The Martingale Central Limit Theorem

Transcription

1 The Martngale Central Lmt Theorem Steven P. Lalley Unversty of Chcago May 27, Lndeberg s Method One of the most useful generalzatons of the central lmt theorem s the martngale central lmt theorem of Paul Lévy. Lévy was n part nspred by Lndeberg s treatment of the central lmt theorem for sums of ndependent but not necessarly dentcally dstrbuted random varables. Lndeberg formulated what, n retrospect, s the rght hypothess, now known as the Lndeberg condton, 1 on the summands for the central lmt theorem, and n addton he proposed a new approach to provng central lmt theorems. The Lndeberg condton plays a central role n the most general form of the martngale central lmt theorem, as well, and as Lévy showed, the Lndeberg method of proof can be adapted to martngales. In ths secton I wll show how Lndeberg s method works n the very smplest context, for sums of ndependent, dentcally dstrbuted random varables. In secton 3, I wll show how the method can be generalzed to martngales. Theorem 1. Central Lmt Theorem Let ξ 1,ξ 2,... be ndependent, dentcally dstrbuted random varables wth mean zero and varance 1. Then for every contnuous, bounded functon f : R R, lm E f where Z s a standard normal random varable. 1 n ξ = E f Z 1 n Proof. It suffces to prove ths for C functons f wth compact support, by a standard approxmaton argument, so I wll assume henceforth that f s such a functon. Lndeberg s method depends on the fact that the famly of normal denstes s closed under convolutons, n partcular, f X and Y are ndependent Gaussan random varables then X + Y s also Gaussan. Consequently, f ζ 1,ζ 2,...,ζ n are ndependent standard normal random varables then Z = D 1 n ζ. n Wthout loss of generalty we may assume that the underlyng probablty space s large enough to support not only the random varables ξ but also an ndependent sequence of..d. standard Gaussan random 1 Feller, and ndependently Lévy, later proved that Lndeberg s condton s n some sense necessary for the valdty of the central lmt theorem. 1

2 varables ζ. The objectve s to show that as n, E f For notatonal ease set 1 n 1 n ξ ζ 0. 2 n n ξ = ξ / n = ζ / n; and then relaton 2 can be re-stated as E f n ξ n 0. Lndeberg s strategy for provng 2 s to replace the summands ξ n the frst expectaton by the correspondng Gaussan summands ζ, one by one, and to bound at each step the change n the expectaton resultng from the replacement of ξ by ζ : n E f ξ n n k E f ξ + k=1 n k 1 ξ + n =k 3 Snce the ndvdual terms ξ and ζ account for only a small fracton of the sums, the dfferences n the value of f can be approxmated by usng two-term Taylor seres approxmatons. Furthermore, snce f has compact support, the dervatves are unformly contnuous, and so the remander terms can be estmated unformly. In partcular, for any ε > 0 there exst δ > 0 and C < such that for any x, y R, Consequently, for each k, k E f ξ + where n f x + y f x f xy f xy 2 /2 εy 2 f y δ and f x + y f x f xy f xy 2 /2 C y 2 otherwse. 4 k 1 n = E f ξ + k 1 ξ + n =k ξ k ζ k + 1 k 1 n 2 E f ξ + ξ k 2 ζ k 2 + ER k A + ER k B 5 R k A εξ k 2 +C ξ k 2 1{ ξ k δ} R k B εζ k 2 +C ζ k 2 1{ ζ k δ}. and The crucal feature of the expanson 5 s the ndependence of the ndvdual terms ξ and ζ ; ths guarantees that the frst two expectatons on the rght sde of 5 splt as products of expectatons, and snce 2

