Endogeny for the Logistic Recursive Distributional Equation

Transcription

1 Zeitschrift für Analysis und ihre Anendungen Journal for Analysis and its Applications Volume 30 20, DOI: 0.47/ZAA/433 c European Mathematical Society Endogeny for the Logistic Recursive Distributional Equation Antar Bandyopadhyay Abstract. In this article e prove the endogeny and bivariate uniqueness property for a particular max-type recursive distributional equation RDE. The RDE e consider is the so called logistic RDE, hich appears in the proof of the ζ2-limit of the random assignment problem using the local eak convergence method proved by D. Aldous [Probab. Theory Related Fields , ]. This article provides a non-trivial application of the general theory developed by D. Aldous and A. Bandyopadhyay [Ann. Appl. Probab , 047 0]. The proofs involves analytic arguments, hich illustrate the need to develop more analytic tools for studying such max-type RDEs. Keyords. Bivariate uniqueness, endogeny, logistic distribution, random assignment problem, recursive distributional equations, recursive tree processes Mathematics Subject Classification Primary 60E05, secondary 60K35, 60J80. Introduction and the main result Fixed-point equations have found many applications in various fields of mathematics. In probability theory they are usually referred as distributional identities. A recent article by Aldous and Bandyopadhyay [5] provides a general frameork to study certain type of distributional identities hich arise in a variety of settings. Given a measurable space S, S rite P S for the set of all probabilities on S. According to [5], a recursive distributional equation RDE is a fixed-point equation on P S defined as X d = g ξ; X j : j N on S, A. Bandyopadhyay: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 7 S. J. S. Sansanal Marg, Ne Delhi 006, India; antar@isid.ac.in

2 238 A. Bandyopadhyay here it is assumed that X j j are S-valued random variables ith identical distribution and are stochastically independent, and their common distribution is same as that of the random variable X, and are independent of the pair ξ, N. Here ξ is a measurable function taking value on some measurable space, say Λ, Z, N is a non-negative integer valued random variable, hich may take the value and g is a given S-valued function. In the above equation by N e mean the left hand side is N if N <, and < N otherise. In the distribution of X is unknon, hile the distribution of the pair ξ, N and the function g are assumed to be knon quantities. Perhaps a more conventional analytic ay of riting the equation ould be here T : P S P S is a function defined as µ = T µ, 2 T µ := dist g ξ; X j : j N here ξ, N is as above and X j j are independent and identically distributed i.i.d. ith common distribution µ P S and these sets of random variables are stochastically independent. As outlined in [5] in many applications RDEs play a very crucial role. Examples include study of Galton-Watson branching processes and related random trees, probabilistic analysis of algorithms ith suitable recursive structure [, 5, 6], statistical physics models on trees [2, 3, 6, 2], and statistical physics and algorithmic questions in the mean-field model of distance [,2,4,7,8,9]. In many of these applications, particularly in the last to types mentioned above, often one needs to construct a particular tree indexed stationary process related to a given RDE, hich is called a recursive tree process RTP [5]. More precisely, suppose the RDE has a solution, say µ. Then as shon in [5], using the consistency theorem of Kolmogorov [9], one can construct a process, say X i i V, indexed by V := d N d { }, such that i X i µ i V ii for each d 0, X i i =d are independent iii X i = g ξ i ; X ij : j N i i V iv X i is independent of { ξ i, N i i < i } i V, 3 here ξ i, N i i V are taken to be i.i.d. copies of the pair ξ, N, and by e mean the length of a finite ord. The process X i i V is called an invariant recursive tree process RTP ith marginal µ. The i.i.d. random variables ξ i, N i i V are called the innovation process. In some sense, an invariant RTP ith marginal µ is an almost sure representation of a solution µ of the RDE. Here e note that there is a natural tree structure on V. Taking V as the vertex set, e join to ords i, i V by an edge, if and only if, i = ij or i = i j, for

