Research Article Asymptotic Time Averages and Frequency Distributions

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1 Ieraioal Sochasic Aalysis Volume 26, Aricle ID 27424, pages hp://dx.doi.org/.55/26/27424 Research Aricle Asympoic Time Averages ad Frequecy Disribuios Muhammad El-Taha Deparme of Mahemaics ad Saisics, Uiversiy of Souher Maie, 96 Falmouh Sree, Porlad, ME 44-93, USA Correspodece should be addressed o Muhammad El-Taha; el-aha@maie.edu Received 7 March 26; Acceped 7 July 26 Academic Edior: MJ Lopez-Herrero Copyrigh 26 Muhammad El-Taha. This is a ope access aricle disribued uder he Creaive Commos Aribuio Licese, which permis uresriced use, disribuio, ad reproducio i ay medium, provided he origial work is properly cied. Cosider a arbirary oegaive deermiisic process (i a sochasic seig {X(), } is a fixed realizaio, i.e., sample-pah of he uderlyig sochasic process) wih sae space S = (, ). Usig a sample-pah approach, we give ecessary ad sufficie codiios for he log-ru ime average of a measurable fucio of process o be equal o he expecaio ake wih respec o he same measurable fucio of is log-ru frequecy disribuio. The resuls are furher exeded o allow uresriced parameer (ime) space. Examples are provided o show ha our codiio is o superfluous ad ha i is weaker ha uiform iegrabiliy. The case of discree-ime processes is also cosidered. The relaioship o previously kow sufficie codiios, usually give i sochasic seigs, will also be discussed. Our approach is applied o regeeraive processes ad a exesio of a well-kow resul is give. For researchers ieresed i sample-pah aalysis, our resuls will give hem he choice o work wih he ime average of a process or is frequecy disribuio fucio ad go back ad forh bewee he wo uder a mild codiio.. Iroducio I his aricle we seek weak codiios, ecessary ad sufficie, for he log-ru ime average of a process or ay measurable fucio of i o be equal o he expecaio ake wih respec o is log-ru frequecy disribuio. Throughou he paper we use a sample-pah approach (see [ 3]) i he sese ha we resric aeio o oe realizaio (sample-pah) of he process of ieres. Our approach reveals ha he sochasic assumpios (regeeraive, semisaioary, ec.) i he mos par are o eeded for his resul o hold, buraheroesurehaheprocessiselfisergodicad ha a saioary disribuio exiss. I oher words, he logru ime averages for differe sample-pahs of he give sochasic process coverge o a commo iig value wih probabiliy oe (see Example 7 i Secio 4). Our approach is iuiive ad he proofs are raher elemeary; ohig beyod Riema-Sieljes iegraio heory is eeded. Le {X(), } be a arbirary real-valued righcoiuous deermiisic process (i a sochasic seig {X(), } cabeafixedrealizaio,hais,samplepah of he uderlyig sochasic process) wih sae space S = (, ).Le X = X (u) du, > ; Y = h (X (u)) du, > ; G (x) = {X (u) x} du, >, x (, ), where {} is a idicaor fucio ad h is a real-valued measurable fucio. I is also assumed ha h is iegrable wih respec o G(x) i he Riema-Sieljes sese. Defie he followig is whe hey exis: ()

2 2 Ieraioal Sochasic Aalysis X= X ; Y= Y ; G (x) = G (x). Le G (x) ad G(x) be he complemes of G (x) ad G(x),respecively(ifollowshaG(x) = G (x)). Noe ha X ad Y are he log-ru ime average of he processes {X(), } ad {h(x()), }, respecively, ad G(x) is he log-ru frequecy disribuio of {X(), }. I a queueig sysem X may represe he log-ru average umber of cusomers i he sysem, L, ad G(x) is he saioary disribuio. A a elemeary level he problem ca be posed as follows. I is of ieres o esablish codiios uder which helog-ruimeaverageofagiveprocessisequalo he expecaio ake wih respec o is log-ru frequecy disribuio; ha is, he followig relaio holds: X= xdg(x). (3) I urs ou ha i our sample-pah seig, whe < X< +, here is a ecessary ad sufficie codiio for relaio (3) o hold. Relaio (3) may also be valid eve whe X = ±. I a sochasic seig relaio (3) may have he followig ierpreaio: for each >,le{x R (), } be a process such ha, for each, X R () has G (x) as is disribuio fucio. The he process {X R (), } represes he saus of he origial process {X(), } as see by a radom observer ha arrives a a radom ime uiformly disribued bewee ad.ifwelex R be a radom variable wih G(x) as is disribuio fucio, he X R describes he behavior of he process {X R (), } i seady sae, ad relaio (3) may be wrie as X=EX R,whereEX R is he expeced value of X R. Thisproblemhasheoreicalaswellaspracicalsigificace. For example, i queueig heory he log-ru average umber of cusomers i a queueig sysem is, someimes, defied as L= L(u)du/,where{L(), } represes he umber of cusomers prese i he sysem (boh i queue ad service) a isa. However,iapplicaios, L is usually calculaed as he expecaio of he saioary disribuio {p },ofheprocess{l(), }, providedi exiss; ha is, L= = p.thequesioarisesasowhehe above wo quaiies are equal, paricularly whe {L(), } is o saioary ad ergodic. Similar quesios arise also whe calculaig oher performace measures for queueig sysems such as W, he log-ru average waiig ime i he sysem per cusomer. Aoher example arises whe dealig wih he ASTA (Arrivals ha See Time Averages) propery (see [, 4 8]). Followig [4, 5] he ASTA problem isposedasfollows:givewosochasicprocessesx {X(), } ad N {N(), } defied o a commo probabiliy space, where X is ieded o represe a queue or a ework of queues, ad N as a arrival poi process, le (2) T = if{ : N() },, be a imbedded poi process, X() X(),adX(T ) X(),where deoes weak covergece[9].theheastaproblemisofidcodiios uder which X() = d X(), where= d deoes equaliy i disribuio. However, raher ha workig o verify X() = d X() direcly, Melamed ad Whi [4, 5] fid codiios such ha V() = W(),where V () = h (X (u)) du, W () = N () N() h(x (T k )), ad h is a bouded measurable real-valued fucio. I geeral V() ad Eh(X()) (equivalely W() ad Eh( X())) are o ecessarily equal uless X is saioary ad ergodic. O he oher had, Sidham Jr. ad El-Taha [] prove ASTA by workig direcly wih sample-pah versios of V() ad W(). I his paper we esablish weak codiios (ecessary ad sufficie) o guaraee he equaliy of ime averages of a process ad he expecaio ake wih respec o is log-ru frequecy disribuio wihou explicily assumig saioariy ad ergodiciy. I a sochasic framework his problem has bee reaed by may auhors, mosly o esablish codiios ha guaraee ergodiciy. For example, he case of a process wih a imbedded saioary sequece (he semisaioary process) is proved by [, ]. Frake e al. [2] derive similar resuls for processes wih imbedded marked poi processes. Rolski [3] iroduces ad explois he oio of ergodically sable processes o prove a similar resul. Wolff [4] proves a similar resul for processes ha are regeeraive or oegaive ad sochasically icreasig. The case for sochasic clearig processes is cosidered by [5]. Releva also are [6 2]. For refereces o sample-pah aalysis, he reader is referred o[,2,6,2 25].Thisaricleshouldbeofiereso readers wih ieres i esablishig relaioships ha ivolve ime averages ad frequecy disribuios i a deermiisic framework. El-Taha ad Sidham Jr. [2] provide a resul i a samplepah seig for he special case of a oegaive deermiisic process. I his aricle he resuls of [2] are exeded o ay fucio of he process; hus all momes of a process ca be reaed wihi oe framework. We also exed he sae space o allow for he process o ake egaive as well as posiive values. Moreover we cosider he case where he parameer space is exeded from [, ) o (, ). The aricle is orgaized as follows. I Secio 2 we cocerae ohe case whe {X(), } is a geeral uresriced process ad give ecessary ad sufficie codiios (see codiios A ad A2 i Secio 2) whe a measurable fucio, saisfyig a weak regulariy codiio, of he process is cosidered. We also provide some isigh io he relaioship bewee our codiio ad uiform iegrabiliy. We also exed our resuls o allow he parameer o be uresriced i sig. Secio 3 provides similar resuls whe X is a discree-ime process. I Secio 4 hree examples are k= (4)

