we say that the sequence of probability measure induced by ST(JT_I weakly converge to the probability measure induced by S(), denoted by Sr(=>SO if all of the following hold: (1). For any finite collection of k particular dates, 0≤1<r2<….<Tk≤1, the sequence of k-dimensional random vector yrta converges in distribution he vector y, wh ere Sr(ry S( S N/2 S(rk) (2). For each E>0, the probability that Sr(ri) differs from Sr(r2) for any dates rI and r2 within 8 of each other goes to zero uniformly in T as 8-0 (3).P{|Sr(0)|>A→0 uniformly in T as A→o This definition applies to sequences of continuous functions, though the func- tion in(9)is a discontinues step function. Fortunately, the discontinuities occur at a countable set of points. Formally, Sr( can be replaced with a similar con- tinuous function, interpolating between the steps The Function Central Limit Theorem(FCLT) provides conditions under which converges to the standard Wiener process, W. The simplest FCLT is a gen- eralization of the Lindeberg-levy clt, known as Donsker's theorem Theorem:(Donsker) Let Et be a sequence of i i d. random scalars with mean zero. If a= Var(Et)< oo, ≠0, then w Because pointwise convergence in distribution Wr( r)-w(, r)for each rE0, 1 is necessary(but not sufficient) for weak convergence Wr=w, the Lindeberg- Levy CLT(Wr(, 1)-w(, 1) follows immediately from Donsker's m is strictly stronger than both use identical assumptions, but Donsker's theorem delivers a much strongerwe say that the sequence of probability measure induced by {ST (·)} ∞ T =1 weakly converge to the probability measure induced by S(·), denoted by ST (·) =⇒ S(·) if all of the following hold: (1). For any finite collection of k particular dates, 0 ≤ r1 < r2 < ... < rk ≤ 1, the sequence of k-dimensional random vector {yT } ∞ T =1 converges in distribution to the vector y, where yT ≡         ST (r1) ST (r2) . . . ST (rk)         y ≡         S(r1) S(r2) . . . S(rk)         ; (2). For each ǫ > 0, the probability that ST (r1) differs from ST (r2) for any dates r1 and r2 within δ of each other goes to zero uniformly in T as δ → 0; (3). P{|ST (0)| > λ} → 0 uniformly in T as λ → ∞. This definition applies to sequences of continuous functions, though the func￾tion in (9) is a discontinues step function. Fortunately, the discountinuities occur at a countable set of points. Formally, ST (·) can be replaced with a similar con￾tinuous function, interpolating between the steps. The Function Central Limit Theorem (FCLT) provides conditions under which WT converges to the standard Wiener process, W. The simplest FCLT is a gen￾eralization of the Lindeberg-L´evy CLT, known as Donsker’s theorem. Theorem: (Donsker) Let εt be a sequence of i.i.d. random scalars with mean zero. If σ 2 ≡ V ar(εt) < ∞, σ 2 6= 0, then WT =⇒ W. Because pointwise convergence in distribution WT (·,r) L−→ W(·,r) for each r ∈ [0, 1] is necessary (but not sufficient) for weak convergence WT =⇒ W, the Lindeberg-L´evy CLT (WT (·, 1) L−→ W(·, 1)) follows immediately from Donsker’s theorem. Donsker’s theorem is strictly stronger than Lindeberg-L´evy however, as both use identical assumptions, but Donsker’s theorem delivers a much stronger 7