Existence of an infinite particle limit of stochastic ranking process

Transcription

1 Exstence of an nfnte partcle lmt of stochastc rankng process Kumko Hattor Tetsuya Hattor February 8, 23 arxv:84.32v2 [math.pr] 25 Feb 29 ABSTRAT We study a stochastc partcle system whch models the tme evoluton of the rankng of books by onlne bookstores (e.g., Amazon. In ths system, partcles are lned n a queue. Each partcle jumps at random jump tmes to the top of the queue, and otherwse stays n the queue, beng pushed toward the tal every tme another partcle jumps to the top. In an nfnte partcle lmt, the random moton of each partcle between ts jumps converges to a determnstc trajectory. (Ths trajectory s actually observed n the rankng data on web stes. We prove that the (random emprcal dstrbuton of ths partcle system converges to a determnstc space-tme dependent dstrbuton. A core of the proof s the law of large numbers for dependent random varables. Key words: stochastc rankng process; hydrodynamc lmt; dependent random varables; law of large numbers MS2 Subject lassfcatons: 8222, 6K35, 9B2 orrespondng author: Tetsuya Hattor, hattor@math.tohoku.ac.jp Mathematcal Insttute, Graduate School of Scence, Tohoku Unversty, Senda , Japan tel+fax: Department of Mathematcs and Informaton Scences, Tokyo Metropoltan Unversty, Hachoj, Tokyo , Japan. emal: khattor@tmu.ac.jp Mathematcal Insttute, Graduate School of Scence, Tohoku Unversty, Senda , Japan. URL: hattor/amazone.htm emal: hattor@math.tohoku.ac.jp

2 Introducton.. Defntons. Let (Ω, B,P be a probablty space, and on ths probablty space we consder a stochastc rankng process {X ( (t t, =,2,,} of partcles, where X ( (t s the rankng of partcle at tme t, defned as follows. For each assume that x (, {,2,,} and w( > are gven. x (, s the ntal value of the rankng of the partcle ; x (, = X( (. We requre that x (,, =,2,,, are dfferent numbers, or n other words, x (, jump rate of the partcle. For each let τ ( {τ (,j and {τ (,j+ τ(,j, =,2,,, s a permutaton of,2,,. w( s the,j, j =,,2,, be an ncreasng sequence of random jump tmes, such that j =,,2, }, =,2,,, are ndependent (ndependence among partcles, τ (, = j =,,2, } are..d. wth the law of τ ( = τ (, beng P[ τ ( t ] = e w( t, t. ( ote that wth probablty, τ (,j, j =,,2,, s strctly ncreasng, and that τ (,j any dfferent par of suffces (,j (,j. For each =,2,, we defne the tme evoluton of X ( by, τ (,j X ( (t = x (, + { {,2,,} x (, > x(,, τ(, t}, t < τ(,, (2 where A denotes the number of elements n the set A, wth =, and for each j =,2,3, X ( (τ ( X (,j =, and (t = { {,2,,} j Z + ; τ (,j < τ (,j t}, τ (,j for < t < τ (,j+. (3 Intutvely speakng, the defnton says that partcle jumps at random tmes τ,j to the top of the queue, and that after the jump t s pushed toward the tal every tme another partcle of larger rankng number jumps to the top. For example, let = 4 and let the ntal rankng be x (4, = 2, x(4 2, = 3, x(4 3, =, x(4 4, = 4. In other words, partcles 4 are ntally algned as 324. For a sample ω such that the confguraton evolves as < τ, (ω < τ 2, (ω < τ 4, (ω < τ,2 (ω <, , where the changes occur at each jump tmes τ,j (ω. The stochastc rankng process may be vewed as a mathematcal model of the tme evoluton of rankngs such as that of books on the onlne bookstores web (e.g., In ths example, stands for the total number of books, represents a specfc ttle of a book, w ( s the average rate wth whch the book s sold, x (, s the ntal poston (rankng of the book, τ (,j s the random tme at whch the book s sold for the j-th tme, and X ( (t s the rankng of the book at tme t. In the tme nterval (τ (,j,τ (,j+ the rankng X( (t ncreases by every tme one of the books n the tal sde of the rankng (.e., wth larger X ( followng. Proposton (2 and (3 are equvalent to the followng: For each, (t s sold. In other words we have the