3 Endogeny for the Logistic RDE 239 some j N. We ill denote this tree by T. The empty-ord ill be taken as the root of the tree T. For simplicity e ill rite j = j for j N. In the applications mentioned above the variables X i i V of a RTP are often used as auxiliary variables to define or to construct some useful random structures. In those cases typically the innovation process defines the internal variables hile the RTP is constructed externally using the consistency theorem. It is then natural to ask hether the RTP is measurable only ith respect to the i.i.d. innovation process ξ i, N i. Definition. An invariant RTP ith marginal µ is called endogenous, if the root variable X is almost surely measurable ith respect to the completion of the σ-algebra G := σ { ξ i, N i i V }. This notion of endogeny has been the main topic of discussion in [5]. The authors provide a necessary and sufficient condition for endogeny in the general setup [5, Theorem ]. Some other concepts similar to endogeny can be found in [6]. In this article e provide a non-trivial application of the theory developed in [5]. The example e consider here arise from the study of the asymptotic limit of random assignment problem using local-eak convergence method [4]. A detailed background of this example is given in Section 2... Main result. The folloing RDE plays the central role in deriving the asymptotic limit of the random assignment problem [4], X d = min j ξ j X j on R, 4 here X j j are i.i.d. ith same la as X and are independent of ξ j j hich are points of a Poisson point process of rate on 0,, that is, ξ ξ 2 ξ k a.s. and the collection ξ i ξ i i are i.i.d. random variables ith Exponential distribution. Here e rite ξ 0 0. It is knon [4] that the RDE 4 has a unique solution as the logistic distribution, given by P X x =, x R. 5 + e For this reason e ill call RDE 4 the logistic RDE. The folloing is our main result. Theorem. The invariant recursive tree process ith logistic marginals associated ith the RDE 4 is endogenous. This result though looks technical, but provides a concrete example falling under the general theory developed in [5]. The proof of Theorem involves analytic techniques, thus this ork also demonstrate the need of developing analytic tools for studying max-type RDEs in general.

4 240 A. Bandyopadhyay.2. Outline of rest of the paper. The next section provides the background and motivation for deriving our main result. In Sections 3 e prove the bivariate uniqueness property of the logistic RDE 4 and finally Section 4 gives the proof the main result. 2. Background and motivation for logistic RDE For a given n n matrix of costs C ij, consider the problem of assigning n jobs to n machines in the most cost effective ay. Thus the task is to find a permutation π of {, 2,..., n}, hich solves the folloing minimization problem A n := min π n C i,πi. 6 This problem has been extensively studied in literature for a fixed cost matrix, and there are various algorithms to find the optimal permutation π. A probabilistic model for the assignment problem can be obtained by assuming that the costs are independent random variables each ith Uniform[0, ] distribution. Although this model appears to be quite simple, careful investigations of it in the last fe decades have shon that it has enormous richness in its structure. See [2, 8] for survey and other related orks. Our interest in this problem is from another perspective. In 200, Aldous [4] shoed lim E[A n] = ζ2 = π2 n 6, 7 confirming the earlier ork of Mézard and Parisi [3], here they computed the same limit using non-rigorous arguments based on the replica method [4]. It is also knon that the limit exists and it is universal for any i.i.d. cost distribution ith density f such that f0 = see []. So for calculation of the limiting constant one can assume that C ij s are i.i.d. ith Exponential distribution ith mean n and then rite the objective function A n in the normalized form, i= A n = d min π n n C i,πi. 8 i= In [4], the limit ζ2 as identified in terms of an optimal matching problem on an infinite tree ith random edge eights, described as follos: Let T := V, E be the canonical infinite rooted labeled tree, as before, here is the root. For every vertex i V, let ξ ij j be points of a Poisson point process of rate on 0,, and they are independent as i varies. Define the eight of the edge e = i, ij E as ξ ij.

