Dog et al. Joural of Iequalities ad Applicatios (208) 208:292 https://doi.org/0.86/s3660-08-884-7 R E S E A R C H Ope Access Some it properties for a hidde ihomogeeous Markov chai Yu Dog *, Fag-qig Dig 2 ad Qi-feg Yao 3 * Correspodece: 405388629@qq.com School of Mathematics, Maasha Teachers College, Maasha, Chia Full list of author iformatio is available at the ed of the article Abstract This paper presets a geeral strog it theorem for delayed sum of fuctios of radom variables for a hidde time ihomogeeous Markov chai (HTIMC), ad as corollaries, some strog laws of large umbers for HTIMC are established thereby. MSC: 60F5; 94A7 Keywords: Ihomogeeous hidde Markov chai; Delayed sum; Law of large umbers Itroductio Hidde Markov chai is a importat brach of Markov chai theory. A classical hidde Markov model was first itroduced by Baum ad Petrie []. It provides a flexible model that is very useful i differet areas of applied probability ad statistics. Examples are foud i machie recogitio, like speech ad optical character recogitio, ad bioiformatics. The power of these models is that they ca be very efficietly implemeted ad simulated. I recet years, may ew theories were itroduced ito hidde time ihomogeeous Markov chai (HTIMC) theory. G.Q. Yag et al. [2]gavealawoflargeumbers for coutable hidde time ihomogeeous Markov models. I additio, delayed sums of radom variables were first discussed by Zygmud [3]. Gut ad Stradtmüller [4] studied the strog law of large umbers for delayed sums of radom fields. Wag ad Yag [5] studied the geeralized etropy ergodic theorem with a.e. ad L covergece for time ihomogeeous Markov chais. Wag [6, 7] discussed the it theorems of delayed sums for row-wise coditioally idepedet stochastic arrays ad a class of asymptotic properties of movig averages for Markov chais i Markovia eviromets. I the classical studies there are two simplest models for predictig: the mea model ad the radom walk model [8].Thesetwomodelsuseallthehistoricaliformatio.Butwe ofte ecouter time series that appear to be locally statioary, so we ca take a average of what has happeed i some widow of the recet past. Based o this idea ad the above researches, the mai focus of this paper is to obtai a geeral strog it theorem of delayed sums of fuctios of radom variables for a HTIMC, ad as corollaries, some strog laws of large umbers for HTIMC are established thereby. The remaider of this paper is orgaized as follows: Sect. 2 gives a brief descriptio of the HTIMC ad related lemmas. Sectio 3 presets the mai results ad the proofs. The Author(s) 208. This article is distributed uder the terms of the Creative Commos Attributio 4.0 Iteratioal Licese (http://creativecommos.org/liceses/by/4.0/), which permits urestricted use, distributio, ad reproductio i ay medium, provided you give appropriate credit to the origial author(s) ad the source, provide a lik to the Creative Commos licese, ad idicate if chages were made.
