conclusion. Donsker called his result an invariance principle. Consequently, the FCLT is often referred as an invariance principle So far, we have assumed that the sequence Et used to construct Wr is i.i.d Nevertheless, just as we can obtain central limit theorems when Et is not necessary id. In fact, versions of the FCLT hold for each CLt previous given in Chapter 4 Theorem: Continuous Mapping Theorem If Sr(=>S( and g() is a continuous functional, then g(Sr()=>9(SO) In the above theorem, continuity of a functional g( means that for any s>0 there exist ad>0 such that if h(r)and k(r) are any continuous bounded functions on [ 0, 1], h:[ 0, 1-R and k: [0, 1]R, such that h(r)-k(r)l< 5 for all r 0, 1, then Ig(h())-g(k()<sconclusion. Donsker called his result an invariance principle. Consequently, the FCLT is often referred as an invariance principle. So far, we have assumed that the sequence εt used to construct WT is i.i.d.. Nevertheless, just as we can obtain central limit theorems when εt is not necessary i.i.d.. In fact, versions of the FCLT hold for each CLT previous given in Chapter 4. Theorem: Continuous Mapping Theorem: If ST (·) =⇒ S(·) and g(·) is a continuous functional, then g(ST (·)) =⇒ g(S(·)). In the above theorem, continuity of a functional g(·) means that for any ς > 0, there exist a δ > 0 such that if h(r) and k(r) are any continuous bounded functions on [0, 1], h : [0, 1] → R 1 and k : [0, 1] → R 1 , such that |h(r) − k(r)| < δ for all r ∈ [0, 1], then |g(h(·)) − g(k(·))| < ς. 8