3 ( X ( (τ (,j =, j =,2,, ( for each and j =,2,, f X ( (τ (,j X ( (τ (,j +, < X ( (τ (,j then X ( (τ (,j = ( otherwse X ( (t s constant n t. As seen n Proposton, each partcle jumps at random tmes to rank, and gradually moves to the rght (ncreasng number wthout outpacng any other partcles on ts rght. Ths mples that for each t there s a boundary poston x ( (t {,,, } such that all the partcles on the left sde have experenced a jump, and that none of the partcles on the rght has jumped by tme t: X ( X ( (t < x ( (t x ( (t τ( t, (t τ( > t. x ( (t s a random varable and s explctly wrtten as: x ( (t = = χ τ ( t. (4 Put y ( (t = x( (t = = χ τ ( t. (5.2 Motvaton. We are nterested n the large lmt of the stochastc rankng process. otng that ( mples E[ χ ( τ ] = P[ τ( t followng: t ] = e w( t, the weak law of large numbers easly leads to the Proposton 2 Assume that the emprcal dstrbuton of jump rates converges to a probablty dstrbuton λ: λ ( (dw = δ(w w ( dw λ(dw,. (6 Then where = lm y( (t = y (t, n probablty, y (t = e wt λ(dw, t. (7 [, In the case of the onlne bookstore Amazon.com, the rankngs seem to be defned n a more nvolved way, but the trajectores of rankngs as predcted by Proposton 2 are actually observed at Amazon.co.jp [3, 4]. As may be seen from ths example, the stochastc rankng process would be of ncreasng practcal nterest and sgnfcance n ths age of onlne retals and web 2. []. In ths paper we go further and prove that n the nfnte partcle lmt, the random emprcal dstrbuton of the partcle system converges to a determnstc space-tme dependent dstrbuton. To consder the lmt, t s natural to use the spacally scaled varables: Y ( (t = (X( (t. (8

4 Y ( (t denotes the spacally scaled poston of the partcle at tme t, takng values n [, Z. In the followng, we wll use Y ( for the ntal confguratons nstead of x (,. (t nstead of X ( (t. orrespondngly, we wll use y (, = (x(, [, Z, Theorem 3 (Theorem 6 Under the assumptons on the ntal confguratons n Secton 2., the jont random emprcal dstrbuton of jump rates (partcle types and postons assocated wth the stochastc rankng process {Y ( } converges as to a dstrbuton (wth determnstc tme evoluton. See Theorem 6 n Secton 2.2 for a precse form of the statement and an explct form of the lmt dstrbuton. In the rankng of books, each tme a book s sold ts rankng jumps to, no matter how unpopular the book may be. At frst thought one mght guess that such a nave rankng wll not be a good ndex for the popularty of books. But thnkng more carefully, one notces that the well sold books (partcles wth large w (, n our defnton are domnant near the top poston, whle books near the tal poston are rarely sold. Hence, though the rankngs of each book s stochastc and has sudden jumps, the spacal dstrbuton of jump rates are more stable, wth the rato of books wth large jump rate hgh near the top poston and low near the tal poston. Seen from the bookstore s sde, t s not a specfc book that really matters, but a totalty of book sales that counts, so the evoluton of dstrbuton of jump rate should be mportant. An ntutve meanng of the Theorem s that we can make ths ntuton rgorous and precse, wth an explct form of the dstrbuton for large lmt. The lmt n the Theorem s mathematcally non-trval n that t nvolves the law of large numbers for dependent varables. Dependence occur because, for each partcle, the tme evoluton between the jump tmes τ ( s a trajectory of a flow caused by the jumps of other partcles n the tal sde of the rankng, and the condtonng on tal sde nduces dependence of stochastc varables. The dea of consderng such a lmt theorem s mathematcally motvated by the celebrated theory of hydrodynamc lmts [5, 6, 7, 8], although the dynamcs (relaxaton to equlbrum and hence the proofs n Secton 3 apparently have lttle n common (and are smpler. A dfference les n that the theory of hydrodynamc lmts (among other thngs evaluate the relaxaton to equlbrum (nvarant measures through entropy and large devaton arguments va local equlbrum, whle the dynamcs of the stochastc rankng process has a specal feature that the queue of the partcles conssts of the tal regme and the head regme, such that the former s the queue of books whch has not been sold up to tme t, and havng no dynamcs for relaxaton, keeps the remnant of ntal data (Secton 3.3. In contrast, n the head regme the statonarty s reached from the begnnng (Secton 3.2. It may also be worthwhle to note that our lmt dstrbutons, unlke the hydrodynamc lmts whch satsfy dffuson equatons, satsfy non-local feld equatons (see remarks to Theorem 6 n Secton 2. In Secton 2 we state our man theorem and n Secton 3 we gve a proof. Acknowledgements. We thank the referee and the assocate edtor and Prof. K. Ishge for constructve dscussons, whch (among other thngs helped refnng the assumptons for the man results. The research of K. Hattor s supported n part by a Grant-n-Ad for Scentfc Research ( 654 from the Mnstry of Educaton, ulture, Sports, Scence and Technology, and the research of T. Hattor s supported n part by a Grant-n-Ad for Scentfc Research (B from the Mnstry of Educaton, ulture, Sports, Scence and Technology.