Dogetal. Joural of Iequalities ad Applicatios (208) 208:292 Page 2 of 9 2 Preiaries I this sectio we list some fudametal defiitios ad related results that are eeded i the ext sectio. Let (, F, P) be the uderlig probability space ad ζ =(ξ, η) a radom vector, where ξ =(ξ 0, ξ,...) ad η =(η 0, η,...) are two differet stochastic processes, η is hidde (η takesvaluesisety = ω 0, ω,...,ω b })adξ is observable (ξ takesvaluesiset X = θ 0, θ,...,θ d }). We first recall the defiitio of a hidde time ihomogeeous Markov chai (HTIMC) ζ =(ξ, η)=ξ, η } =0 with hidde chai η } =0 ad observable process ξ } =0. Defiitio The process ζ =(ξ, η) is called a HTIMC if it follows the followig form ad coditios:. Suppose that a give time ihomogeeous Markov chai takes values i state space Y, its startig distributio is ( q(ω0 ), q(ω );...;q(ω b ) ), q(ω i )>0,ω i Y, (2.) ad trasitio matrices are Q k = ( q k (ω j ω i ) ), q k (ω j ω i )>0,ω i, ω j Y, k, (2.2) where q k (ω j ω i )=P(η k = ω j η k = ω i ), k. 2. For ay positive iteger, P(ξ 0 = x 0,...,ξ = x η)= P(ξ k = x k η k ) a.s. (2.3) k=0 Some ecessary ad sufficiet coditios for (2.3) have bee give by G.Q.Yag et al. [2]. (a) (2.3) holds if, for ay, P(ξ 0 = x 0,...,ξ = x η 0 = y 0,...,η = y )= P(ξ k = x k η k = y k ) (2.4) holds. (b) ζ =(ξ, η) is a hidde time ihomogeeous Markov chai if ad oly if 0, p(x 0, y 0,...,x, y )=q(y 0 ) q k (y k y k ) p k (x k y k ),. (2.5) k= (c) ζ =(ξ, η) is a hidde time ihomogeeous Markov chai if ad oly if 0, P(η = y ξ 0 = x 0,...,ξ = x, η 0 = y 0,...,η = y ) k=0 = P(η = y η = y ), (2.6) k=0
Dogetal. Joural of Iequalities ad Applicatios (208) 208:292 Page 3 of 9 P(ξ = x ξ 0 = x 0,...,ξ = x, η 0 = y 0,...,η = y ) = P(ξ = x η = y ). (2.7) Let a, } be two sequeces of oegative itegers with covergig to ifiity as.lets a, (θ i, ω j ), W a, (ω i ), T a, (θ i ), θ i X, ω j Y be the umber of ordered couples (θ i, ω j )i(ξ a +, η a +), (ξ a +2, η a +2),...,(ξ a, η a ), with ω i amog η a +, η a +2,...,η a ad θ i amog ξ a +, ξ a +2,...,ξ a,respectively. It is easy to verify that ad S a, (θ i, ω j )= W a, (ω i )= T a, (θ i )= θi }(ξ k ) ωj }(η k ), (2.8) ωi }(η k ), (2.9) θi }(ξ k ), (2.0) where A ( ) deotes the idicator fuctio of set A. Lemma Let ζ =(ξ, η) =(ξ k, η k )} k=0 be a HTIMC which takes values i X Y, let f k (x, y)} k=0 be a sequece of fuctios o X Y, let F m, = σ (ξ m, η m,...,ξ, η ), 0 m Z + }, ad let a, } be a sequece of pairs of positive itegers with = exp[ ε]<, where ε >0is arbitrary. Defie A(α)= ω : sup E [ } fk 2 (ξ k, η k )e α f k(ξ k,η k ) ] F a,k = M(α, ω)< (α > 0). (2.) The fk (ξ k, η k ) E [ ]} f k (ξ k, η k ) F a,k =0 a.s. ω A(α). (2.2) Proof Let λ be a real umber. We first defie Note that t a, (λ, ω)= e λ a+b k=a+ f k(ξ k,η k ) a E[eλf k(ξ k,η k) F a,k ]. (2.3) e λf a+b (ξ a+b,η a+b ) t a, (λ, ω)=t a, (λ, ω) E[e λf a+b (ξ a+b,η a+b ) F a,a ]
Dogetal. Joural of Iequalities ad Applicatios (208) 208:292 Page 4 of 9 ad E [ t a, (λ, ω) ] = E E [ t a, (λ, ω) ] F a,a }. Hece, we have E [ t a, (λ, ω) F a,a ] = ta, (λ, ω) a.s. It is easy to show that E[t a, (λ,ω)]=;. This ad the Markov iequality imply that, for every ε >0, [ ] P log t a,b (λ, ω) ε = P [ t a, (λ, ω) exp(ε) ] exp( ε ). Hece = [ ] P log t a,b (λ, ω) ε exp( ε )<, = which, by the first Borel Catelli Lemma, allows us to coclude that sup log t a, (s, ω)<ε a.s., sice ε is arbitrary, thus sup log t a, (λ, ω) 0 a.s. (2.4) follows sice log 2 = 2 log 0(). We have by Eqs. (2.3)ad(2.4)that sup λ f k (ξ k, η k ) log E [ e λf k(ξ k,η k ) F a,k ] } 0 a.s. (2.5) Takig 0 < λ α, ad dividig both sides of Eq. (2.5)byλ,weget sup a + f k (ξ k, η k ) } log E[e λf k(ξ k,η k ) F a,k ] 0 a.s. (2.6) λ We have by Eq. (2.6) ad iequalities log x x (x >0),0 e x x 2 x2 e x that sup sup sup fk (ξ k, η k ) E [ f k (ξ k, η k ) F a,k ]} log E[e λf k (ξ k,η k ) F a,k ] λ E[e λf k (ξ k,η k ) F a,k ] λ E [ ] } f k (ξ k, η k ) F a,k E [ ] } f k (ξ k, η k ) F a,k
