A Weak Law of Large Numbers Under Weak Mixing

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1 A Weak Law of Large Numbers Uder Weak Mixig Bruce E. Hase Uiversity of Wiscosi Jauary 209 Abstract This paper presets a ew weak law of large umbers (WLLN) for heterogeous depedet processes ad arrays. The depedece requiremets are otably weaker tha the best available curret results (due to Adrews (988)). Specifically, we show that the WLLN holds whe the process is weak mixig, oly requirig that the mixig coeffi ciets Cesàro sum to zero. This is weaker tha the covetioal assumptio of strog mixig. Research supported by the Natioal Sciece Foudatio ad the Phipps Chair. Departmet of Ecoomics, 80 Observatory Drive, Uiversity of Wiscosi, Madiso, WI

2 Itroductio Oe of the foudatios for asymptotic iferece is the weak law of large umbers (WLLN). For depedet ad strictly statioary time series, the most flexible ad powerful result is the Ergodic Theorem, which states that sample meas coverge almost surely to the populatio mea uder the miimal coditio that the process is ergodic. The latter oly requires that separated evets are o average asymptotically idepedet. For may applicatios, however, the assumptio of strict statioarity is too restrictive. It does ot allow for heterogeeous time series, or allow for radom array structures. I such settigs the best available depedece coditios for the WLLN are due to Adrews (988), who showed that uiform itegrability plus strog mixig are suffi ciet for the WLLN. His result is particularly powerful as it does ot require ay rate of covergece for the mixig coeffi ciets. A limitatio with Adrews WLLN is that strog mixig may be uecessarily restrictive. I classical ergodic theory, distictios are made betwee ergodic processes, weak mixig processes, ad strog mixig processes. Ergodicity is the weakest requiremet (ad broadest class), strog mixig the strogest requiremet (ad most arrow class). The distictio betwee weak mixig ad strog mixig is that the former requires that the Cesàro sum of mixig coeffi ciets is zero, while strog mixig requires the coeffi ciets to limit to zero. The Cesàro limit requiremet is weaker, as it ca hold eve whe the mixig coeffi ciets do ot coverge to zero. This may seem like a mior differece i practice, but classical ergodic theory has argued that it is a major differece. Specifically, the view is that weak mixig processes are relatively geeric, while strog mixig processes are relatively special. For example, the textbook of Peterse (983, p. 7-72) states that there is a sese i which almost every measure preservig trasformatio is weakly mixig but ot strog mixig... Halmos (944) proved that with respect to the weak topology, the set of weakly mixig measure preservig trasformatio systems is residual (i.e. the complemet of a first category set); while Rokhli (948) showed that with respect to the weak topology the set of all strogly mixig trasformatios is of the first category. Thus, i this particular sese, the geeric measure preservig trasformatio is weakly mixig but ot strogly mixig. (emphasis as quoted). For examples of processes which are weak mixig but ot strog mixig, see Sectio 4.5 of Peterse (983), which describes the examples of Kakutai (973) ad Chaco (969). Other examples are provided by Maruyama (949) ad Katok ad Stepi (967). By showig that the WLLN holds for heterogeous weak mixig processes ad arrays, our result brigs the theory for heterogeeous depedet processes closer to the classical Ergodic Theorem. Oe of the iterestig features of our result is that the proof is elemetary. It uses the stadard represetatio of the variace of the trimmed mea as the weighted Cesàro sum of the covariaces, ad bouds the latter usig the mixig iequality for bouded radom variables. The deviatio of the mea from the trimmed mea is bouded covetioally. This proof method is otably differet from that of Adrews (988) who approximated the sample mea by the sum of M martigale differece meas, bouded the latter usig momet bouds for martigale differece