3 Ieraioal Sochasic Aalysis 3 giveohelpclarifyheeedforcodiioa, herelaio of codiio A o uiform iegrabiliy, ad he relaioship bewee our pure sample-pah seig ad sochasic seigs. I Secio 5 we close by lookig a regeeraive processes. A process wih a imbedded sequece is ivesigaed ad he used o exed a well-kow resul for regeeraive processes. 2. Mai Resuls ad Relaed Aalysis I his secio we prove he mai resul, provide several applicaios, sudy he coecio of he codiios eeded o uiform iegrabiliy, ad poi ou a exesio of he parameer space o allow egaive ime. Give ay deermiisic process, we show ha for ay measurable fucio of heprocessheasympoicimeaverageofafucioofa give process is equal o he expecaio ake wih respec o is asympoic frequecy disribuio fucio uder weak codiios. The several special cases of ieres will be saed. Our objecive is o seek weak codiios uder which he asympoic ime average of a fucio of a give process is equal o he expecaio ake wih respec o i asympoic frequecy disribuio fucio; ha is, h (X (u)) du = h (x) dg (x). (5) Noe ha (5) reduces o (3) whe h is a ideiy fucio. Firs we esablish he followig ime depede relaioships ilemmas 3,keyoprovighemairesul. Lemma. Le h(x) be ay real-valued measurable fucio ad G (x) a disribuio fucio defied o (, +).The for all, > (i) (ii) (iii) h (x) dg (x) = G (x) dh (x) +h() G (), (6) h (x) dg (x) = G ( x) dh ( x) +h( ) G ( ), h (x) dg (x) = G (x) dh (x) + G ( x) dh ( x) +h(), providedalliegralsarewelldefied(i.e.,h(x) is iegrable wih respec o G (x) i he Riema-Sieljes sese). (7) (8) Proof. Noe ha h (x) dg (x) = = x [ dh (y) + h ()]dg (x) dg (x) dh (y) + dg (x) dh (y) y +h() G () = G (y)dh(y)+h() G () which proves par (i). The proof of (ii) is similar. Par (iii) follows by akig =i (i) ad (ii) ad combiig boh resuls. The resuls i Lemma remai valid if we replace G (x) by G(x) whe he i exiss. Now we give he secod parial resul. Lemma 2. For all > Proof. Noe ha G (x) dh (x) = = (9) h (X (u)) du = h (x) dg (x). () {X (u) >x} du dh (x) X(u) {X (u) >} dh (x) du = h (X (u)) {X (u) >} du h () G (). Similarly G ( x) dh ( x) = {X (u) x} du dh ( x) = h (X (u)) {X (u) } du h () G (). The he resul follows by appealig o Lemma (iii). () (2) By akig is, as of boh sides of Lemma 2, showigrelaio(5)holdsisseeobeequivaleoa problem of ierchagig is ad iegrals. The ex resul is a key lemma. Lemma 3. For all >ad h (x) dg (x) {x: x >} = h (X (u)) { X (u) >} du. (3)

4 4 Ieraioal Sochasic Aalysis Proof. The resul will follow if we show ha (i) Now (ii) h (x) dg (x) = h (X (u)) {X (u) >} du, (4) h (x) dg (x) = h (X (u)) {X (u) < } du. G (x) dh (x) = = {X (u) >x} du dh (x) X(u) {X (u) >} dh (x) du = h (X (u)) {X (u) >} du h () G (). Therefore, usig Lemma (i), we obai (5) (6) h (x) dg (x) = h (X (u)) {X (u) >} du (7) which proves par (i). Par (ii) is proved similarly usig Lemma (ii). Because Lemmas 3 are ime depede, hey ca be useful i ime depede aalysis of osaioary sochasic sysems. Now we give he mai resul. Theorem 4. Cosider he deermiisic process {X(), }, wih sae space S = (, ),adleh( ) be ay real-valued measurable fucio. The, he followig are equivale. (i) Codiio A: (ii) Codiio A2: h (X (u)) { X (u) >} du =. (8) h (x) dg (x) =. (9) {x: x >} Proof. Takig he is as o boh sides of Lemma 3 proves he equivalece of (i) ad (ii). Now, usig Lemmas 3 ad 2, we wrie + h (X (u)) du h (x) dg (x) = h (x) dg (x), {x: x >} + h (X (u)) du h (x) dg (x) = h (X (u)) { X (u) >} du. (2) Suppose A (equivalely A2) holds.now,akig is as ad assumig hey exis give + h (X (u)) du h (x) dg (x) = h (x) dg (x), {x: x >} + h (X (u)) du h (x) dg (x) = h (X (u)) { X (u) >} du. (22) (23) The par (iii) of he heorem follows by akig is of boh sides i (22) ad (23) as. Coversely if par (iii) holds, i is sraighforward o see ha A ad A2 hold. I follows from Theorem 4 ha he followig codiios are sufficie for (5) o hold. Corollary 5. Cosider he process {X(), }. Suppose G (x) G(x) uiformly i x as. Suppose also ha codiio (i) (equivalely (ii)) of Theorem 4 holds. The for ay measurable real-valued fucio h, h(x(u))du is well defied if ad oly if h(x)dg(x) is well defied, i which case relaio (5) holds; ha is, h (X (u)) du = h (x) dg (x). (24) Proof. The proof follows by akig is, as,i(22) ad(23)adusigasimilarargumeasitheorem4. Corollary 6. Cosider he process {X(), } ad suppose ha he codiios of Corollary 5 hold. Addiioally, le h( ) be differeiable ad h() =.The (iii) [ h (X (u)) du h (x) dg (x)] =. (2) h (X (u)) du = h (x) G (x) dx + h (x) G (x) dx. (25)

5 Ieraioal Sochasic Aalysis 5 Proof. By Corollary 5, i suffices o show ha h (x) dg (x) = h (x) G (x) dx So, he resul follows by oig ha ad usig Lemma (iii). + h (x) G (x) dx. (26) G (x) dh ( x) = h (x) G (x) dx (27) I Corollary 6, le h(x) = x β,he(26)givesheβh mome of a disribuio fucio x β dg (x) = βx β G (x) dx + βx β G (x) dx which is familiar whe G(x) is defied oly o [, ). (28) Remarks 7. (i) I follows immediaely from Corollary 5 ha whe < Y<+(recall Y() := h(x())) codiio A (equivalely A2) is ecessary ad sufficie for relaio (5) o hold. The requireme ha < Y<isoeeded o provehesufficiecyparofheasserio. (ii) From (23), i is clear ha if + h(x)dg(x) is well defied ad fiie for all <<,hey is well defied iff h(x(u)){x(u) > }du is well defied for all >. Now ake is i (23) as o obai h (X (u)) du = h (x) dg (x) + h (X (u)) { X (u) >} du. (29) If ay wo of he hree erms i (29) are well defied ad a leas oe of hem is fiie he he hird erm exiss ad Corollary 5 holds. (iii) Assume ha all he releva is i Corollary 5 are well defied. From (23) oe cocludes ha, for all, Y=± iff h (X (u)) { X (u) >} du = ±. (3) Moreover, if Y = + ad codiio A is o saisfied, we ca disiguish o possibiliies (a) xdg(x)=; he relaio (5) holds; (b) xdg(x)<; he relaio (5) does o hold ad codiio A akes he value +.Asimilarargume applies whe Y=.Theabovediscussioshould o imply ha if codiio A does o hold i shouldbeifiie.example5givebelowshowsha relaio (5) fails wih codiio A assumig a fiie value. (iv) Oe ca cosruc sufficie codiios for codiio A o hold. Followig a argume by Billigsley [26, page 86], oe ca easily see ha [h(x(u))]+ε du < is sufficie for (5) o hold. Nex we explore he relaioship bewee codiio A ad uiform iegrabiliy of he process {X R (), }. Relaio o Uiform Iegrabiliy. We discuss he coecio bewee codiio A ad uiform iegrabiliy whe h is a ideiy fucio. Codiio A of Theorem 4 requires, roughly speakig, ha he area, for, bewee ad X(){X() > } mius he area bewee ad X(){X() < } goes o zero as approaches ifiiy. We poi ou here ha here is a differece bewee codiio A ad uiform iegrabiliy (of X R ()) which requires ha he above wo areas add up o zero as approaches ifiiy. Wolff [4] suggess ha provig relaio (5), i a sochasic seig, is equivale o showig ha he process {X R (), } ad X are uiformly iegrable (u.i.) i. Codiio A is weaker ha beig u.i.; i oly coicides wih uiform iegrabiliy of he process {X R (), } whe he process {X(), } is oegaive. To shed more ligh o his differece, we show ha he followig modified codiio is he equivale o uiform iegrabiliy. Codiio A3. sup X(u) { X(u) > }du =. Noe ha i A3 we ake he absolue value of X().Iour sample-pah seig he defiiio of uiform iegrabiliy [26, 27] of he process {X R (), } is equivale o sup x dg (x) =. (3) {x: x >} Now, usig a argume similar o ha used i provig Lemma 3, we obai x dg (x) {x: x >} = X (u) { X (u) >} du. (32) Thus codiio A3 ad (3) are equivale. We oe ha codiio A3 (equivalely (3)) is sufficie for relaio (5) o hold. Example 6 give below shows he exisece of a process ha obeys relaio (5), ye codiio A3 is o saisfied. 