1、Convex ordering of integrated telegraphers processesAntonio Di CrescenzoDipartimento di Matematica e InformaticaUniversita di SalernoVia S. AllendeI-84081 Baronissi (SA)Italyadicrescenzounisa.itFranco PellereyDipartimento di MatematicaPolitecnico di TorinoC.so Duca degli Abruzzi, 24I-10129 TorinoIta
2、lypellereycalvino.polito.itAbstractWe show that for every xed time t 0 the marginal of the classical integrated telegraphersprocess increases in the convex order if the rate of the undelying Poisson process decreases.1 IntroductionLet W = fW(t);t 0g be a continuous-time stochastic process that descr
3、ibes the wear accumulated byan item during its lifetime, and assume that wear alternately increases and decreases between randomtime points, with linear wear rates. In Di Crescenzo and Pellerey (2000) we obtained conditions onthe time intervals between the random points such that two processes of th
4、e given class are orderedaccording to the usual stochastic order. However, the results stated therein can not be applied when thetime intervals of each of the two underlying alternating counting processes are identically distributed.Thus, in particular, they can not be applied to compare two integra
5、ted telegraphers processes W1andW2, de ned asWi(t) = W0Zt0(?1)Ni(s)ds; t 0; i = 1;2;the underlying counting processes N1and N2being homogeneous Poisson processes with rates 1and2, respectively, with initial velocity W0being either +1 or ?1 with equal probability.Since integrated telegraphers process
6、es are of interest in various domains of industry and technology(see, e.g., Kolesnik (1996), Foong and Kanno (1994) and Orsingher (1990) for some recent references),an interesting problem is to compare two such processes according to some other stochastic order.Hereafter we prove that, even when W1a
7、nd W2are not ordered according to the usual stochasticorder, for every xed t 0 the marginals of such processes are comparable in the well-known convexorder. In other words we show that if 1 2then W1(t) cxW2(t), i.e.E(W1(t) E(W2(t) (1)for every t 0 and for every convex function such that the above ex
8、pectations exist. (See Shaked andShanthikumar (1994) for a whole list of applications of the convex order.)2 Main AnalysisTo prove inequality (1) let us setW(t) = W0Zt0(?1)N(s)ds; t 0; (2)where Nis a homogeneous Poisson process having rate 0.First of all we note that the conditional distribution of
9、W(t), given N(t) = n, does not dependon . Indeed, it just depends on W0and on the epoch-times of Nup to time t, which, in turns, givenN(t) = n, are distributed as the order statistics ofn independent randomvariablesuniformlydistributedon the interval 0;t (see, e.g., Ross (1983), page 37).Lemma 1 Let
10、 the process fW(t);t 0g be de ned as in (2), and letht(n) =E(W(t)jN(t) = n: (3)For any convex function one has ht(n) ht(n +2) for all n = 0;1;: and t 0.Proof. Let us denote byUi:nthe i-th epoch-timeofthe process NgivenN(t) = n,with i = 1;2;:;n.It is not hard to see that for all n = 0;1;: we haveht(2
11、n) =EfW0(2V2n? t)g and ht(2n+1) =EfW0(2V2n+1? t)g;whereV2n=nXi=1U2i?1:2n?nXi=1U2i:2n+t and V2n+1=n+1Xi=1U2i?1:2n+1?nXi=1U2i:2n+1:We note that random variables Vm, m = 0;1;:, represent the portion of time 0;t during whichddtW(t) = W0conditioned on knowledge that N(t) = m.Since the variables Ui:nare d
12、istributed as the order statistics corresponding to a sample of n inde-pendent random variables uniformly distributed on 0;t, by induction it is not hard to verify that therandom variables V2nand V2n+1have probability densitiesf2n(s) =2(2n?1)!t(n?1)!2h1?stin?1hstin10;t(s) (4)andf2n+1(s) =(2n+1)!tn!2
13、hstinh1?stin10;t(s); (5)respectively, where 1A(s) is the indicator function of A.Let us now setS2n= W0(2V2n? t) and S2n+1= W0(2V2n+1? t);where W0is such thatP(W0= 1) =12. From (4) and (5) it follows that the densities of S2nand S2n+1are given byg2n(s) =(2n? 1)!22nt(n?1)!2t?stn?1t+stn?11?t;t(s)andg2n