5 Endogeny for the Logistic RDE 24 This structure is called Poisson eighted infinite tree and henceforth abbreviated as PWIT. Let K r n,n be the complete bipartite graph on 2n vertices ith a root selected uniformly at random. Suppose e also equip it ith i.i.d. Exponential edge eights ith mean n. Then one can sho [2,4] that in the sense of Aldous-Steel local eak convergence K r n,n converges to the PWIT. Moreover heuristically the random assignment problem on K r n,n has a natural analog to the limit structure, hich is to consider the optimal in sense of minimizing the total cost matching problem on PWIT. Naturally PWIT being an infinite graph ith edge eights each having mean at least, the total cost of any matching is infinite a.s., and hence minimization total cost is not quite meaningful. Hoever, Aldous [4] shoed that it is possible to make a sensible definition of optimal matching on PWIT hich minimizes the average edge eight. This construction is quite hard, and e refer the readers to [2, 4] for the technical details. Here e only provide the basic essentials to understand the motivation for our ork. Consider the heuristic description of the optimal matching problem on PWIT and suppose e define variables X i for each vertex i as follos X i = Total cost of a maximal matching on the subtree T i Total cost of a maximal matching on the forest T i \ {i}, here T i is the subtree rooted at the vertex i. Here by total cost e mean the sum total of all the edge eights in the matching. As noted above, both the total costs appearing in 9 are infinity almost surely. Thus rigorously speaking X i is not ell defined. But at the heuristic level if e ignore this, then simple manipulation yields that they must satisfy the folloing recurrence relation see [4, Section 4.2] 9 X i = min j ξ ij X ij. 0 This is of course the recurrence relation for a RTP associated ith the logistic RDE 4. Thus one can no construct the X i -variables externally as the RTP associated ith the logistic RDE and then use them to redefine the optimal matching on PWIT. This is the construction given in [4] hich also provides a characterization of the optimal matching on the PWIT. Finally one can then derive the ζ2-limit for the random assignment problem. Once again a natural question ould be to figure out hether the random variables X i s are truly external or not, in other ords to see hether the RTP is endogenous or not see [4, Remarks 4.2.d and 4.2.e]. This is our main motivation for this ork. Theorem proves that the X i -variables can be defined using only the edge-eights. In other ords they have no external randomness in them. Some other significance of this result has also been pointed out in [5, Section 7.5].

6 242 A. Bandyopadhyay 3. Bivariate uniqueness for the logistic RDE In this section e prove the bivariate uniqueness property for the logistic RDE 4. Theorem 2. Consider the folloing bivariate RDE X d minj ξ = j X j Y min j ξ j Y j, here X j, Y j j are i.i.d. pairs ith same joint distribution as X, Y and are independent of ξ j j hich are points of a Poisson process of rate on 0,. Then the unique solution of this RDE is given by a diagonal measure, namely, µ := dist X, X here X has the logistic distribution. 3.. Proof of Theorem 2. First observe that if the equation has a solution then, the marginal distributions of X and Y solve the logistic RDE 4, and hence they are both logistic. Further by inspection µ is a solution of. So it is enough to prove that µ is the only solution of. Let µ 2 be a solution of. No consider the points {ξ j ; X j, Y j j } as a point process, say P on 0, R 2. Since ξ j j is a Poisson point process on 0, of rate and X j, Y j j are i.i.d. random vectors on R 2 ith distribution µ 2 hich are independent of the Poisson process ξ j j, thus P is a Compound Poisson process on 0, R 2 ith mean intensity ρt; x, y dt dx, y := dt µ 2 dx, y see [0, Lemma 6.4.VI]. Thus if Gx, y := P X > x, Y > y, for x, y R, then Gx, y = P min = P = exp ξ j Y j > y j ξ j X j > x, and, min j { No points of P are in t; u, v t u x, or, t v y } = exp { t;u,v 0 t u x, or, t v y } ρt; u, v dt d u, v [ Ht x + Ht y Gt x, t y ] dt = Hx Hy exp Gt x, t y dt, here H is the right tail of logistic distribution, defined as 0 Hx = e + e 2