3 sequeces, ad bouded the deviatio by a mixigale iequality. I the ecoometrics literature, Adrews WLLN has bee geeralized to allow for tredig momets by Davidso (993) ad De Jog (995, 998). These papers do ot weake, however, the depedece coditios. For strog laws of large umbers for depedet heterogeous processes, see Hase (99, 992), De Jog (995, 996), ad Davidso ad De Jog (997). For a textbook treatmet see Davidso (994). I Sectio 2 we review the cocepts of ergodicity ad mixig for statioary processes. I Sectio 3 we discuss mixig for heterogeeous arrays. Sectio 4 presets our WLLN. Sectio 5 presets the proof of the result. 2 Ergodicity ad Mixig Let (Ω, F, P ) deote a probability space. A stochastic process {X t R : < t < } is a measurable mappig from Ω to R. Lettig ω Ω, we ca write X t (ω) to idicate that the process depeds o the elemet ω. We ca the defie the shift trasformatio T by X t+ (ω) = X t (T ω). A evet E F is ivariat if E = T E. The process X t is called ergodic if all ivariat evets have probability either 0 or. Ituitively, a ergodic process visits all parts of the probability space, ad ever gets stuck i a subspace. There are several equivalet ways to characterize a ergodic process. Oe is that X t is ergodic if ad oly if for all A, B F, lim ( P (T m A B) P (A)P (B) ) = 0. m= See, for example, Theorem.4 of Billigsley (965) or Corollary.4.2 of Walters (982). This meas, ituitively, that the traslatio T m A becomes idepedet of B o average. For some purposes it is desirable to work with somewhat stroger characterizatios of asymptotic idepedece. The process X t is called weakly mixig if for all A, B F, lim ad is called strogly mixig if lim m m= P (T m A B) P (A)P (B) = 0 P (T m A B) P (A)P (B) = 0. Weak mixig also ca be iterpreted as statig that T m A becomes idepedet of B provided we eglect a few istaces. Strog mixig ca be iterpreted as statig that T m A is asymptotically idepedet of B. From these expressios it is evidet that ergodicity implies weak mixig, ad weak mixig implies strog mixig. It is kow that this estig is strict, as there are examples of ergodic 2

4 trasformatios which are ot weak mixig, ad weak mixig trasformatios which are ot strog mixig. Strog mixig requires that the probabilities P (T m A B) P (A)P (B) limit to zero, but this is ot required by the Cesàro summability of weak mixig. For cocrete examples of trasformatios which are weak but ot strog mixig see the refereces i the itroductio. 3 Mixig for Heterogeous Arrays I ecoometrics we are frequetly iterested i heterogeous stochastic processes ad arrays (idexed by sample size ). For this purpose the most commoly used depedece tool are mixig coeffi ciets. For a radom array {X t : t =,..., } the mixig coeffi cets are defied as α (m) = sup sup <t< A F,t,B F t+m, P (A B) P (A)P (B) where F,t = σ (..., X,t, X,t ) ad F t+m, = σ (X,t+m, X,t+m+,...). The latter are σ-fields geerated by the past ad future values of the stochastic process, respectively, separated by m time periods. The mixig coeffi ciets α (m) measure the serial depedece as the degree of separatio is icreased. For statioary stochastic processes the coeffi ciets do ot deped o. It is stadard to say that the stochastic process X t is strog mixig if sup α (m) 0 as m. This is a aalog of the ergodic theory cocept of strog mixig. We ow itroduce a aalog of the ergodic theory cocept of weak mixig. Defiitio. X t is weak mixig if lim α (m) = 0. () The expressio () states that the Cesàro sum of the mixig coeffi ciets is zero. m= Sice covergece implies Cesàro covergece, weak mixig implies strog mixig. Thus weak mixig is a strictly broader class of stochastic processes tha strog mixig. For example, cosider the mixig coeffi ciet sequece α (m) = ( m = [ m]) = {, 0, 0,, 0, 0, 0, 0,,...}. This does ot have a limit, but its Cesàro sum limits to zero. Hece it is weak mixig but ot strog mixig. For aother example usig arrays, take the process X t = e t + e t q() with e t i.i.d. If q() yet q()/ 0 as the this process is weak mixig but ot strog mixig. 3

5 4 Weak Law of Large Numbers Defie the sample mea Theorem. If X t is weak mixig ad lim sup B X = X t. t= E X t ( X t > B) = 0 (2) t= the ad E X E ( X ) 0 (3) X E ( X ) p 0 (4) as. Theorem shows that the sample mea coverges i L ad coverges i probability. The coditio (2) is a average uiform itegrability coditio. It is implied if X t is uiformly itegrable: lim sup sup E X t ( X t > B) = 0 B t or if X t has a uiformly bouded momet: lim sup sup E X t r < B t for some r >. Theorem geeralizes the WLLN for strog mixig processes of Adrews (988) (his Theorem 2, example 4). Primarily, Theorem relaxes the assumptio of strog mixig to that of weak mixig. Theorem shows that weak mixig is suffi ciet for cosistet estimatio. 5 Proof We show (3). (4) follows by Markov s iequality. Without loss of geerality assume E(X t ) = 0. Fix ε > 0. Pick B large eough such that sup E X t ( X t > B) ε 4 t= (5) 4