14、+1(s) =(2n+1)!22ntn!2t?stnt+ stn1?t;t(s);respectively.Evaluating the expression g2n(s)?g2n+2(s), s 2 ?t;t, and observing thatES2n =ES2n+2 = 0 forall n = 0;1;:, by Theorem 2.A.17 in Shaked and Shanthikumar (1994) it follows thatS2ncxS2n+2:Similarly, noting thatES2n+1 =ES2n+3 = 0 for all n = 0;1;:, ev
15、aluating the di erence g2n+1(s) ?g2n+3(s), s 2 0;t, by Theorem 2.A.17 in Shaked and Shanthikumar (1994) we getS2n+1cxS2n+3;for all n = 0;1;:.Therefore, for a convex function we haveht(2n+1) =E(S2n+1) E(S2n+3) = ht(2n+3)andht(2n) =E(S2n) E(S2n+2) = ht(2n+ 2);i.e., the assertion.The following result i
16、s a technical lemma which will be used in the main theorem. Its proof requiressome notion of total positivity theory, such as the preservation properties of totally positive of order 2(TP2) functions. We refer the reader to Karlin (1968) for de nitions and properties of TP2functions.Lemma 2 Let z1(n
17、) and z2(n) be two non-increasing functions in n = 0;1;:, and, for any xed t 0,letZ1;t() =1Xn=0z1(n)(t)2n(2n)!and Z2;t() =1Xn=0z2(n)(t)2n+1(2n+ 1)!: (6)Then, the ratioZ1;t() + Z2;t()exp(t)is non-increasing in 0 for all t 0.Proof. Fix t 0. First of all, we prove that functionsG1;t() =Z1;t()cosh(t)and
18、 G2;t() =Z2;t()sinh(t)are non-increasing in 0. For it de nez1(n;k) =z1(n) if k = 11 if k = 2;and observe that the function z1(n;k) is TP2in (n;k), since by assumption z1(n) is non-increasing. Letus setZ1;t(;k) =1Xn=0z1(n;k)(t)2n(2n)!=8:1Xn=0z1(n)(t)2n(2n)!= Z1;t() if k = 11Xn=0(t)2n(2n)!= cosh(t) if
19、 k = 2:Since the ratio(t)2n(2n)!is TP2in (;n), from the Basic Composition Formula (see Karkin (1968), page 17)it follows thatZ1;t(;k) is TP2in (;k), i.e., thatZ1;t(;1)Z1;t(;2)=Z1;t()cosh(t)is non-increasing in 0:In a similar way it can be shown thatZ2;t()sinh(t)is non-increasing in 0:Now, since both
20、 functions cosh(t) and sinh(t) are non-decreasing in 0, for xed t 0, it easilyfollows thatZ1;t()cosh(t) + sinh(t)andZ2;t()cosh(t)+ sinh(t)are non-increasing in 0. Therefore, also the ratioZ1;t() + Z2;t()exp(t)=Z1;t() +Z2;t()cosh(t) + sinh(t)is non-increasing in 0, this giving the proof.We can now st
21、ate the main result.Theorem 1 Let the process fW(t);t 0g be de ned as in (2). If the function is convex then theexpected valueE(W(t) is non-increasing in 0 for all t 0.Proof. Let the function htbe de ned as in (3). We haveE(W(t) =EE(W(t)jN(t)=Eht(N(t)=1Xn=0ht(n)PN(t) = n=1Xn=0fht(2n)PN(t) = 2n+ht(2n
22、+1)PN(t) = 2n+ 1g=1exp(t)“1Xn=0ht(2n)(t)2n(2n)!+1Xn=0ht(2n+1)(t)2n+1(2n+ 1)!#:Let us now set z1(n) = ht(2n) and z2(n) = ht(2n+1), n = 0;1;:. ThenE(W(t) =Z1;t() + Z2;t()exp(t);where Zi;t();i = 1;2; are de ned in (6). By Lemma 1 the functions z1(n) and z2(n) are non-increasingin n = 0;1;:, and therefo
23、re, by Lemma 2, the last term is non-increasing in 0 for every t 0.We nally state that inequality (1) is a direct consequence of Theorem 1. We also remark that someresults presented above have been used in Di Crescenzo and Pellerey (2002) to obtain comparisons forprice processes related to the integ
24、rated telegraphers process.ReferencesDi Crescenzo A. and Pellerey F. (2000) Stochastic comparison of wear processes characterized byrandom linear wear rates. In M. Nikulin and N. Limnios (Eds.), Abstracts Book of the Second In-ternational Conference on Mathematical Methods in Reliability,pp. 339342.
25、 Bordeaux: UniversiteVictor Segalen.DiCrescenzo A. and Pellerey F. (2002) On prices evolutions based on geometric telegraphers process.Appl. Stoch. Mod. Bus. Ind. (to appear).Foong S.K. and Kanno S. (1994) Properties of the telegraphers random process with or without atrap. Stoch. Proc. Appl. 53, 14
26、7173.Karlin S. (1968) Total Positivity, Vol. 1. Stanford: Stanford University Press.Kolesnik A. (1996) Applied models of random evolutions. Z. Angew. Math. Mech. 76 S3, 477478.Orsingher E. (1990) Probability law, ow function, maximum distribution of wave-governed randommotions and their connections with Kircho s laws. Stoch. Proc. Appl. 34, 4966.Ross S.M. (1983) Stochastic Processes. New York: John Wiley & Sons.Shaked M. and Shanthikumar J.G. (1994) Stochastic Orders and Their Applications. San Diego:Academic Press.