2.. Momes. A impora special case is whe h(x) = x β, β >, ha provides a relaio bewee ime averages of a process momes ad he momes obaied by usig is

6 6 Ieraioal Sochasic Aalysis asympoic disribuio fucio. For h(x) = x β,wehavehe followig resul. Corollary 8. Le h(x) = x β. The he followig are equivale: (i) (ii) (iii) [X (u)] β { X (u) >} du = ; (33) x β dg (x) =; (34) {x: x >} [X (u)] β du = x β dg (x), Y <, where Y= [X(u)]β du. (35) Aoher impora special case is whe cosiderig absolue momes, ha is, whe h(x) = x β Exesio. Here, a geeralizaio of Theorem 4 will be cosidered. We allow he parameer space [, ) o exed o (, ) ad show ha, for ay measurable fucio of he ew process, relaio (5) remais valid uder codiios similar o A ad A2. Theorem 4 is exeded o he case where he parameer ca also be egaive. Le {X(), < < } be a arbirary (deermiisic) process wih sae space S = (, ),adle G (x) = 2 {X (u) <x} du, (36) x (, ), >. Here we seek weak codiios uder which he followig relaio holds: 2 h (X (u)) du = h (x) dg (x). (37) Theorem 9. Cosider he deermiisic process {X(), < <}, X() R, adleh( ) be ay real-valued measurable fucio. The, he followig are equivale. (i) Codiio B: 2 h (X (u)) { X (u) >} du =. (38) (ii) Codiio B2: (iii) [ 2 h (x) dg (x) =. (39) {x: x >} h (X (u)) du h (x) dg (x)] =. (4) Proof. SimilaroLemma3wecashowhaforall>ad h (x) dg (x) {x: x } = 2 h (X (u)) { X (u) } du. Now, le =i(4);hewehaveforall> 2 (4) h (X (u)) du = h (x) dg (x). (42) Now a argume similar o ha of he proof of Theorem 4 will yield his resul. Noe ha proof of Theorem 9 follows he same lies of argume as ha of Theorem 4. Noe also ha codiios B ad B2 are similar o codiios A ad A2,buwih2 i he deomiaor i B. 3. Discree-Time Processes I his secio we cosider a discree-ime process; specifically le {X, } be ay deermiisic discree-ime process, ha is, a ifiie sequece of real umbers, ad le h( ) be a real-valued measurable fucio. Moreover, le X = G (x) = k= k= X k ; {X k x}; x (, ) ; ad defie he followig is whe hey exis: X= X, G (x) = G (x), x (, ). (43) (44) Similar o he coiuous ime model, we have he followig resuls. Theorem. Le {X, } be ay discree-ime process, ad le h( ) be ay a real-valued measurable fucio. The, he followig are equivale: (i) k= h(x k){ X k >}=; (ii) {x: x >} h(x)dg (x) = ; (iii) [ k= h(x k) h(x)dg(x)] =. Noe ha whe h( ) is a ideiy fucio, codiio (i) of Theorem is someimes referred o i he lieraure as he codiio for uiform iegrabiliy. Corollary. Cosider he process {X, }. Suppose G (x) G(x) uiformly i x as. Suppose also ha codiio (i) (equivalely (ii)) of Theoremholds.Theforaymeasurablereal-valuedfucio

