Binomial approximation to the Markov binomial distribution

Transcription

1 Biomial approximatio to the Markov biomial distributio V. Čekaavičius ad B. Roos Vilius Uiversity ad Uiversity of Hamburg 2d November 2006 Abstract The Markov biomial distributio is approximated by the biomial distributio. Estimates of accuracy are obtaied for the total variatio ad local orms. The results iclude secod-order estimates ad asymptotically sharp costats. Keywords: Markov biomial distributio, biomial approximatio, local orm, total variatio orm. MSC 2000 Subject Classificatio: Primary 60J0; Secodary 60F05. Itroductio ad otatio The Markov biomial distributio is a geeralizatio of the biomial oe. Depedig o the choice of parameters, both distributios ca be close or eve equal. To the best of our kowledge, the closeess of both distributios was ot ivestigated i detail, though there are some geeral results for the biomial approximatio of the sum of depedet variables, see Serflig 975, Soo 996, ad Boutsikas ad Koutras Apparetly, the results of Soo 996 ad Boutsikas ad Koutras 2000 caot be applied to the Markov biomial distributio directly. Note also that umerous papers are devoted to compoud Poisso approximatios of the Markov biomial distributio, see Dobrushi 953, Serflig 975, Wag 98, Serfozo 986, Čekaavičius ad Mikalauskas 999, ad the refereces therei. For papers dealig with related problems, see, for example, Campbell et al. 994, Erhardsso 999, ad Vellaisamy We eed the followig otatio. Let I k deote the distributio cocetrated at a iteger k Z ad set I = I 0. Throughout this paper, we use the abbreviatio U = I I. Departmet of Mathematics ad Iformatics, Vilius Uiversity, Naugarduko 24, Vilius 03225, Lithuaia. vydas.cekaavicius@maf.vu.lt Departmet of Mathematics, SPST, Uiversity of Hamburg, Budesstr. 55, 2046 Hamburg, Germay. roos@math.ui-hamburg.de

2 2 V. Čekaavičius ad B. Roos I what follows, let V ad W be two fiite siged measures o Z. Products ad powers of V, W are uderstood i the covolutio sese, i.e. V W {A} = k= V {A k} W {k} for a set A Z; further W 0 = I. Here ad heceforth, we write W {k} for W {{k}}, k Z. The total variatio orm ad the local orm of W are deoted by W = k= W {k}, W = sup W {k}, k Z respectively. The logarithm ad expoetial of W are give by Note that li + W = k+ W k if W <, e W = exp{w } = k k= k=0 k! W k. V W V W, V W V W, e W e W. We deote by C positive absolute costats. Sometimes, to avoid possible cofusio, we supply costats C with idices. The letter Θ stads for ay fiite siged measure o Z satisfyig Θ. The values of C ad Θ ca vary from lie to lie, or eve withi the same lie. For x R ad k N = {, 2, 3,... }, we set x = x k k! xx... x k +, =. 0 Let Bi, p deote the biomial distributio with parameters N ad p [0, ]. Let ξ 0, ξ,..., ξ,... be a Markov chai with the iitial distributio Pξ 0 = = p 0, Pξ 0 = 0 = p 0, p 0 [0, ] ad trasitio probabilities Pξ i = ξ i = = p, Pξ i = 0 ξ i = = q, Pξ i = ξ i = 0 = q, Pξ i = 0 ξ i = 0 = p, p + q = q + p =, p, q 0,, i N. The distributio of S = ξ + + ξ N is called the Markov biomial distributio. We deote it by F, that is PS = m = F {m} for m Z + = N {0}. We should ote that the defiitio of the Markov biomial distributio slightly varies from paper to paper, see Dobrushi 953, Serflig 975, ad Wag 98. Sometimes ξ 0 is added to S or statioarity of the chai is assumed. For example, Dobrushi 953 assumed that p 0 = ad cosidered S +. However, if p = q, the Dobrushi s Markov biomial distributio becomes a biomial distributio shifted by uity. Therefore, we use the defiitio above

