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Unformatted text preview: Economics 241B Review of Limit Theorems for Sequences of Random Variables Convergence in Distribution The previous de&nitions of convergence focus on the outcome sequences of a ran- dom variable. Convergence in distribution refers to the probability distribution of each element of the sequence directly. De&nition. A sequence of random variables & & Y 1 ; & Y 2 ;::: is said to converge in distribution to a random variable Y if lim n !1 P & Y n < c = P ( Y < c ) at all c such that F Y is continuous. We express this as & Y n D ! Y . An equivalent de&nition is De&nition. A sequence of random variables & & Y 1 ; & Y 2 ;::: is said to converge in distribution to F Y if lim n !1 F & Y n ( c ) = F Y ( c ) at all c such that F Y is continuous. We express this as & Y n D ! F Y . The distribution F Y is the asymptotic (or limiting) distribution of & Y n . Convergence in distribution is simply the pointwise conver- gence of F & Y n to F Y . ( The requirement that F Y be continuous at all c will hold for all applications in this course, with the exception of binary dependent variables.) In most cases F Y is a Gaussian distribution, for which we write & Y n D ! N &; 2 : Convergence in distribution is closely related to convergence in probability, if the limit quantity is changed. In the discussion above, a sequence of random variables was shown to converge (in probability) to a constant. One can also establish that a sequence of random variables converges in probability to a random variable. Recall, for convergence (i.p.) to a constant, the probability that & Y n is in an neighborhood of & must be high, P ( & Y n & & < ) > 1 & . That is P & ! : & Y n ( ! ) & & < > 1 & : For convergence to a random variable, Y , we need P & ! : & Y n ( ! ) & Y ( ! ) < > 1 & ; (1) that is, for large n we want the probability that the histogram for & Y n is close to the histogram for Y . There is no natural measure of distance here, although we think of & as de&ning the histogram bin width. If the two histograms are arbitrarily close as n increases, then we write & Y n P ! Y: Because the two histograms are arbitrarily close, it should not be surprising that & Y n P ! Y ) & Y n D ! F Y (sometimes written & Y n D ! Y ). Why doesnt the reverse hold? Because of the basic workings of probability spaces, about which we rarely concern ourselves. By the fact that both & Y n and Y are indexed by the same ! in (1), both quantities are de&ned on the same probability space. No such requirement is made in establishing convergence of distribution, we simply discuss a sequence of distribution functions.distribution, we simply discuss a sequence of distribution functions....
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- Fall '08