Dogetal. Joural of Iequalities ad Applicatios (208) 208:292 Page 5 of 9 λ 2 sup E [ fk 2 (ξ k, η k )e α f k(ξ k,η k ) ] F a,k = λ M(α, ω) a.s.ω A(α). (2.7) 2 Lettig λ 0 + i Eq. (2.7), we get sup fk (ξ k, η k ) E [ ]} f k (ξ k, η k ) F a,k 0 a.s.ω A(α). (2.8) Takig α < λ 0, similarly, we have if fk (ξ k, η k ) E [ f k (ξ k, η k ) F a,k ]} λ M(α, ω) 2 a.s.ω A(α). Puttig λ 0,wehave if fk (ξ k, η k ) E [ ]} f k (ξ k, η k ) F a,k 0 a.s.ω A(α). (2.9) From Eqs. (2.8) ad(2.9), we obtai fk (ξ k, η k ) E [ ]} f k (ξ k, η k ) F a,k =0 a.s.ω A(α). Thus we complete the proof of Lemma. Lemma 2 Assume that ζ =(ξ, η)=(ξ k, η k )} k=0 is a HTIMC defied as i Lemma. The, for every j < k; k, E [ f k (ξ k, η k ) F j,k ] = E [ fk (ξ k, η k ) η k ] a.s. (2.20) Proof From defiitio of Hidde Markov chai, we have, for every x i X, y j Y, m ;, p(ξ = x, η = y ξ m = x m, η m = y m,...,ξ - = x -, η - = y - ) = p(ξ = x ξ m = x m, η m = y m,...,ξ - = x -, η - = y -, η = y ) p(η = y ξ m = x m, η m = y m,...,ξ - = x -, η - = y -, ξ = x ) = p(ξ = x η = y ) p(η = y η - = y -, ξ = x ) = p(ξ = x, η = y η - = y - ). Hece, we have E [ f k (ξ k, η k ) F j,k ]
Dogetal. Joural of Iequalities ad Applicatios (208) 208:292 Page 6 of 9 = f k (x k, y k )p(ξ k = x k, η k = y k ξ j = x j, η j = y j,...,ξ k = x k, η k = y k ) x k X y k Y = x k X y k Y f k (x k, y k )p(ξ k = x k, η k = y k η k = y k ) = E [ ] f k (ξ k, η k ) η k. AccordigtoTheoremofWag[5], it is easy to verify the followig lemma. Lemma 3 Suppose that η =(η 0, η,...)is a time ihomogeeous Markov chai which takes value i state space Y, its startig distributio is ( q(ω0 ), q(ω );...;q(ω b ) ), q(ω i )>0,ω i Y, (2.2) ad trasitio matrices are Q k = ( q k (ω j ω i ) ), q k (ω j ω i )>0,ω i, ω j Y, k, (2.22) where q k (ω j ω i )=P(η k = ω j η k = ω i ), k. Assume that =(q(ω i, ω j )), q(ω i, ω j )>0,ω i, ω j Y is aother trasitio matrix which satisfies the followig coditio: q k (ω i, ω j ) q(ω i, ω j ) =0 ω i, ω j Y. (2.23) The, for each ω s Y, ωs }(η k )=π s a.s., (2.24) where (π 0, π, π 2,...,π b ) is the statioary distributio determied by. 3 Mai results Theorem Let ζ =(ξ, η)=(ξ k, η k )} k=0 be a HTIMC which takes values i X Y, f (x, y) be a fuctio o X Y. Let =(q(ω i, ω j )), q(ω i, ω j )>0,ω i, ω j Y be aother trasitio matrix ad p(θ i ω j ), (θ i, ω j ) X Y be coditioal probabilities which satisfy q k (ω i, ω j ) q(ω i, ω j ) =0 ω i, ω j Y, (3.) pk (θ i ω j ) p(θ i ω j ) =0 (θi, ω j ) X Y. (3.2)