5 2 Man result. 2. Assumptons on ntal confguraton. We consder the lmt of the emprcal dstrbuton on the product space of jump rate and spacal poston R + [,, µ ( t (dw,dy = δ(w w ( δ(y Y ( (tdw dy. (9 We mpose that the ntal dstrbuton µ ( (dw,dy = δ(w w ( δ(y y (, dw dy ( converges as unformly n y to a probablty dstrbuton whch s absolutely contnuous wth respect to the Lebesgue measure n y. (For the set of probablty measures on R + [, we work wth the topology of weak convergence. Explctly, we assume that for each y < there exsts a probablty measure µ y, (dw on R + such that for any bounded contnuous functon g : R + R, the quantty lm sup y [, = g(wµ y, (dw s Lebesgue measurable n y and y ( χ ( y, y g(wµ z, (dw dz =, ( where (and also n the followng we use a notaton χ A whch s f A s true and f A s false. That each µ y, (dw s a probablty measure on R + s consstent wth (, whch may be seen by lettng g(w =, w, n ( to fnd µ y, (R + =, y <. (2 ote that λ ( and λ n (6 are the margnal dstrbuton of µ ( y, and µ y, of the jump rate; λ ( (dw = λ(dw = = µ y, (dwdy. δ(w w ( dw = µ ( (dw,[,, ote also that ( and Fubn s Theorem mply (6. We assume that the average of λ s fnte, wλ(dw <, (4 and Ths complets the assumptons for our man results. (3 λ({} =. (5 Remarks. ( The assumpton (4 assures that µ,t s well-defned (see (24. The man results on the exstence of the nfnte partcle lmt wll hold wthout (4 for y >, but we keep ths assumpton to nclude y =. ( We assume (5 to assure that y : [, [, defned n (7 s onto (see Proposton 4. The basc results n ths paper wll hold (wth extra complexty n notatons and arguments wthout (5, but we prefer to keep notatons and arguments smple by keepng ths assumpton. (5 mples n the actual bookstore rankng, that all the books sell (almost surely.

