Pa the lln justifies the fre quency interpretation of

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= P(A). The LLN justifies the fre- quency interpretation of probabilities and much statistical estimation theory where it underlies the notion of consistency of an estimator. Central limit theorem (CLT): The central limit theorem assures us that sam- ple averages when centered and scaled to have mean 0 and variance 1 have a distribution that is approximately normal. If {X n, n ::::: 1} are iid with
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2 1. Sets and Events common mean E(Xn) = Jl and variance Var(Xn) = a 2 , then X; - nJ.l ] lx e- u2;z p '- 1 .jii .::: x .... N(x) := du. a n -oo v2rr This result is arguably the most important and most frequently applied re- sult of probability and statistics. How is this result and its variants proved? Martingale convergence theorems and optional stopping : A martingale is a stochastic process {Xn, n 2: 0} used to model a fair sequence of gam- bles (or, as we say today, investments). The conditional expectation of your wealth Xn+l after the next gamble or investment given the past equals the current wealth X n. The martingale results on convergence and optimal stop- ping underlie the modern theory of stochastic processes and are essential tools in application areas such as mathematical finance. What are the basic results and why do they have such far reaching applicability? Historical references to the CLT and LLN can be found in such texts as Breiman (1968), Chapter I; Feller, volume I (1968) (see the background on coin tossing and the de Moivre-Laplace CLT); Billingsley (1995), Chapter 1; Port (1994), Chapter 17. 1.2 Basic Set Theory Here we review some basic set theory which is necessary before we can proceed to carve a path through classical probability theory. We start by listing some basic notation. • Q: An abstract set representing the sample space of some experiment. The points of Q correspond to the outcomes of an experiment (possibly only a thought experiment) that we want to consider. • 'P(Q): The power set of n, that is, the set of all subsets of Q. • Subsets A , B, . . . of Q which will usually be written with roman letters at the beginning of the alphabet. Most (but maybe not all) subsets will be thought of as events, that is, collections of simple events (points of Q). The necessity of restricting the class of subsets which will have probabili- ties assigned to them to something perhaps smaller than 'P(Q) is one of the sophistications of modern probability which separates it from a treatment of discrete sample spaces. • Collections of subsets A, B, ... which will usually be written by calligraphic letters from the beginning of the alphabet. • An individual element of Q: w e Q.
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1.2 Basic Set Theory 3 The empty set 0, not to be confused with the Greek letter¢. P(Q) has the structure of a Boolean algebra. This is an abstract way of saying that the usual set operations perform in the usual way. We will proceed using naive set theory rather than by axioms. The set operations which you should know and will be commonly used are listed next. These are often used to manipulate sets in a way that parallels the construction of complex events from simple ones.
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