If x n x as then for any e 0 pixn xi e io plimsupixn

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If X n X a.s. then for any E, 0 = P([IXn- XI > E] i.o.) = P(limsup[IXn -XI> E]) n-+00 = lim P(U£1Xn -XI> E]) N-+oo ::: lim P[IXn- XI > E] . n-+oo D Remark. The definition of convergence i.p . and convergence a.s . can be read - ily extended to random elements of metric spaces. If {Xn, n ::: 1, X} are ran - dom elements of a metric spaceS with metric d, then Xn X a.s . means that p p d(Xn. 0 a.s. and Xn means d(Xn. 0. 6.2.1 Statistical Terminology In statistical estimation theory, almost sure and in probability convergence have analogues as strong or weak consistency. Given a family of probability models (Q , B, Pe), 0 E 9). Suppose the statis- tician gets to observe random variables X 1, . . . , X n defined on n and based on
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6.3 Connections Between a.s. and i.p. Convergence 171 these observations must decide which is the correct model; that is, which is the correct value of(}. Statistical estimation means: select the correct model. For example, suppose S1 = lR 00 , B = B(lR 00 ). Let w = (x 1 , xz, ... ) and define Xn(w) = Xn. For each(} E IR, let Pe be product measure on lR 00 which makes {Xn, n 1} iid with common N(O, 1) distribution. Based on observing X 1, ... , X n, one estimates (} with an appropriate function of the observations On = On(Xt. ... , Xn). On(X1, ... , Xn) is called a statistic and is also an estimator. When one actually does the experiment and observes, then O(x1, ... xn) is called the estimate. So the estimator is a random element while the estimate is a number or maybe a vector if(} is multidimensional. In this example, the usual choice of estimator is On = 2:7 = 1 X; In. The estima- tor On is weakly consistent if for all (} E 8 Pe[IOn- 01 > E]-+ 0, n-+ oo; that is, A p8 On -+ 0. This indicates that no matter what the true parameter is or to put it another way, no matter what the true (but unknown) state of nature is, 0 does a good job estimating the true parameter. On is strongly consistent if for all(} E e, On -+ (}, Pe-a.s. This is obviously stronger than weak consistency. 6.3 Connections Between a.s. and i.p. Convergence Here we discuss the basic relations between convergence in probability and almost sure convergence. These relations have certain ramifications such as extension of the dominated convergence principle to convergence in probability. Theorem 6.3.1 (Relations between i.p. and a.s. convergence) Suppose that {Xn, X, n 1} are real-valued random variables. (a) Cauchy criterion: {X n} converges in probability iff {X n} is Cauchy in prob- ability. Cauchy in probability means p Xn -Xm-+ 0, asn,m-+ 00. or more precisely, given any E > 0, 8 > 0, there exists no = no(E, 8) such that for all r, s no we have P[IXr- Xsl > E] < 8. (6.1)
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172 6. Convergence Concepts (b) Xn X iff each subsequence {Xntl contains a further subsequence {X nk(iJ} which converges almost surely to X. Proof. (i) We first show that if Xn then {Xn} is Cauchy i.p. For any E > 0, [IXr- Xsl >E) C [IXr- XI > 2) U [IXs- XI > 2). (6.2) To see this, take complements of both sides and it is clear that if E E IX,- XI ::: 2 and IXs- XI ::: 2' then by the triangle inequality IXr- Xsl::; E.
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