6 2.2 Man theorem. Wth (5, t s straghtforward to show Proposton 4 Assume (5. Then y : [, [, defned n (7 s a contnuous, strctly ncreasng, bjectve functon of t. Proposton 4 mples the exstence of the nverse functon t : [, [,, satsfyng y (t (y = y, y <, (6 or y = Dfferentatng (7 and (6, we have e wt (y λ(dw. (7 dy dt (t = we wt λ(dw =. (8 dt dy (y (t We generalze (7 and defne (wth slght abuse of notatons y (y,t = y e wt µ z, (dwdz, t, y <. (9 In partcular, y (t = y (,t. In the nfnte partcle lmt, y (y,t denotes the poston of a partcle at tme t (f t does not jump up to tme t whose ntal poston s y (Proposton 8. Proposton 5 y (,t : [, [y (t, s a contnuous, strctly ncreasng, bjectve functon of y. Proof. It s straghtforward from the defnton of y (y,t n (9 to see that y (,t s contnuous and non-decreasng n y. To see that t s strctly ncreasng, let z 2 < z <. Then (9 mples y (z,t y (z 2,t = z M >, for a.e. z [z 2,z ], whch contradcts (2. z 2 e wt µ z, (dwdz. If ths s, then µ z, ([,M] = for any Proposton 5 mples that the nverse functon ŷ(,t : [y (t, [, exsts: y = ŷ(y,t e wt µ z, (dwdz, t, y (t y <. (2 ŷ(y,t denotes the ntal poston of a partcle located at y (> y (t at tme t. It holds that ow we return to our -partcle process. ŷ y (y,t = e wt µŷ(y,t, (dw. (2 Theorem 6 onsder the stochastc rankng process {Y ( } defned by ( and (8. Assume ( (4 and (5. Then the jont emprcal dstrbuton of partcle types and postons at tme t µ ( t (dw,dy = δ(w w ( δ(y Y ( (t dw dy (22

7 converges as to a dstrbuton µ y,t (dwdy on R + [,; for any bounded contnuous functon f : R + [, R lm f(w (,Y ( (y = ( f(w,yµ y,t (dw dy, n probablty. (23 The lmt dstrbuton s absolutely contnuous wth respect to the Lebesgue measure on [,. The densty µ y,t (dw wth regard to y s gven by µ y,t (dw = we wt(y λ(dw, y < y (t, we wt(y λ(dw e wt µŷ(y,t, (dw, y > y (t. e wt µŷ(y,t, (dw Remarks. ( (22 and (3 mply λ ( ( = µ ( t (,[,. Moreover, f n (23 we take f wthout y dependence and use (6, we have as a generalzaton of (3 λ = (24 µ y,t dy. (25 ( Our results state that a random phenomenon approaches a determnstc one as the partcle number s ncreased. We state the results n terms of convergence n probablty, but snce the lmt quantty s determnstc, ths lmt s equvalent to convergence n law. ( The explct forms n (24 dffer drastcally for y > y (t and y < y (t. As we have ponted out n the Introducton, and also as we wll see n the proofs n the next secton, the dynamcs for large y and small y are totally dfferent. (v By drect calculatons, one sees that µ y,t (dw satsfes the followng (non-local equatons: µ y,t (dw t + (v(y,tµ y,t(dw y = wµ y,t (dw, (26 where v(y,t = y ( w µ z,t (dw dz = y t (t (y, y < y (t, y t (ŷ(y,t,t, y > y (t. The partal dfferental equaton (26 can be seen as the equatons of contnuty (conservaton of mass for the one-dmensonal ncompressble mxed fluds, wth w standng for the rate of evaporaton of specfc type of flud n the mxture [3]. v(y,t s the velocty of the flud at poston y and tme t, and (27, the source of non-localty, means that the flow s drven by evaporaton. Intutvely, the equatons are natural classcal lmt of the stochastc processes consdered n ths paper. In fact, we can drectly prove (wthout referrng to stochastc processes that (24 s the unque classcal soluton to the auchy problem of partal dfferental equaton (26 wth sutable boundary condtons. See [3] for detals. (27

8 3 Proof of Theorem 6. It s suffcent to consder the case that f : R + [, R n (23 s expressed as f(w,z = g(wχ z [,y], wth a bounded contnuous functon g : R + R and < y <. Thus we prove n ths secton lm y χ ( = Y dz (t y for any bounded contnuous functon g : R + R. 3. ase y = y (t. g(wµ z,t (dw, n probablty, (28 Lemma 7 For each t > and each bounded contnuous functon g : R + R, t holds that and lm lm χ Y ( χ ( Y (t y ( = (t (t y ( (t = lm g(w( e wt λ(dw, n probablty, (29 χ ( Y (t y (t, n probablty. (3 Proof. Snce we have from (5 and the defnton of Y ( (t we have from ( χ Y ( (t y ( (t = χ τ ( t, (3 E[ χ ( Y (t y ( ] = g(w( (t ( e w( t. Snce g s bounded, n a smlar manner as the proof of Proposton 2, we apply the weak law of large numbers to obtan lm ( χ Y ( (t y ( (t ( e w( t =, n probablty. Wth (6 we have (29. ext, Proposton 2 mples that, for any ǫ >, wth large enough, t holds that P[ y (t y ( (t ǫ ] > ǫ. For such, notng that χ Y ( (t y ( (t χ Y ( the probablty of the event (t y (t = χ Y ( sup g(w w k= (t y ( (t χ Y ( (t y ( χ ( k y (t χ k y (t y ( (t y (t +, (t χ Y ( χ ( Y (t y (t (t y (t sup w g(w (ǫ + s larger than ǫ, thus the left-hand sde converges to n probablty as, whch mples (3.