3 Biomial approximatio 3 which cotais the biomial distributio as a special case. Moreover, it obviously allows the rewritig of our results for S +. Further o, we eed various characteristics of S. Let q ν = q + q, ν 2qqp q 2 = q + q 3, A = q p q + q 2 q p 0q + q, a = ν + A, A 2 = q p q 2q + qq p Note that q + q > 0. It is kow that q + q 4 p 0 q p q + qq p q + q 3. see ES = ν + A A p q, VarS = ν 2 + ν ν 2 + A A 2 + 2A 2 + p q [ q q ] 2A q + q + A2 2 p q A 2A 2, Čekaavičius ad Roos 2006b. If p ad q are uiformly bouded away from uity, the p q is at least of expoetially vaishig order. Therefore, a ca be viewed as the mai part of ES. 2 Results It is kow that S has seve differet limit laws, see Dobrushi 953, Table. Typical limit distributios are the compoud Poisso ad the ormal oe. Cosequetly, we caot expect that the biomial approximatio is good for all values of parameters p ad q. However, if p = q the the Markov biomial distributio coicides with the biomial oe. Therefore, our aim is to get bouds, which are equal to zero if p = q. What is kow about the closeess of F to the biomial distributio? We formulate a cosequece of a more geeral result of Serflig 975, Eq. 2.4b. For arbitrary p 0,, the estimate F Bi, p 2 E Pξ j = ξ j p j= holds. For a better uderstadig of, let us take p = a. The from, we obtai F Bi, a 2 p q +. 2 q + q Estimate 2 may be good i the case p q = o oly. Note that the factor is due to the summatio i. Our purpose is to show that, uder additioal assumptios, it ca be replaced by a smaller factor. Let p 20, ν 30. 3

4 4 V. Čekaavičius ad B. Roos Assumptio 3 was itroduced by Čekaavičius ad Mikalauskas 999. Though certai smalless of p ad q is required, evertheless both parameters ca be costats. Thus, eve if 3 is satisfied, the limit distributio of S ca be the ormal oe. O the other had, it also allows a compoud Poisso limit distributio, which occurs whe q ˆλ ad p ˆp. If ˆp = 0, we have the Poisso limit distributio, see Dobrushi 953, Table. Our first result is the followig theorem. Theorem 2. Let coditio 3 be satisfied. The F Bi, a C p q mi, p + q, F Bi, a C 2 p q mi, p + q. q The right-had side of 2 is always less or equal to C p q. Thus, i compariso to 2, we get a estimate which has o factor. Due to the method of proof, absolute costats i Theorem 2. are ot give explicitly. However, we ca calculate asymptotically sharp costats. Theorem 2.2 Let coditio 3 be satisfied. The F Bi, a 4 p q C 3 p q p q +, 4 2πe q + q q F Bi, a p q p q C4 p q π qq q q Note that if, i additio, p q = o, q, the the right had sides of 4 ad 5 are of order o p q ad o p q / q, respectively. The accuracy of approximatio ca be improved by the secod-order estimates. Theorem 2.3 Let coditio 3 be satisfied. Set W = Bi, ai + 2 ν 2 U 2. The F W C 5 p q p q mi 2 q 2, + p + q mi,, 6 q F W C 6 p q p q mi 2 q 2, + p + q mi, q Note that the estimate i 6 is always less or equal to C p q p + q.. 7 q So far we cosidered oe-parametric biomial approximatio. It is possible to make use of both parameters of the biomial distributio BiN, p, where N ad p are chose i order to match two momets of S. The mai beefit of two-parametric biomial approximatio is that the estimates become comparable with the oes obtaied i the ormal approximatio. Such two-parametric approach was used for idepedet ad depedet idicators by Barbour et al. 992, p. 88, Čekaavičius ad Vaitkus 200, ad Soo 996, respectively. However, i Soo s paper, N ad p deped o the variace of idepedet idicators rather tha o the variace of the approximated sum.

