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09 Limit Theorems

# 09 Limit Theorems - Economics 241B Review of Limit Theorems...

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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 ² ° Y n < c ³ = 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 °; ± 2 ³ : 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 ² neighborhood of ° must be high, P ( ´ ´ ° Y n ° ° ´ ´ < ² ) > 1 ° ³ . That is P ° ! : ´ ´ ° Y n ( ! ) ° ° ´ ´ < ² ± > 1 ° ³: For convergence to a random variable, Y , we need P ° ! : ´ ´ ° Y n ( ! ) ° Y ( ! ) ´ ´ < ² ± > 1 ° ³; (1)

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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 ² 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 doesn±t 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. Not only could ° Y n and Y be de°ned on di/erent probability spaces, for each n , ° Y n could be de°ned on a di/erent probability space.
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09 Limit Theorems - Economics 241B Review of Limit Theorems...

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