7 Endogeny for the Logistic RDE 243 for x R. The last equality follos from properties of the logistic distribution see Proposition 3c. For notational convenience in this paper e ill rite F := F for any distribution function F. The folloing simple lemma reduces the bivariate problem to a univariate problem. Lemma. For any to random variables U and V, U = V a.s. if and only if U d = V d = U V. Proof. First of all, if U = V a.s. then U V = U a.s. Conversely suppose that U d = V d = U V. Fix a rational q, then under our assumption, PU q < V = PV > q PU > q, V > q = PV > q PU V > q = 0 A similar calculation ill sho that P V q < U = 0. These are true for any rational q, thus P U V = 0. Thus if e can sho that X Y also has logistic distribution, then from the lemma above e ill be able to conclude that X = Y a.s., and hence the proof ill be complete. Put g := P X Y >, e ill sho g = H. No, for every fixed x R, by definition gx = Gx, x. So using 2 e get gx = H 2 x exp gs ds, x R. 3 Notice that from 26 see Proposition 3c g = H is a solution of this nonlinear integral equation 3, hich corresponds to the solution µ 2 = µ of the original equation. To complete the proof of Theorem 2 e need to sho that this is the only solution. For that e ill prove that the operator associated ith 3 defined on an appropriate space is monotone and has unique fixed-point as H. The techniques e ill use here are similar to Eulerian recursion [7], and are heavily based on analytic arguments. Let F be the set of all functions f : R [0, ] such that H 2 x fx Hx for all x R, f is continuous and non-increasing. Observe that by definition H F. Further from 3 it follos that gx H 2 x, as ell as, gx = P X Y > x P X > x = Hx for all x R. Note also that g being the tail of the random variable X Y, is continuous because both X and Y are continuous random variables and non-increasing. So it is appropriate to search for solutions of 3 in F. Let T : F F be defined as T fx := H 2 x exp fs ds, x R. 4

8 244 A. Bandyopadhyay Note that this operator T is not same as the general operator defined in Section, henceforth by T e ill mean the specific operator defined above. Proposition of Section 3.2 shos that T does indeed map F into itself. Observe that the equation 3 is nothing but the fixed-point equation associated ith the operator T, that is, g = T g on F. 5 We here note that using 26 see Proposition 3c T can also be ritten as T fx := Hx exp Hs fs ds, x R, 6 hich ill be used in the subsequent discussion. Define a partial order on F as, f f 2 in F if f x f 2 x for all x R, then the folloing result holds. Lemma 2. T is a monotone operator on the partially ordered set F,. Proof. Let f f 2 be to elements of F, so from definition f x f 2 x, for all x R. Hence f s ds f 2 s ds x R T f x T f 2 x x R T f T f 2. Put f 0 = H 2, and for n N, define f n F recursively as, f n = T f n. No from Lemma 2 e get that if g is a fixed-point of T in F then, f n g n 0. 7 If e can sho f n H point ise, then using 7 e ill get H g, so from definition of F it ill follo that g = H, and our proof ill be complete. For that, the folloing lemma gives an explicit recursion for the functions {f n } n 0. Lemma 3. Let β 0 s = s, 0 s. Define recursively Then for n, β n s := s β n d, 0 < s. 8 f n x = Hx exp β n Hx, x R. 9

9 Endogeny for the Logistic RDE 245 Proof. We ill prove this by induction on n. Fix x R, for n = e get f x = T f 0 x = Hx exp = Hx exp = Hx exp Hs H2 s ds Hs Hs ds H s ds = Hx exp Hx = Hx exp β 0 Hx [using 6] [using Proposition 3a] No, assume that the assertion of the Lemma is true for n {, 2,..., k}, for some k, then from definition e have f k+ x = T f k x = Hx exp = Hx exp = Hx exp Hs fk s ds Hs β k Hs ds Hx β k d [using 6] 20 The last equality follos by substituting = Hs and thus from parts a and b of Proposition 3 e get that d = Hs ds and H = Hx. Finally by definition of β n s and using 20 e get f k+ = T f k. To complete the proof it is no enough to sho that β n 0 point ise, hich ill imply by Lemma 3 that f n H point ise, as n. Using Proposition 2 see Section 3.2 e get the folloing characterization of the point ise limit of these β n s. Lemma 4. There exists a function L : [0, ] [0, ] ith L = 0, such that Ls = s and Ls = lim n β n s for all 0 s. L d s [0,, 2 Proof. From the Proposition 2 e kno that for any s [0, ] the sequence {β n s} is decreasing, and hence there exists a function L : [0, ] [0, ] such