7 Ieraioal Sochasic Aalysis 7 h, (/) k= h(x k) is well defied if ad oly if h(x)dg(x) is well defied, i which case h(x k )= k= h (x) dg (x). (45) Thiscorollaryhasbeefoudusefulihelieraure. Corollary 2. Le {X, } be ay oegaive deermiisic discree-ime process, ha is, ifiie sequece of oegaive real umbers. Suppose ha ay of he codiios (i),(ii),ad(iii)oftheoremissaisfiedadg(x) is a proper disribuio fucio. Now, suppose here exiss y>such ha if h( ) is a odecreasig real-valued measurable fucio o (y, ) he x h (x) Gc (x) = (46) ad if h( ) is a oicreasig real-valued measurable fucio o (, y),he Proof. Noe ha x h (x) G (x) =. (47) x h (x) Gc (x) = x h (x) G c (x) = x x x x h (x) dg (u) h (u) dg (u) =, (48) by codiio (ii) of Theorem. The proof of he secod saeme is similar. The resuls i Corollary 2 apply o he coiuous ime process as well. Whe he sequece {X k ;k }represes, say, service imes i queueig model, le h(x) = x 2 i Theorem o, immediaely, obai he followig useful resuls. Corollary 3. Le EX 2 = k= X2 k <.Thehe followig are equivale: (i) k= X2 k {X k >}=; (ii) x2 dg (x) = ; (iii) EX 2 = x 2 dg(x) <. Corollary 4. Suppose ha eiher (i) EX 2 = x 2 dg(x) < or (ii) k= X2 k {X k >}=.The x x2 G c (x) =. (49) Corollary 4 follows from Corollary 2 by leig h(x) = x 2. This resul is used i Ayesa [28]. 4. Examples ad Discussio I his secio, we give hree examples. The firs example shows ha codiio A is o superfluous. The secod example, a modificaio of he firs oe, shows ha he ew modified process does o saisfy he uiform iegrabiliy codiio (codiio A3), ye codiios A ad A2 are saisfied ad herefore relaio (5) holds. I he hird example, we verify ha whe codiio A is saisfied, for a saioary oergodic sochasic process, eve hough relaio (5) is o saisfied i a sochasic seig, i remais valid for every idividual sample-pah of he process. Example 5. Cosider a process {X, }i discree ime, adlehesequece{x } represe oe sample-pah such ha 2 m, = 4 m +3k2 m { k=,2,...,2 m x = m =,,... { { oherwise. (5) For example if m=,hex =, x 2 =, x 3 =;if m=,hex 4 = = x 8 =, x 9 =2, x = =x 4 =, x 5 =2;ifm=2,hex 6 = = x 26 =, x 27 =4, x 28 = =x 38 =, x 39 =4, x 4 = =x 5 =, x 5 =4, x 52 = = x 62 =, x 63 =4;ifm=3,hex 64 =,...,ad so o. Figure plos he cumulaive sequece j= x j for all, showig he jumps a posiive erms of {x }. Now, cosider he sum of he sequece {x } a posiive erms; he sup x j = 2 2m +3k 2 m 2 2m +3 k 2 m j= x j = (m ) +k 2 m 4 m +3k 2 m = 3. Now, cosider he sum jus before posiive erms o obai if x j = 2 2m +3k 2 m 2 2 2m +3 k 2 m 2 j= = (m ) + (k ) 2 m 4 m +3k 2 m 2 = 3 +( 3 ) 3/2 m 3 2 m +3k 2/2 m 3 x j as m for ay fixed k. (5) (52) Therefore if(/) j= x j = /3, ad hece (/) j= x j = /3. We also coclude, usig Lemma 2, ha x= xdg (x) = 3. (53)

8 8 Ieraioal Sochasic Aalysis x j j= h cycle s cycle 2d cycle S lope = / Usig Example 5 ad he fac ha z = z + +z,we coclude z k { z k >}=2 3 ; (58) k= hus he sequece {z } does o saisfy he uiform iegrabiliy codiio (A3). Now, usig he fac ha z =z + z,we coclude ha codiio A is saisfied, ad herefore relaio (5) holds. Figure : Cumulaive sequece for {x }. Example 7. Le he saioary sochasic process {X, } be defied as The ex sep is o calculae he r.h.s. of relaio (5) ad show ha i differs from /3.Now xdg(x) = G (x) dx, G (x) = k= {x k >x}. (54) Successive values of m deermie successive cycles. Now cosider G () a he begiig of such cycles o obai G 4 m () = 4 m 4 m k= {x k >}= m = 2m 4 m, as m. 4 m (55) Sice (/) k= {x k >x}decreases i as log as x =,i suffices o cosider G () a a subsequece of posiive erms. The G 4 m +3k 2 m () = m +k 4 m +3k 2 m 2 m +k = 4 m +3k2 m, as m. (56) Therefore G (x) as for all x ad G (x) as for all x ;hais,g(x) is well defied ad G(x) = G (x) = for all x.thus x = /3 = G(x)dx =, so relaio (5) fails. I his example codiio A akes he value /3. The ex example gives a sequece ha does o saisfy he uiform iegrabiliy codiio A3, ye i saisfies relaio (5). Example 6. Now, we modify Example 5 as follows: le wih z =. z =x x, =,2,... (57) P{X =k}=p k, EX =μ<, where p k, k= p k =,adx =X for all. (59) The sequece {X, } is a example of a saioary oergodic process (see, e.g., [29]). The sample space is give by S={ω,ω,...