Dogetal. Joural of Iequalities ad Applicatios (208) 208:292 Page 7 of 9 If the trasitio matrix has a statioary distributio π =(π 0, π, π 2,...,π b ), the f (ξ k, η k )= θ i X a.s. (3.3) Proof Sice f (x, y) is bouded, we have by Lemmas ad 2 that f (ξk, η k ) E [ f (ξ k, η k ) η k ]} =0 a.s. (3.4) Observe that E [ ] f (ξ k, η k ) η k = f (θ i, ω j )q k (η k, ω j )p k (θ i η k ). θ i X ω j Y We have that, by Eq. (3.4), sup sup = sup f (ξ k, η k ) θ i X E [ ] f (ξ k, η k ) η k θ i X ω j Y θ i X = sup θ i X f (θ i, ω j )q k (η k, ω j )p k (θ i η k ) θ i X θ i X = sup ωs }(η k )π s f (θ i, ω j )q k (ω s, ω j )p k (θ i ω s ) θ i X ωs }(η k )f (θ i, ω j ) [( q k (ω s, ω j ) q(ω s, ω j ) ) p k (θ i ω s ) + q(ω s, ω j ) ( p k (θ i ω s ) p(θ i ω s ) ) + p(θ i ω s )q(ω s, ω j ) ] θ i X sup sup θ i X θ i X,ω j Y f (θi, ω j ) qk (ω s, ω j ) q(ω s, ω j )
Dogetal. Joural of Iequalities ad Applicatios (208) 208:292 Page 8 of 9 =0. + p k (θ i ω s ) p(θ i ω s ) } k=a Therefore Eq. (3.3) holds. Corollary Uder the coditios of Theorem, we have for each θ i X, ω j, ω s Y, S a,b (θ i, ω j )= π s q(ω s, ω j )p(θ i ω s ) a.s. (3.5) ω s Y Proof Put f (x, y)= θi }(x) ωj }(y), (θ i, ω j ) X Y i Theorem.The f (ξ k, η k ) θ i X = θ i X θi }(ξ k ) ωj }(η k ) θi }(θ i ) ωj }(ω j )π s q(ω s, ω j )p(θ i ω s ) = S a,b (θ i, ω j ) π s q(ω s, ω j )p(θ i ω s )=0 ω s Y a.s. Corollary 2 Uder the assumptios of Theorem, we have, for each θ i X, ω s Y, T a,b (θ i )= π s p(θ i ω s ) a.s. (3.6) ω s Y Proof Put f (x, y)= θi }(x), (x, y) X Y i Theorem.The f (ξ k, η k ) θ i X = = θi }(ξ k ) θ i X T a,b (θ i ) π s p(θ i ω s )=0 ω s Y θi }(θ i )π s q(ω s, ω j )p(θ i ω s ) a.s. Fudig This research is supported i part by the RP of AHui Provicial Departmet of Educatio (KJ207A85, KJ207A547). Competig iterests The authors declare that they have o competig iterests. Authors cotributios All authors carried out the proof. All authors coceived of the study, ad participated i its desig ad coordiatio. All authors read ad approved the fial mauscript.
Dogetal. Joural of Iequalities ad Applicatios (208) 208:292 Page 9 of 9 Author details School of Mathematics, Maasha Teachers College, Maasha, Chia. 2 Departmet of Mathematics ad Physics, HeFei Uiversity, Ahui, P.R. Chia. 3 School of Mathematics & Physics, AHui Uiversity of Techology, Ma asha, Chia. Publisher s Note Spriger Nature remais eutral with regard to jurisdictioal claims i published maps ad istitutioal affiliatios. Received: March 208 Accepted: 7 October 208 Refereces. Baum, L.E., Petrie, T.: Statistical iferece for probabilistic fuctios of fiite state Markov chais. A. Math. Stat. 37, 554 563 (966) 2. Yag, G.Q., Yag, W.G., Wu, X.T.: The strog laws of large umbers for coutable ohomogeeous hidde Markov models. Commu. Stat., Theory Methods 46(7),8808 889 (207) 3. Zygmud, A.: Trigoometric Series, vol.. Cambridge Uiversity Press, New York (959) 4. Gut, A., Stradtmüller, U.: O the strog law of large umbers for delayed sums ad radom fields. Acta Math. Hug. 29( 2), 82 203 (200) 5. Wag, Z.Z., Yag, W.G.: The geeralized etropy ergodic theorem for ohomogeeous Markov chais. J. Theor. Probab. 29,76 775 (206) 6. Wag, Z.Z.: A kid of asymptotic properties of movig averages for Markov chais i Markovia eviromets. Commu. Stat., Theory Methods 46(22),0926 0942 (207) 7. Wag, Z.Z.: Some it theorems of delayed sums for row-wise coditioally idepedet stochastic arrays. Commu. Stat., Theory Methods 46(),5265 5272 (207) 8. Isaacso, D., Madse, R.: Markov Chais Theory ad Applicatios. Wiley, New York (976)