6 which is feasible uder (2). Defie W t = X t ( X t B) E (X t ( X t B)) Z t = X t ( X t > B) E (X t ( X t > B)) so that E X = E W + Z E W + E Z. (6) By the triagle iequality ad (5) E Z 2 E Z t t= E X i ( X t > B) t= ε 2. (7) It is useful to observe that W t satisfies the boud W t 2B ad has the same mixig coeffi ciets as X t. By the mixig iequality for bouded radom variables (e.g. Theorem A.5 of Hall ad Heyde (980)), ad the fact that W t are mea zero, By Jese s iequality, (8) ad α (0) /4, E (W t W j ) 6B 2 α ( t j ). (8) ( E W ) 2 E W 2 = 2 2 6B2 2 t= j= t= j= E (W t W j ) E (W t W j ) t= j= α ( t j ) ( = 6B 2 α (0) + 2 m= ( 6B 2 ε m= ) ( m ) α (m) α (m) The fial iequality holds for large eough sice X t is weak mixig. Thus ) 5

7 E W ε 2. (9) Together, (6), (7) ad (9) show that E X ε which establishes (3) as claimed. Refereces [] Adrews, Doald W. K. (988): Laws of large umbers for depedet o-idetically distributed radom variables, Ecoometric Theory, 4, [2] Billigsley, Patrick (965): Ergodic Theory ad Iformatio, Wiley. [3] Chaco, R. V. (969): Weakly mixig trasformatios which are ot strogly mixig, Proceedigs of the America Mathematical Society, 22, [4] Davidso, James (993) A L-covergece theorem for heterogeeous mixigale arrays with tredig momets, Statistics ad Probability Letters, 6, [5] Davidso, James (994): Stochastic Limit Theory, Oxford Uiversity Press. [6] Davidso, James ad Robert M. De Jog (997): Strog laws for ear epoch depedet fuctios of mixig processes: a sythesis of ew results, Ecoometric Reviews, 6, [7] De Jog, Robert M. (995): Laws of large umbers for depedet heterogeeous processes, Ecoometric Theory,, [8] De Jog, Robert M. (996): A strog law of large umbers for triagular mixigale arrays, Statistics ad Probability Letters, 27, -9. [9] De Jog, Robert M. (998): Weak laws of large umbers for mixigales, Aales d Ecoomie et de Statistiques, 5, [0] Hall, Peter ad C. C. Heyde (980): Martigale Limit Theory ad Its Applicatio, Academic Press. [] Hase, Bruce E. (99): Strog laws for depedet heterogeeous processes, Ecoometric Theory, 7, [2] Hase, Bruce E. (992): Erratum, Ecoometric Theory, 8, [3] Halmos, Paul R. (944): I geeral a measure preservig trasformatio is mixig, Aals of Mathematics, 45, [4] Kakutai, Shizuo (973): Examples of ergodic measure preservig trasformatios which are weakly mixig but ot strogly mixig, Recet Advaces i Topological Dyamics, Lecture Notes i Mathematics 38, Spriger-Verlag,

8 [5] Katok, A. B. ad Stepi, Aatolii M. (967): Approximatios i ergodic theory, Russia Mathematical Surveys, 22, [6] Maruyama, Gisiro (949): The harmoic aalysis of statioary stochastic processes, Memoirs of the Faculty of Sciece Kyushu Uiversity Series A, 4, [7] Peterse, Karl (983): Ergodic Theory, Cambridge Uiversity Press. [8] Rokhli, Vladimir Abramovich (948): A geeral measure-preservig trasformatio is ot mixig, Doklady Akademii Nauk SSSR, 60, [9] Walters, Peter (982): A Itroductio to Ergodic Theory, Spriger-Verlag. 7

Lecture 19: Convergence

Lecture 19: Convergence Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may