},whereω represes he sequece {} k=, =,,... I is clear ha X(ω ) = ad xdg(x,ω )=. I should be clear ha oe of he ime average values X(w), exceppossiblyoe,isequaloμ. So relaio (5) is rue for every realizaio of he process. O he oher had, X =(/) k= X k =X is a radom variable for all =, 2,.... Therefore X is a radom variable, while xdg(x)=ex =μ; hus relaio (5) does o hold wih probabiliy oe. To exed our resuls, ha is, relaio (5), o sochasic seigs a ergodiciy assumpio is eeded; more precisely, we eed o isure ha he ime average X i relaio (5) assumes a cosa value, say, m, wihprobabiliyoe. To illusrae, leheevea = {w Ω : relaio (5) fails}, he A coais he followig realizaios: (i) sample-pahs for which he is i relaio (5) do o exis; (ii) samplepahs for which he is exis bu relaio (5) does o hold (as i Example 5); (iii) sample-pahs for which relaio (5) holds, bu X =m. Relaio (5) holds i a sochasic seig if P(A) =. I coras, i Example 7, P(A) =. Thushefollowig corollary follows: Corollary 8. Le {X()} be a sochasic process defied o a probabiliy space (Ω, F, P). If here exiss a cosa m such ha X=mwih probabiliy oe, he (wih probabiliy oe) Codiio A2 is saisfied iff X <, ad relaio (5) holds. 5. Applicaio o Regeeraive Processes I his secio, we give a simple proof of relaio (5) for regeeraive processes. Cosider a geeral process wih a imbedded poi process. Here we eed oly o cosider he i i codiio A aheimbeddedsequeceofimepois ad he specialize o regeeraive processes ad exed a resul by Wolff [4]. Le {X(), } be ay process wih

9 Ieraioal Sochasic Aalysis 9 {T }, =,,...,T =, be a imbedded poi process. Defie N () = max { : T }, N () (6) λ=, λ (, ). Assume ha iff.leg T (x) be he log-ru frequecydisribuioofhe process {X()} a he imbedded pois T, ad assume ha G(x) = T G T (x) exiss. Usig he relaio Y = λx [], Corollary 5 remais valid whe codiio A is replaced by he followig equivale codiio: T i h (X (u)) { X (u) >} du =, i= T i (6) T =. Now, we shif aeio o sochasic seigs ad assume ha {X(), } is a sochasic regeeraive process defied o a probabiliy space (Ω, F,P).Lehesequece{Y ()}, where Y () = T h(x(u)){ X(u) > }du, besuchha T EY () = EY () < for all =,,.... Thehe codiio E(Y ()) = (62) is sufficie for relaio (5) o hold wih probabiliy oe. Noe ha λ=/et wih probabiliy oe. Moreover he followig resulcabeeasilyderived. Theorem 9. Le {X(), } be a regeeraive sochasic process wih regeeraio pois T,T 2,... such ha E(T )<,ade T h(x(u))du < ;hewihprobabiliyoe h (X (u)) du = EX, (63) where X has disribuio fucio G( ); ha is, relaio (5) holds wih probabiliy oe. Proof. I follows from he regeeraive processes heory, for example, Theorem B.4 i [2], ha codiio A ca be wrie as h (X (u)) { X (u) >} du T E h (X (u)) { X (u) >} du = =. E[T ] (64) So, we eed oly o verify ha codiio (62) is saisfied. Usig he domiaed covergece heorem [26] ad he hypohesis of he heorem, i follows ha E(Y ()) T =E h (X (u)) { X (u) >} du T =E h (X (u)) { X (u) >} du =. (65) Whe h is a ideiy fucio, Theorem 9 slighly exeds Theorem of Wolff [4, page 92], i he sese ha he absolue value of X() is o eeded i he hypohesis of he heorem. Compeig Ieress The auhor declares ha hey have o compeig ieress. Ackowledgmes The auhor wishes o hak Professor Shaler Sidham, Jr., for helpful commes, i paricular for suggesig Example 5. Refereces []S.SidhamJr.adM.El-Taha, Sample-pahaalysisofprocesses wih imbedded poi processes, Queueig Sysems, vol. 5, o. 3, pp. 3 65, 989. [2] M. El-Taha ad S. Sidham Jr., Sample-Pah Aalysis of Queueig Sysems, Kluwer Academic Publishers, Boso, Mass, USA, 999. [3] M.El-Taha, PASTAadrelaedresuls, iwiley EORMS,Joh Wiley ad Sos, New York, NY, USA, 2. [4] B. Melamed ad W. Whi, O arrivals ha see ime averages, Operaios Research,vol.38,o.,pp.56 72,99. [5] B. Melamed ad W. Whi, O arrivals ha see ime averages: a marigale