9 3.2 ase y < y (t. Frst note that, to prove (28 for y < y (t, t s suffcent to prove that for each bounded contnuous functon g : R + R, lm χ ( Y (t y = g(w( e wt (y λ(dw, n probablty, (32 where t (y s as n (6. To see that (32 mples (28, dfferentate the rght-hand sde of (32 wth respect to y, use (8 and (6, and ntegrate from to y, keepng n mnd t ( =, and fnally rewrte usng µ y,t (dw n (24, to obtan lm = y y χ ( Y (t y = g(wµ z,t (dwdz, n probablty, whch gves (28. To prove (32, fx y < y (t and let t = t (y < t. Denote by {Ỹ ( (s, τ ( by the amount t t >. amely, let Ỹ ( g(wwe wt(z λ(dw dz we wt(z λ(dw,j } the scaled stochastc rankng process wth the tme orgn shfted (s = Y ( (s + t t. In partcular, we have χ ( = Y (t y χỹ (. (33 (t y For j =,,2,, defne τ (.j by τ (, = and where Put, n analogy to (5, τ (.j = τ (, j(,t t +j (t t, (34 j(,t t = nf{j τ (,j > t t }. (35 ỹ ( (s = = χ τ ( s. ote that {τ (,j+ τ(,j j =,,2, } are ndependent and have exponental dstrbutons. The loss of memory property of exponental dstrbutons then mples that { τ (,j } have the same dstrbutons as {τ (,j }, and {Ỹ ( (s} s a scaled stochastc rankng process wth jump tmes { τ (,j } and ntal confguraton {Y ( (t t }. Snce y = y (t, (3 for the tme shfted rankng process mples lm = lm χỹ ( = lm (t y χỹ ( (t y (t χỹ ( (t ỹ ( (t, n probablty. (36

10 Usng (3 for the orgnal process and the tme shfted process, and recallng that {τ ( } and { τ ( } have the same dstrbuton, and then usng (29, we arrve at lm = lm = χỹ ( (t ỹ ( g(w( e wt λ(dw, (33 (36 and (37 together mply ( ase y > y (t. χ ( τ = lm t (t = lm n probablty. ( χ τ t χ ( Y (t y ( (t Frst we make some preparatons for the proof. We generalze y ( (t n (5 and defne (wth slght abuse of notatons for y < y ( (y,t = y + = χ τ ( (37 χ t y (, y. (38 Proposton 8 For y < and t, lm y( (y,t = y (y,t, n probablty. (39 amely, the (random poston y ( (y,t of a partcle at tme t whose ntal poston s y converges n probablty to a determnstc trajectory y (y,t defned by (9 n the nfnte partcle lmt. Remark. The proof below shows that the convergence n (39 holds unformly n y. Proof. (38 and ( and the ndependence of {τ ( } mply ( E[ y (y,t y ( 2 (y,t ] ( = y (y,t 2 2y (y,t + y 2 + 2y = 2 ( + 2 = j= = y + = = y + ( e w( ( e w( ( e w( = t χ y (, y t ( e w( t ( e w( ( e w( j t χ y (, y t χ y (, yχ y ( j, y t 2 χ y ( ( e w( t χ ( y y (y,t, y = ( e w( t ( e w(, y 2 t 2 χ y (, y.