10 246 A. Bandyopadhyay that Ls = lim n β n s. No observe that β n β 0 = for all 0, and hence 0 β n β n 0. Thus by taking limit as n in 8 and using the dominated convergence theorem along ith part a of Proposition 2 e get that Ls = s L d 0 s <. The above lemma basically translates the non-linear integral equation 3 to the non-linear integral equation 2, here the solution g = H of 3 is given by the solution L 0 of 2. So at first sight this may not lead us to the conclusion. But fortunately, something nice happens for equation 2, and e have the folloing result hich is enough to complete the proof of Theorem 2. Lemma 5. If L : [0, ] [0, ] is a function hich satisfies the non-linear integral equation 2, namely, Ls = and if L = 0, then L 0. s L d 0 s <, Proof. First note that L 0 is a solution. No let L be any solution of 2, then L is infinitely differentiable on the open interval 0,, by repetitive application of Fundamental Theorem of Calculus. Consider η := L + e L, [0, ]. Observe, that η0 = η = 0 as L = 0. No, from 2 e get that L = L, 0,. Thus differentiating the function η e get η = e L [ 2 e L + e L ] 0 0,. 22 So the function η is decreasing in 0, and is continuous in [0, ] ith boundary values as 0, hence η 0. Thus e must have η 0, so from equation 22 e get that e Ls + e Ls = 2 for all s 0,. This implies L 0 on [0, ].

11 Endogeny for the Logistic RDE Some technical details. This section provides some of the technical results hich ere needed in the previous section. Proposition. The operator T maps F into F. Proof. First note that if f F, then by definition T fx H 2 x for all x R. Next by definition of F e get that f F f H, thus fs ds Hs ds x R T fx H 2 x exp Hs ds = Hx x R. The last equality follos from 26 see Proposition 3c. So, H 2 x T fx Hx x R. 23 No e need to sho that for any f F e must have T f continuous and non-increasing. From the definition T f is continuous in fact, infinitely differentiable. Moreover if x y be to real numbers, then Hs fs ds Hs fs ds, y because f H. Also Hx Hy, thus using 6 e get T fx T fy 24 So using 23 and 24 e conclude that T f F if f F. Proposition 2. The folloing are true for the sequence of functions {β n } n 0 defined in 8. a For every fixed s 0, ], the sequence {β n s} is decreasing. b For every n, lim s 0+ β n s exists, and is given by 0 β n d, e ill rite this as β n 0. c The sequence of numbers {β n 0} is also decreasing. Proof. a Notice that β 0 s = s for s [0, ], thus β s = s d < s = β 0 s s 0, ].

12 248 A. Bandyopadhyay No assume that for some n e have β n s β n s β 0 s for all s 0, ], if e sho that β n+ s β n s for all s 0, ] then by induction the proof ill be complete. For that, fix s 0, ] then β n+ s = s β n d s β n d = β n s. This proves the part a. b, c First note that by trivial induction β n s 0 for every s 0, ], n 0. Thus from definition for every n 0, the limit lim s 0+ β n s exists in [0, ] and is given by β n d. No using a above e conclude 0 β n+ 0 = lim s 0+ β n+s lim s 0+ β ns = β n 0, 25 for every n 0. Since β 0 0 =, so e get β n 0 < for all n 0, and the sequence is decreasing. This proves parts b and c. Finally, the folloing proposition lists some basic facts about the logistic distribution function. Proposition 3. Suppose X is a random variable ith logistic distribution and H x := P X x for x R be the distribution function of X. Then a H is infinitely differentiable and H = H H, here H = H. b H is symmetric around 0, that is, H = Hx for all x R. c H is the unique solution of the non-linear integral equation Hx = exp Hs ds x R. 26 Proof. a and b trivially follos from the definition of H. For c notice that the equation 26 is nothing but the Logistic RDE, this is because P min ξ j X j > x = exp Hs ds x R j here X j j are i.i.d. ith distribution function H and are independent of ξ j j, hich are points of a Poisson point process of rate on 0,. Thus from the fact that H is the unique solution of logistic RDE see [4, Lemma 5] e conclude that H is unique solution of equation Proof of Theorem We note that by Theorem 2 the logistic RDE 4 has bivariate uniqueness property. Thus by [5, Theorem b] to prove endogeny for the logistic RDE 4 all remains is to check the folloing technical continuity assumption.