approach, Applied Probabiliy, vol. 27, o. 2,pp ,99. [6]R.W.Wolff, Poissoarrivalsseeimeaverages, Operaios Research,vol.3,o.2,pp ,982. [7] P. Brémaud, Characerisics of queueig sysems observed a eves ad he coecio bewee sochasic iesiy ad palm probabiliy, Queueig Sysems, vol. 5, o., pp. 99, 989. [8] P. Brémaud, R. Kaurpai, ad R. Mazumdar, Eve ad ime averages: a review, Advaces i Applied Probabiliy,vol.27,pp , 992. [9] P. Billigsley, Weak Covergece, Joh Wiley & Sos, New York, NY, USA, 968. [] R. Serfozo, Semi-saioary processes, Zeischrif für Wahrscheilichkeisheorie ud Verwade Gebiee, vol. 23, o. 2, pp , 972. [] R. F. Serfozo, Fucioal i heorems for sochasic processes based o embedded processes, Advaces i Applied Probabiliy,vol.7,pp.23 39,975. [2] P. Frake, D. Köig, U. Ard, ad V. Schmid, Queues ad Poi Processes, Joh Wiley & Sos, Chicheser, UK, 982. [3] T. Rolski, Saioary Radom Processes Associaed wih Poi Processes, vol. 5 of Lecure Noes i Saisics, Spriger, New York, NY, USA, 98. [4] R. W. Wolff, Sochasic Modelig ad he Theory of Queues, Preice Hall Ieraioal Series i Idusrial ad Sysems Egieerig, Preice Hall, Eglewood Cliffs, NJ, USA, 989. [5] R. Serfozo ad S. Sidham Jr., Semi-saioary clearig processes, Sochasic Processes ad Their Applicaios, vol. 6, o. 2, pp.65 78,978. [6] J. Sidham Jr., Regeeraive processes i he heory of queues, wih applicaios o he aleraig-prioriy queue, Advaces i Applied Probabiliy, vol. 4, pp , 972.

10 Ieraioal Sochasic Aalysis [7] M. Brow ad S. M. Ross, Asympoic properies of cumulaive processes, SIAM Joural o Applied Mahemaics, vol. 22, o., pp. 93 5, 972. [8] D. R. Miller, Exisece of is i regeeraive processes, Aals of Mahemaical Saisics,vol.43,pp ,972. [9] H. Toyoizumi, Sample pah aalysis of coribuio ad reward i cooperaive groups, JouralofTheoreicalBiology, vol. 256, o. 3, pp. 3 34, 29. [2] E. Orsigher, C. Ricciui, ad B. Toaldo, Populaio models a sochasic imes, Advaces i Applied Probabiliy, vol. 48, o. 2, pp , 26. [2] S. Sidham Jr., A las word o L=λW, Operaios Research, vol.22,o.2,pp.47 42,974. [22] S. Sidham Jr., Sample-pah aalysis of queues, i Applied Probabiliy ad Compuer Sciece: The ierface, R. Disey ad T. O, Eds., pp. 4 7, Spriger, Berli, Germay, 982. [23] E. Gelebe, Saioary deermiisic flows i discree sysems. I, Theoreical Compuer Sciece,vol.23,o.2,pp.7 27,983. [24] E. Gelebe ad D. Fikel, Saioary deermiisic flows: II. The sigle-server queue, Theoreical Compuer Sciece,vol.52, o. 3, pp , 987. [25] J.-P. Haddad, R. R. Mazumdar, ad F. J. Piera, Pahwise compariso resuls for sochasic fluid eworks, Queueig Sysems: Theory ad Applicaios, vol.66,o.2,pp.55 68, 2. [26] P. Billigsley, Probabiliy ad Measure,JohWiley&Sos,New York, NY, USA, 2d ediio, 985. [27] K. L. Chug, ACourseiProbabiliyTheory, Academic Press, New York, NY, USA, 2d ediio, 974. [28] U. Ayesa, A uifyig coservaio law for sigle-server queues, Applied Probabiliy,vol.44,o.4,pp.78 87, 27. [29] D. P. Heyma ad M. J. Sobel, Sochasic Models i Operaios Research, vol.ofmcgraw-hill Series i Quaiaive Mehods for Maageme,McGraw-Hill,NewYork,NY,USA,982.

11 Advaces i Operaios Research Volume 24 Advaces i Decisio Scieces Volume 24 Applied Mahemaics Algebra Volume 24 Probabiliy ad Saisics Volume 24 The Scieific World Joural Volume 24 Ieraioal Differeial Equaios Volume 24 Volume 24 Submi your mauscrips a Ieraioal Advaces i Combiaorics Mahemaical Physics Volume 24 Complex Aalysis Volume 24 Ieraioal Mahemaics ad Mahemaical Scieces Mahemaical Problems i Egieerig Mahemaics Volume 24 Volume 24 Volume 24 Volume 24 Discree Mahemaics Volume 24 Discree Dyamics i Naure ad Sociey Fucio Spaces Absrac ad Applied Aalysis Volume 24 Volume 24 Volume 24 Ieraioal Sochasic Aalysis Opimizaio Volume 24 Volume 24