3 ξ k and ζ have the same mean and varance, t follows that the frst two expectatons on the rght sde are k 0. Consequently, for each k, k E f ξ + n k 1 ξ + n =k E R k A + E R k B εeξ k 2 + εeζ k 2 +C Eξ k 2 1{ ξ k δ} +C Eζ k 2 1{ ζ k δ} n 1 εeξ k 2 + Eζ k 2 + n 1 C Eξ k 2 1{ ξ k nδ} + n 1 C Eζ k 2 1{ ζ k nδ} 2εn 1 + n 1 C Eξ k 2 1{ ξ k nδ} + n 1 C Eζ k 2 1{ ζ k nδ}. Substtutng ths bound n nequalty 3 now yelds E f 1 n 1 n ξ ζ 2ε +C Eξ 1 2 1{ ξ 1 nδ} +C Eζ 1 2 1{ ζ 1 nδ}. n n Snce Eξ 2 1 = 1 < and Eζ2 1 = 1 <, the domnated convergence theorem mples that the last two expectatons converge to zero as n, and so lm sup E f 1 n 1 n ξ ζ 2ε. n n Fnally, snce ε > 0 s arbtrary, the convergence 2 must hold. 2 The Lndeberg Condton Lndeberg s second nsght was that a smlar parng of non-gaussan, mean-zero random varables ξ wth Gaussan, mean-zero random varables ζ of the same varance could be carred out even when the random varables ξ are not dentcally dstrbuted, because sums of ndependent Gaussan random varables are stll Gaussan, even f the summands have dfferent varances. When varances are matched, so that Eξ 2 = Eζ2, most of the proof gven above carres through drectly: n partcular, the frst two terms n the Taylor seres would after takng expectatons cancel, leavng only the remander terms ER k A and ER k B. Thus, the real ssue n generalzng the central lmt theorem s to formulate a hypothess that wll guarantee that the sum of the expectatons ER k A and ER k B wll be small. Trangular Arrays: A trangular array s a doubly-ndexed famly {ξ n, } n 1,1 mn of random varables. Lndeberg s Condtons: A trangular array {ξ n, } n 1,1 mn of ndependent random varables satsfes Lndeberg s condtons f A Eξ n, = 0 for all n,. B mn Eξ 2 n, = 1. C For every δ > 0, mn lm Eξ 2 n, 1{ ξ n, δ} =

4 Theorem 2. Lndeberg s Central Lmt Theorem If {ξ n, } s a trangular array that satsfes Lndeberg s condtons, then as n mn ξ n, D Normal0,1. 7 The proof s very nearly dentcal to Lndeberg s proof of the central lmt theorem. As an exercse, you should fll n the detals. 3 Martngale Central Lmt Theorem Independence s used n the proof of the central lmt theorem and of Lndeberg s generalzaton to trangular arrays of ndependent random varables only n the evaluaton of the frst two expectatons on the rght sde of equaton 5. These nvolve only the frst two condtonal moments of the random varables ξ gven the σ algebra generated by ξ 1,ξ 2,...,ξ 1. Lévy realzed that ndependence s more than s needed for ths purpose: n fact, only the martngale property s needed. Assume now that each row of the trangular array {ξ n, } mn s a martngale dfference sequence, that s, for each row n there s a fltraton {F n, } 0 mn such that the sequence {ξ n, } mn s adapted to the fltraton and Eξ n, F n, 1 = 0. 8 Wrte S n,k = k k ξ n, and V 2 n,k = Eξ 2 n, F n, 1. 9 Theorem 3. P. Lévy Assume n addton to 8 that the sum of the condtonal varances n each row s 1, that s, V 2 n,mn = 1, and assume that the trangular array {ξ n, } mn satsfes the Lndeberg condton, that s, for every δ > 0, Then as n, mn lm Eξ 2 n, 1{ ξ n, δ} = S n,mn D Normal0,1. 11 REMARK. There are many varants of ths theorem. In one of the more useful of these, The Lndeberg condton s replaced by the hypothess mn lm Eξ 2 n, 1{ ξ n, δ} F n, 1 = 0 n probablty. 12 See the book by HEYDE & HALL for a more detaled dscusson. In many applcatons the rather strong hypothess that V n,mn = 1 s not satsfed. For ths reason, the followng varant of Lévy s theorem whch we wll not prove s often cted. Theorem 4. Martngale Central Lmt Theorem Assume n addton to 8 that 12 holds, and that V n,mn P 1 as n. Then S n,mn D Normal0,