5 Biomial approximatio 5 Sice we wat to fit two momets of the Markov biomial ad biomial distributios, oe should ote that this is ot always possible. Ideed, oe ca check that the Markov biomial distributio ca be so close to a compoud Poisso limit distributio that its secod factorial cumulat becomes positive. Meawhile the biomial distributio has egative secod factorial cumulat. Therefore, we use some additioal assumptios. Let p q ad ν. The ν 2 0 ad ν 2 ν 2 ν 2 > 0. Now we ca defie N N ad p [0, ] i the followig way: a 2 N = ν 2 ν = a2 2 ν 2 ν 2 δ, 0 δ <, N p = a. From the defiitio of VarS, it follows that the mai part of the secod factorial cumulat of S is equal to ν 2 ν 2 /2. Now 2 ν 2 ν 2 + N p 2 = δν 2 ν 2 2 2a 2 + δν 2 ν 2 / Cδq2. Thus, we see that N ad p are ideed chose to match two factorial cumulats ad cosequetly momets of S closely. Theorem 2.4 Let coditio 3 be satisfied, p q ad ν. The q F BiN, p C 7 q p + δq + δ q F BiN, p C 8 q p, 8. 9 It is clear that 8 is at least of order O /2. Thus, i this case, it becomes comparable to the classical Berry-Essee boud i the cotext of idepedet summads. If p = q, the the right-had sides of 8 ad 9 are equal to zero. Therefore, the closeess of p ad q is also reflected i the bouds. 3 Auxiliary results I what follows, Ck deotes a positive costat depedig o k. Lemma 3. Let t 0, ad k Z +. The we have U 2 e tu 3 te, U k e tu 2k k/2, U k e tu Ck. 0 te t k+/2 The first iequality was proved i Roos 200, Lemma 3. The secod boud follows from formula 3.8 i Deheuvels ad Pfeifer 988 ad the properties of the total variatio orm. Here ad throughout this paper, we set 0 0 =. The third relatio is a simple cosequece of the formula of iversio.

6 6 V. Čekaavičius ad B. Roos Lemma 3.2 For N ad p = q 0,, we have U 2 I + p U 4 C 2πe p q p q 2, U 2 I + p U C 2π p q 3/2 p q. 2 5/2 Lemma 3.2 was proved i Roos 2000, Lemma 8, see also Čekaavičius ad Roos 2006a, Prop. 3.5 ad Rem. 3.. We ow give some facts about F. It is kow that, if coditio 3 is satisfied, the F ca be expressed as F = Λ W + Λ 2 W 2, see Čekaavičius ad Mikalauskas 999, p. 25. lemma. Lemma 3.3 Let coditio 3 be satisfied. The The properties of Λ,2 ad W,2 are give i the followig Λ = I + ν U + ν 2 U 2 Θ, 3 Λ = I + ν U + ν 2 2 U 2 + Cq p q p + qu 3 Θ, 4 l Λ = ν U + ν 2 2 U 2 j+ + ν j j U j + Cq p q p + qu 3 Θ, 5 j=2 Λ 2 = 2 p q Θ, Λ 2 = Cs p q s e Cs Θ if s 0, 6 W = I + A U + C p q p + qu 2 Θ, W = I + Θ, 7 2 l W = A U + C p q p + qu 2 Θ, W 2 = C p q UΘ. 8 For ay fiite siged measure V o Z ad ay t 0,, we have V e t l Λ C V e 0.tν U. 9 Estimate 9 also holds if the total variatio orm o both sides is replaced by the local oe. Proof. Estimates 3, 4, 6, 7, 8, 9 ca be obtaied from the explicit formulas for Λ,2, W,2 i Čekaavičius ad Mikalauskas 999, p.p ad are already proved i Čekaavičius ad Roos 2006b. For the proof of 5 ote that 3, 4 ad the trivial fact U = 2 imply that Cosequetly, for j 2, Λ I j ν j U j = Λ I = 3ν 2 UΘ, Λ I ν U = C p q qu 2 Θ. j i= Λ I i ν j i U j i Λ I ν U j = C p q qu 2 3ν 2 i= = Cj p q qu 3 ν 3ν 2 i ν j i U j Θ j 22 j 2 Θ = Cj p q q 2 U 3 j 2Θ 0

7 Biomial approximatio 7 ad j=2 j+ Λ I j ν j j U j = C p q q 2 U 3 j=2 j 2Θ = C p q q 2 U 3 Θ Estimate 20 combied with 4 completes the proof of 5. Lemma 3.4 Let coditio 3 be satisfied. The j+ a j U j = j j=3 j=3 Proof. Note that, due to 3, we have j+ ν j U j p q p + q2 + C U 3 Θ. j a 0.2, ν 30, aj ν j p q Cj q + q 0.2j 3 p + q 2. Now the proof is obvious. Lemma 3.5 Let coditio 3 be satisfied. The, for ay fiite siged measure V o Z ad ay t 0,, we have V exp{t li + au} C V exp{0.tν U}, 2 V exp{t li + au + 0.5tν 2 U 2 } C V exp{0.tν U}. 22 The estimates remai valid if the total variatio orm o both sides is replaced by the local oe ad a is replaced by ν. Proof. Notig that a 0., we obtai t li + au + tν 2 2 U 2 = tau + ta2 2 U 2 Moreover, applyig 0, we get j=2 0.2 j 2 Θ + tν 9 U 2 Θ = CΘ + tν U tν U 2 Θ. exp{0.9tν U tν U 2 Θ} { tν U 2 0.9tν } r exp U r! r r= e r 0.24r r + C. r r 2πr 0.90e The last estimate ad the properties of the orms are sufficiet for the proof of 22. Estimate 2 is proved similarly. r= 4 Proofs Proof of Theorem 2.. Let B = I + au, M = l Λ + l W, ad M 2 = l B. We have F Bi, a Λ W B + Λ 2 W 2.

