We say that the sequence {X n} converges in distribution to X if … Suppose B is … (a) Xn a:s:! (This is because convergence in distribution is a property only of their marginal distributions.) Theorem 2.11 If X n →P X, then X n →d X. Convergence in probability implies convergence in distribution — so convergence in distribution is the weakest form of convergence we discuss in distribution is the weakest form of convergence … There are several different modes of convergence. We begin with convergence in probability. However, this random variable might be a constant, so it also makes sense to talk about convergence to a real number. In fact, a sequence of random variables (X n) n2N can converge in distribution even if they are not jointly de ned on the same sample space! X =)Xn p! The converse is not true: convergence in distribution does not imply convergence in probability. Thus X„ £ X implies ^„{B} — V{B) for all Borel sets B = (a,b] whose boundaries {a,6} have probability zero with respect to the measur We V.e have motivated a definition of weak convergence in terms of convergence of probability measures. No other relationships hold in general. However, it is clear that for >0, P[|X|< ] = 1 −(1 − )n→1 as n→∞, so it is correct to say X n →d X, where P[X= 0] = 1, Assume that X n →P X. Proof: Let F n(x) and F(x) denote the distribution functions of X n and X, respectively. Suppose Xn a:s:! the same sample space. Convergence in probability is also the type of convergence established by the weak ... Convergence in quadratic mean implies convergence of 2nd. so almost sure convergence and convergence in rth mean for some r both imply convergence in probability, which in turn implies convergence in distribution to random variable X. convergence in distribution to a random variable does not imply convergence in probability dY. The vector case of the above lemma can be proved using the Cramér-Wold Device, the CMT, and the scalar case proof above. X, and let >0. Convergence in Distribution • Recall: in probability if • Definition Let X 1, X 2,…be a sequence of random variables with cumulative distribution functions F 1, F 2,… and let X be a random variable with cdf F X (x). By the de nition of convergence in distribution, Y n! In general, convergence will be to some limiting random variable. X. Proof. most sure convergence, while the common notation for convergence in probability is X n →p X or plim n→∞X = X. Convergence in distribution and convergence in the rth mean are the easiest to distinguish from the other two. However, we now prove that convergence in probability does imply convergence in distribution. The Cramér-Wold device is a device to obtain the convergence in distribution of random vectors from that of real random ariables.v The the-4 Definition B.1.3. This limiting form is not continuous at x= 0 and the ordinary definition of convergence in distribution cannot be immediately applied to deduce convergence in distribution or otherwise. The former says that the distribution function of X n converges to the distribution function of X as n goes to infinity. Note that the convergence in is completely characterized in terms of the distributions and .Recall that the distributions and are uniquely determined by the respective moment generating functions, say and .Furthermore, we have an ``equivalent'' version of the convergence in terms of the m.g.f's

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