11 The second term n the rght-hand sde of the equaton above vanshes n the lmt because of the factor 2 n the denomnator. oncernng the frst term, as n the proof of Proposton 2 and Lemma 7, ( mples that lm = y ( e w( t χ ( y = y =, y y e wt µ z, (dwdz, ( e wt µ z, (dwdz unformly n y. Ths combned wth (9 mples that the frst term also vanshes. Thus we have ( lm E[ y (y,t y ( 2 (y,t ] =. Wth hebyshev s nequalty follows (39. As an equvalent statement to (28, we wll prove lm χ ( = Y (t>y y dz g(wµ z,t (dw, n probablty. (4 Let Ω ( = {y > y ( (t}. Then Proposton 2 mples that f y > y (t, For t > and y > y ( (t, let lm P[ Ω( ] =. (4 ŷ ( (y,t = nf{y (, =,,, Y ( (t > y}. (42 ote that Ths follows because hence wth y > y ( (t Y ( y y ( (ŷ( (y,t,t 2. (43 (t = y ( (y(,,t, f Y ( (t > y ( (t, (44 y y( (ŷ( (y,t,t Y ( (t + for all wth Y ( (t > y. Untl a partcle jumps to the top of the queue, changes of ts poston are caused only by the jumps of other partcles that st on ts rght (Proposton, hence χ ( Y (t>y = χ ( y, ŷ( (y,t χ τ (, on Ω( >t. (45 ote that ŷ ( (y,t depends on τ s. Ths means that the summands on the rght-hand sde are not ndependent random varables, and that we can not apply the law of large numbers as t s. In contrast, snce ŷ(y,t s determnstc, the law of large numbers yelds, as n the proofs of Proposton 2 and Lemma 7, lm χ ( y ŷ(y,tχ, τ ( >t = ŷ(y,t g(we wt µ z, (dwdz, n probablty. (46

12 The rght-hand sde concdes wth that of (4 through a change of varables ŷ(z,t z (note (2. ombnng (4, (4, (45 and (46, we see that t s suffcent to prove that lm P[ y > y( (t, χ y (, ŷ(y,t χ y (, ŷ( (y,t χ τ ( >t ǫ ] = (47 holds for any ǫ (,. ote that f y > y ( (t and ŷ( (y,t ŷ(y,t ǫ, then χ y (, ŷ(y,t χ y (, ŷ( (y,t χ τ >t M χ y (, ŷ(y,t χ y ( M (ǫ +,, ŷ( (y,t where M > s a constant satsfyng g(w M for all w. Thus to prove (47, t s suffcent to show lm P[ y > y( (t, ŷ( (y,t ŷ(y,t > ǫ ] =, for an arbtrary ǫ >. Let us assume otherwse, that s, wth (4 n mnd, assume that there s ǫ >, ρ > and an ncreasng sequence of postve ntegers { } such that or P[ y > y ( (t, ŷ( (y,t ŷ(y,t > ǫ ] > ρ (48 P[ y > y ( (t, ŷ( (y,t ŷ(y,t < ǫ ] > ρ holds. We consder the case (48, snce the second case s dealt wth smlarly. Let y = ŷ(y,t. Wth Proposton 5, we have y = y (y,t < y (y + ǫ,t. Let ǫ 2 = y (y + ǫ,t y >. Proposton 8 mples P[ y ( (y + ǫ,t y (y + ǫ,t ǫ 2 4 ] ρ 2 holds for suffcently large. ombnng (48, y = ŷ(y,t and (49, we have (49 P[ y > y ( (t, y + ǫ < ŷ ( (y,t, y ( (y + ǫ,t y (y + ǫ,t < ǫ 2 4 ] > ρ 2 > for suffcently large. ote that (38 mples that f y y then y ( (y,t y( (y,t +. Ths combned wth (43 mples that f y > y ( (t and y + ǫ < ŷ ( (y,t, then y ( (y + ǫ,t < y + ǫ , (5 for suffcently large. On the other hand, f y ( (y + ǫ,t y (y + ǫ,t < ǫ 2 4, then But (5 and (5 put together mply y + ǫ 2 = y (y + ǫ,t < y ( (y + ǫ,t + ǫ 2 4. (5 y + ǫ 2 < y + ǫ , whch s a contradcton for large. Thus the assumpton (48 s false, whch completes the proof of Theorem 6.

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