Chapter 7 Mixigales I this sectio we prove some of the results stated i the previous sectios usig mixigales. We first defie a mixigale, otig that the defiitio we give is ot the most geeral defiitio. Defiitio 7.0. Mixigale) Let F t σx t, X t,...), {X t } is called a mixigale if it satisfies ρ t,k { 2 } /2 E EX t F t k ) EX t )), where ρ t,k 0 as k. We ote if {X t } is a statioary process the ρ t,k ρ k. Lemma 7.0.3 Suppose {X t } is a mixigale. he {X t } almost surely satisfies the decompositio { } X t EX t F t j ) EX t F t j ). 7.) j0 PROOF. We first ote that by usig a telescopig argumet that X t EX t ) m { EXt F t k ) EX t F t k ) } + { EX t F t m ) EX t ) }. By defiitio of a martigale E EX t F t m ) EX t ) ) 2 0 as k, hece the remaider term i the above expasio becomes egligable as m ad we have almost surely X t EX t ) { EXt F t k ) EX t F t k ) }. hus givig the required result. We observe that 7.) resembles the Wold decompositio. he differece is that the Wolds decompositio decomposes a statioary process ito elemets which are the errors i the best 74
liear predictors. Whereas the result above decomposes a process ito sums of martigale differeces. It ca be show that fuctios of several ARCH-type processes are mixigales where ρ t,k Kρ k rho < )), ad Subba Rao 2006) ad Dahlhaus ad Subba Rao 2007) used these properties to obtai the rate of covergece for various types of ARCH parameter estimators. I a series of papers, Wei Biao Wu cosidered properties of a geeral class of statioary processes which satisfied Defiitio 7.0., where k ρ k <. I Sectio 7.2 we use the mixigale property to prove heorem 6..3. his is a simple illustratio of how useful mixigales ca be. I the followig sectio we give a result o the rate of covergece of some radom variables. 7. Obtaiig almost sure rates of covergece for some sums he followig lemma is a simple variat o a result proved i Móricz 976), heorem 6. Lemma 7.. Let {S } be a radom sequece where E t S t 2 ) φ) ad {phit)} is a mootoically icreasig sequece where φ2 j )/φ2 j ) K < for all j. he we have almost surely S O φ)log )log log ) +δ ). PROOF. he idea behid the proof is to that we fid a subsequece of the atural umbers ad defie a radom variables o this subsequece. his radom variable, should domiate i some sese) S. We the obtai a rate of covergece for the subsequece you will see that for the subsequece its quite easy by usig the Borel-Catelli lemma), which, due to the domiace, ca be trasfered over to S. We make this argumet precise below. Defie the sequece V j t 2 j S t. Usig Chebyshev s iequality we have PV j > ε) φ2j ). ε Let εt) φt)log log t) +δ log t. It is clear that PV j > ε2 j )) j j Cφ2 j ) φ2 j )log j) +δ j <, where C is a fiite costat. Now by Borel Catelli, this meas that almost surely V j ε2 j ). Let us ow retur to the orgial sequece S. Suppose 2 j 2 j, the by defiitio of V j we have S ε) V j a.s ε2 j ε2j ) ) ε2 j ) < uder the stated assumptios. herefore almost surely we have S Oε)), which gives us the required result. 75
We observe that the above result resembles the law of iterated logarithms. he above result is very simple ad ice way of obtaiig a almost sure rate of covergece. he mai problem is obtaiig bouds for E t S t 2 ). here is o exceptio to this, whe S t is the sum of martigale differeces the oe ca simply apply Doob s iequality, where E t S t 2 ) E S 2 ). I the case that S is ot the sum of martigale differeces the its ot so straightforward. However if we ca show that S is the sum of mixigales the with some modificatios a boud for E t S t 2 ) ca be obtaied. We will use this result i the sectio below. 7.2 Proof of heorem 6..3 We summarise heorem 6..3 below. heorem Let us pose that {X t } has a ARMA represetatio where the roots of the characteristic polyomials φz) ad θz) lie are greater tha + δ. he i) ii) for ay γ > 0. tr+ tmaxi,j) log log ) ε t X t r O +γ log ) 7.2) log log ) X t i X t j O +γ log ). 7.3) By usig Lemma??, ad that tr+ ε tx t r is the sum of martigale differeces, we prove heorem 6..3i) below. PROOF of heorem 6..3. We first observe that {ε t X t r } are martigale differeces, hece we ca use Doob s iequality to give E r+ s s tr+ ε tx t r ) 2 ) r)eε 2 t)ext 2 ). Now we ca apply Lemma?? to obtai the result. We ow show that tmaxi,j) log log ) X t i X t j O +δ log ). However the proof is more complex, sice {X t i X t j } are ot martigale differeces ad we caot directly use Doob s iequality. However by showig that {X t i X t j } is a mixigale we ca still show the result. o prove the result let F t σx t, X t,...) ad G t σx t i X t j, X t i X t j i,...). We observe that if i > j, the G t F t i. 76
Lemma 7.2. Let F t σx t, X t,...) ad pose X t comes from a ARMA process, where the roots are greater tha + δ. he if Eε 4 t) < we have E EX t i X t j F t mii,j) k ) EX t i X t j ) ) 2 Cρ k. PROOF. By expadig X t as a MA ) process we have EX t i X t j F t mii,j) k ) EX t i X t j ) { a j a j2 Eεt i j ε t j j2 F t k mii,j) ) Eε t i j ε t j j2 ) }. j,j 2 0 Now i the case that t i j > t k mii, j) ad t j j 2 > t k mii, j), Eε t i j ε t j j2 F t k mii,j) ) Eε t i j ε t j j2 ). Now by cosiderig whe t i j t k mii, j) or t j j 2 t k mii, j) we have have the result. Lemma 7.2.2 Suppose {X t } comes from a ARMA process. he i) he sequece {X t i X t j } t satisfies the mixigale property E EX t i X t j F t mii,j) k ) EX t i X t j F t k ) ) 2 Kρ k, 7.4) ad almost surely we ca write X t i X t j as X t i X t j EX t i X t j ) tmii,j) 7.5) where EX t i X t j F t k mii,j) ) EX t i X t j F t k mii,j) ), are martigale differeces. ii) Furthermore EV 2 t,k ) Kρk ad mii,j) s tmii,j) where K is some fiite costat. {X t i X t j EX t i X t j )}) 2} K, 7.6) PROOF. o prove i) we ote that by usig Lemma 7.2. we have 7.4). o prove 7.5) we use the same telescopig argumet used to prove Lemma 7.0.3. o prove ii) we use the above expasio to give {X t i X t j EX t i X t j )}) 2} 7.7) mii,j) s mii,j) s tmii,j) tmii,j) k 0 k 2 0 mii,j) s { E mii,j) s ) 2 } tmii,j) tmii,j) 77 tmii,j) )} /2 ) 2 2 2 }
Now we see that { } t {EX t i X t j F t k mii,j) ) EX t i X t j F t k mii,j) )} t, therefore { } t are also martigale differeces. Hece we ca apply Doob s iequality to s mii,j) s tmii,j) V ) t,k ad by usig 7.4) we have mii,j) s tmii,j) ) 2 } E herefore ow by usig 7.7) we have mii,j) s tmii,j) tmii,j) ) 2 tmii,j) {X t i X t j EX t i X t j )}) 2} K. EV 2 t,k ) K ρk. hus givig 7.6). We ow use the above to prove heorem 6..3ii). PROOF of heorem 6..3ii). o prove the result we use 7.6) ad Lemma 7... 78