where 0<r< l and Tr* denotes the largest integer that is less than or equal Applying the same rescaling, we define Wr()≡T-1/Yr(r)/a (9) Now Trl W()=T1()()∑e/ and for a given r, the term in the brackets again obeys the CLT and converges in distribution to N(0, 1), whereas T-12([Tr]*1/2 converges to r1/2. It follows from standard arguments that Wr(r) converges in distribution to N(O, r) We have written Wr(r) so that it is clear that Wr can be considered to be a function of r. Also, because Wr(r) depends on the E s, it is random. There- fore, we can think of Wr(r) as defining a random function of r, which we write Wr(. Just as the CLT provides conditions ensuring that the rescaled random walk T-1/Yr/o(which we can now write as Wr(1) converges, as T become large, to a well-defined limiting random variables(the standard normal), the function central limit theorem(FCLt) provides conditions ensuring that the random function Wr( converge, as T become large, to a well-defined limit ran- dom function, say W(. The word "Functional"in Functional Central Limit theorem appears because this limit is a function of r Some further properties of random walk, suitably rescaled, are in the follow P If Yt is a random walk, then Yta -Yis is independent of Yt2 -Yt for all ti <t2< t3 <t4. Consequently, W(ra)-Wr(r3) is independent of Wi(r2)-Wr(ri) for all T·r=t1,i=1where 0 ≤ r ≤ 1 and [Tr] ∗ denotes the largest integer that is less than or equal to Tr. Applying the same rescaling, we define WT (r) ≡ T −1/2YT (r)/σ (9) = T −1/2 [T r] X∗ t=1 εt/σ. (10) Now WT (r) = T −1/2 ([Tr] ∗ ) 1/2    ([Tr] ∗ ) −1/2 [T r] X∗ t=1 εt/σ    , and for a given r, the term in the brackets {·} again obeys the CLT and converges in distribution to N(0, 1), whereas T −1/2 ([Tr] ∗ ) 1/2 converges to r 1/2 . It follows from standard arguments that WT (r) converges in distribution to N(0,r). We have written WT (r) so that it is clear that WT can be considered to be a function of r. Also, because WT (r) depends on the ε ′ t s, it is random. There￾fore, we can think of WT (r) as defining a random function of r, which we write WT (·). Just as the CLT provides conditions ensuring that the rescaled random walk T −1/2YT /σ (which we can now write as WT (1)) converges, as T become large, to a well-defined limiting random variables (the standard normal), the function central limit theorem (FCLT) provides conditions ensuring that the random function WT (·) converge, as T become large, to a well-defined limit ran￾dom function, say W(·). The word ”Functional” in Functional Central Limit theorem appears because this limit is a function of r. Some further properties of random walk, suitably rescaled, are in the follow￾ing. Proposition: If Yt is a random walk, then Yt4 − Yt3 is independent of Yt2 − Yt1 for all t1 < t2 < t3 < t4. Consequently, Wt(r4) − WT (r3) is independent of Wt(r2) − WT (r1) for all [T · ri ] ∗ = ti ,i = 1,..., 4. 4