N 0 1 2 are orthonormal that is show e t 1 17r iktd 1

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n = 0, ±1, ±2, .. . } are orthonormal; that is, show - e t= 1 17r iktd ,1, 2rr -1r 0, if k = 0, if k ¥= 0. (b) Suppose X is integer valued with chf t/J. Show 1 17r P[X = k] = - 2 e-iktf/J(t)dt. 1T -]f (c) If X1, ... , Xn are iid, integer valued, with common chf f/J(t), show 43. Suppose {Xn, n ::: 1} are independent random variables, satisfying E(Xn) = 0, Var(Xn) =a; < 00. (a) Sn/Sn N(O, 1), (b) Gn/Sn-+
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332 9. Characteristic Functions and the Central Limit Theorem Prove Xnfsn => N(O, p 2 ). (Hint: Assume as known a theorem of Cramer and Uvy which says that if X, Y are independent and the sum X + Y is normally distribute, then each of X andY is normally distributed.) 44. Approximating roulette probabilities. The probability of winning $1 in roulette is 18/38 and the probability of losing $1 is thus 20/38 . Let {X n, n 1} be the outcomes of successive plays; so each random variable has range ±1 with probabilities 18/38, 20/38. Find an approximation by the central limit theorem for P[Sn OJ, the probability that after n plays, the gambler is not worse off than when he/she started. 45. Suppose f(x) is an even probability density so that f(x) = f(-x) . Define g(x) = Jx s s, I roo Md g(-x), Why is g a probability density? if X> 0, if X< 0. Iff has chf l/>(t), how can you express the chf of gin terms of 4>? 46 . Suppose {X n, n 1} are independent gamma distributed random variables and that the shape parameter of Xn is an . Give conditions on {an} which guarantee that the Lindeberg condition satisfied.
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S.I. Resnick, A Probability Path , Modern Birkhäuser Classics, ss Media New York 2014 DOI 10.1007/978-0-8176-8409-9_10, © Springer Science+Busine 333 10 Martingales Martingales are a class of stochastic processes which has had profound influence on the development of probability and stochastic processes. There are few areas of the subject untouched by martingales. We will survey the theory and appli- cations of discrete time martingales and end with some recent developments in mathematical finance. Here is what to expect in this chapter: Absolute continuity and the Radon-Nikodym Theorem. • Conditional expectation. Martingale definitions and elementary properties and examples. Martingale stopping theorems and applications. Martingale convergence theorems and applications. The fundamental theorems of mathematical finance. 10.1 Prelude to Conditional Expectation: The Radon-Nikodym Theorem We begin with absolute continuity and relate this to differentiation of measures. These concepts are necessary for a full appreciation of the mathematics of condi- tional expectations. Let (Q, 13) be a measurable space. Let f..L and). be positive bounded measures on (Q, 13). We say that A. is absolutely continuous (AC) with respect to f..L, written
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334 10. Martingales ). < < J.t, if J.t(A) = 0 implies J...(A) = 0. We say that). concentrates on A e B if J...(A c) = 0. We say that ). and J.t are mutually singular, written ).
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