8 8 V. Čekaavičius ad B. Roos Applyig 9 ad Lemma 3.5, we get Λ W B = e M e M 2 = 0 e M 2 0 e M M 2 τ dτ M M 2 e M τ+m 2 τ dτ C M M 2 e 0.ν U. Now it suffices to apply Lemmas 3., 3.3, ad 3.4. The estimate for the local orm is proved similarly. Proof of Theorem 2.3. Let B ad M be defied as i the proof of Theorem 2.. Takig ito accout 22 ad arguig as i the proof of Theorem 2., we get { Λ W B ν2 exp Further, 2 U 2} [ e C 0.ν U M l B ν 2 2 U 2] C p q p + q mi q, B { ν2 exp 2 U 2} I ν 2 2 U 2 ν 2 2 = U q + mi { B exp τ ν 2 Cν 2 2 U 4 e 0.ν U C q p q 2 mi,,. q 2 U 2} τ dτ q 2. Combiig the last two estimates, we get 6. The local estimate is proved usig 0. Proof of Theorem 2.2. that q. Let b = b = Due to Theorem 2., without loss of geerality, we ca assume 4 p q = ν 2 4 2πe q + q 2 2πe ν ν, p q = ν 2 2π qq 2 2π ν ν. 3/2 The F Bi, a b F Bi, a + ν 2 I + ν 2 2 U 2 2 U 2 Bi, ν Bi, a ν U 2 Bi, ν b. Takig ito accout Lemma 3.5, similarly to the proof of Theorem 2., we get ν 2 2 U 2 Bi, ν Bi, a A C ν 2 U 3 e 0.ν U p q 2 C. q Now the proof of 4 follows from ad Theorem 2.2. The estimate 5 is obtaied with b replaced by b ad the total variatio orm replaced by the local orm.

9 Biomial approximatio 9 Proof of Theorem 2.4. We give oly a sketch of the proof. Due to assumptio p q we have ν 2 0. The followig estimates ca be obtaied: ν 2 ν 2 0.4, p Cp + q, p ν 4, ν + A 8 9, p a = p ν 2 + 2ν A / + A 2 /2 ν 2 ν + pδ C p q + δq, 2 N p j a j = a p j a j j 3 Cq 2 j p a 4 Cjq 2 q p + δq j 3, j 3, 4 j+ U j N p j a j = Cq 2 q p + δq U 3 Θ, j j=3 l Λ + l W N li + pu = Cq 2 q p + δq U 3 Θ +Cqq pu 2 Θ + Cδq 2 U 2 Θ. The proof of the theorem is ow similar to the oe of Theorem 2.. Ackowledgmet This work was fiished durig the secod author s stay at the Istitute of Mathematics at Carl vo Ossietzky Uiversity of Oldeburg i Summer term The secod author would like to thak Professor Dietmar Pfeifer for the ivitatio to Oldeburg ad the Istitute for its hospitality. We are grateful to the referee for valuable remarks, which helped to improve the results of the paper. Refereces [] Barbour, A.D., Holst, L., Jaso, S.: Poisso Approximatio. Claredo Press, Oxford 992 [2] Boutsikas, M.V., Koutras, M.V.: A boud for the distributio of the sum of discrete associated or egatively associated radom variables. A. Appl. Probab. 0, [3] Campbell, S.A., Godbole, A.P., Schaller, S.: Discrimiatig betwee sequeces of Beroulli ad Markov-Beroulli trials. Commu. Statist. Theory Meth. 23, [4] Čekaavičius, V., Mikalauskas, M.: Siged Poisso approximatios for Markov chais. Stochastic. Proc. Appl. 82,

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