5 Proof. As n the proof of the central lmt theorem, t suffces to prove that for every C functon f, lm E f S n,mn = E f Z 14 where Z s standard normal. The strategy wll be the same as n Lndeberg s proof of the central lmt theorem: we wll match the martngale dfferences ξ n, wth ndependent, mean-zero, normal random varables n such a way that the condtonal varances gven F n, 1 agree. Assume that the probablty space s large enough to support all of the random varables ξ n, and n addton ndependent standard normal random varables Z n,. Set Then E f mn mn ξ n, = σ n, Z n, where σ 2 n, = Eξ2 n, F n, 1. mn k=1 k E f ξ n, + mn k 1 mn ξ n, To bound the terms on the rght sde of 14, we wll use Taylor s theorem 4 n much the same way as earler: k mn k 1 mn f ξ n, + f ξ n, + =k k 1 = f mn ξ n, + ξ k ζ k + 1 k 1 2 f mn ξ n, + ξ 2 k ζ2 k where =k + R n,k A + R n,k B 16 R n,k A εξ n,k 2 +C ξ n,k 2 1{ ξ n,k δ} R n,k B εζ n,k 2 +C ζ n,k 2 1{ ζ n,k δ}. and Our next objectve s to show that the expectatons of the frst two terms on the rght sde of equaton 15 both vansh. In Lndeberg s proof of the central lmt theorem the correspondng step was trval, because the random varables were all ndependent, but here we must take care n sortng out the dependence of the varous terms n the expectatons. By constructon, = σ n, Z n, where the random varables Z n, are standard normal and ndependent of F n,mn. Consequently, the condtonal dstrbuton of the random varable mn Z n,k := 1 V 2 n,k gven F n,mn s the standard normal dstrbuton, and so n partcular Z n,k s ndependent of F n,mn. Moreover, snce the random varables Z n, are condtonally ndependent gven F n,mn, the random varable Z n,k s condtonally ndependent of ζ n,k = σ n,k Z n,k. Hence, for any bounded, Borel measurable functon g and any nonnegatve Borel measurable h, E g k 1 ξ n, + mn hζ n,k F n,mn = k 1 g ξ n, + 1 V 2 n,k z 1 hσ n,k z 2 ϕz 1 ϕz 2 d z 1 d z 2, 5

6 where ϕz s the standard normal probablty densty functon. Applyng ths wth h = 1 and g = f, and usng the fact that V 2 n,k s measurable wth respect to F n,k 1, gves k 1 E f ξ n, + mn ξ k = E = = 0, and then wth h + x = x 0 and h = x 0, k 1 E f ξ n, + mn ζ k = E k 1 f ξ n, + 1 V ξ 2 n,k z k ϕzd z k 1 Ef ξ n, + 1 V Eξ 2 n,k z k,f n,k 1 ϕzd z k 1 f ξ n, + 1 V 2 n,k z 1 Smlar calculatons exercse: fll n the detals wth g = f show that k 1 E f ξ n, + mn Thus, equaton 15 smplfes to and so by 14, k E f ξ n, + E f ξ 2 k ζ2 k = E mn k 1 f ξ n, + 1 V 2 n,k z 1 σ 2 n,k 1 z 2 ϕz 1 ϕz 2 d z 1 d z 2 = 0. k 1 mn ξ n, + = ER n,k A + ER n,k B, =k mn mn mn ξ n, k=1 ER n,k A + ER n,k B. z 2 2 ϕz 2d z 2 = 0. The Lndeberg condton 10 mples that for any δ > 0 the lmsup of the rght sde s 2δ. Snce δ > 0 s arbtrary, the result 13 follows. 6