ps and .pdf format Information about the course .ps file. Information about the course .pdf file Chapter 1 of Notes .ps file

Probability Theory S.R.S.Varadhan Courant Institute of Mathematical Sciences New York University August 31, 2000

154 CHAPTER 5. MARTINGALES. 5.2 Martingale Convergence Theorems. If F nis an increasing family of σ-fields and X nis a martingale sequence with respect to F n,one can always assume without loss of generality that the full σ-field Fis the smallest σ-field generated by ∪ nF n.If for some p≥1, X ∈L p, and we define X n = E[X|F n]thenX n is a martingale and by Jensen’s …

400 S. R. S. VARADHAN 2. Rate functions, duality and generating functions. We start with i.i.d. random variables and look at the sample mean Yn = Sn n = X1 +···+Xn n. According to a theorem of Cramér, its distribution Pn satisfies a large deviation principle with a rate function h(a) given by h(a)=sup θ θa−logE[eθX].

THERE IS NO CLASS ON NOV 7. Chapters available now: Reserve List . Introduction .pdf file. Chapter 1 of notes .ps file Chapter 1 of notes .pdf file

Real variables II Syllabus . Part 1. Part 2. Part 3. Part 4. Problems.2.25. Problems.3.3