4. Define subintervals Jn, n = 0, 1, 2, . . ., of [0, 1] as follows: Jo = 1, J = [0, 1 2 J5 J2 [0,1], = [,1], 2 3. 3 44 4- 4-4 4-6 J3 = [0, J = J6 = [4, 1], __J7 = [0, and so on. Together they form the dyadic intervals of [0,1]. Each Jn has the form In = [2] for some i and k. Let U be a U(0, 1) random variable. For n 1, define Xn = I(U) be the indicator of interval Jn, i.e., Xn 1 if U Jn and otherwise Xn = 0. = (a) Does Xn converge in probability? If so, identify the limit. (b) Does Xn converge in L? If so, identify the limit. = IJn (u). (c) Consider a fixed realization U = u. Given U = u (0,1), we have Xn Does the (deterministic) sequence I (u) converge? This is related to the concept of almost sure convergence (see Remark 2.13(iv)). (iv) Almost sure convergence is fundamental in probability theory but plays a minor role in the context of our course. Let X1, X2,... and X be Rd. valued random variables defined on the same probability space (Q, A, P). By definition, we say that X converges to X P-almost surely, written Xn X P-a.s., if P 6 lim Xn = x) = 1. 8u. Explicitly, if we let En A = {w N: _lim_Xn(w) = X(w)} 8u be the event that X converges to X, then A E A (so A is indeed an event) and P(A) = 1. It can be shown that almost sure convergence implies convergence in probability (and hence convergence in distribution). However, almost sure convergence does not imply, and is not implied by, convergence in L. Definition 2.9 (Convergence in L). Let X1, X2,... and X be Rd-valued random variables defined on the same probability space. We say that Xn converges to X in L, written XnX, if L2 lim E[|XX|] = 0. NX Example 2.10. Suppose Xn ~ n N(0, 1). Then X 0. This is because 1 - E[|Xn 0|] = E[|X|] = E[X] = Var(Xn) = 0. n (2.10) We proceed to study some relations among the three types of convergence. Recall that if A is an event, the indicator of A is the random variable I which takes value 1 on A and 0 elsewhere. Definition 2.3 (Convergence in probability). Let X1, X2,.. and X be Rd- valued random variables defined on the same probability space. We say that Xn converges in probability to X, written Xn X, if for any e > 0 we have lim P(X X| ) = 0. NX (2.5) Example 2.4. Let Z and be independent N(0, 1) random variables. Let XZ and Xn Z + 1/2 . Then Xn PX. == n Convergence in distribution, to be introduced next, is mathematically more advanced but is required in fundamental results such as the central limit theorem.