13 Endogeny for the Logistic RDE 249 Proposition 4. Let S be the set of all probabilities on R 2 and let Γ : S S be the operator associated ith the RDE, that is, Γ µ 2 d = minj ξ j X j min j ξ j Y j, 27 here X j, Y j j are i.i.d. ith joint la µ 2 S and are independent of ξ j j hich are points of a Poisson point process of rate on 0,. Then Γ is continuous ith respect to the eak convergence topology hen restricted to the subspace S defined as { } S := µ 2 both the marginals of µ 2 are logistic distribution. 28 Before e prove this proposition, it is orth mentioning that the operator Γ is not continuous ith respect to the eak convergence topology on the hole space S. In fact, it is not difficult to see that the operator Γ is every here discontinuous on S. But fortunately for applying [5, Theorem b] e only need the continuity of Γ hen restricted to the subspace S. Proof of Proposition 4. Let { µ 2 } n n= S be such that µ 2 n µ 2 S. We ill sho that Γµ 2 d n Γµ 2. d µ 2 here Let Ω, F, P be a probability space such that there exists {X n, Y n } n= and X, Y random vectors taking values in R 2, ith X n, Y n µ 2 n, n, and X, Y µ 2. Notice that by definition X n d = Y n d = X d = Y, and each has logistic distribution. Fix x, y R, let G n x, y := Γµ 2 n x, y,, then by exactly the same argument as in the derivation of 2 e get G n x, y = Hx Hy exp = Hx Hy exp = Hx Hy exp G n t x, t y dt P X n > t x, Y n > t y dt P X n + x Y n + y > t dt = Hx Hy exp E [ X n + x + Y n + y +], and a similar calculation ill also give that Gx, y := Γµ 2 x, y, = Hx Hy exp E [ X + x + Y + y +].

14 250 A. Bandyopadhyay No to complete the proof all e need is to sho E [ X n + x + Y n + y +] E [ X + x + Y + y +]. Since e assumed that X n, Y n d X, Y it follos X n + x + Y n + y + d X + x + Y + y + x, y R. Fix x, y R, define Zn x,y Y + y +. Observe that := X n + x + Y n + y +, and Z x,y := X + x + 0 Z x,y n X n + x + X n + x n. But, X n + x = d X + x for all n. So clearly {Zn x,y } n= is uniformly integrable. Hence e conclude using the theorem of Billingsley [9, Theorem 25.2] that E [Zn x,y ] E [Z x,y ]. This completes the proof. Acknoledgments. The author ould like to thank David Aldous for suggesting the the problem and for many illuminating discussions. Thanks are also due to the Department of Statistics, University of California, Berkeley here most of the ork as done during the author s doctoral study. References [] Aldous, D. Asymptotics in the random assignment problem. Probab. Theory Related Fields , [2] Aldous, D. and Steele, J. M., The objective method: probabilistic combinatorial optimization and local eak convergence. In: Probability on Discrete Structures ed.: H. Kesten. Encyclopaedia Math. Sci. 0. Berlin: Springer 2004, pp. 72. [3] Aldous, D. J., The percolation process on a tree here infinite clusters are frozen. Math. Proc. Cambridge Philos. Soc , [4] Aldous, D. J., The ζ2 Limit in the Random Assignment Problem. Random Structures Algorithms , [5] Aldous, D. J. and Bandyopadhyay, A., A survey of max-type recursive distributional equations. Ann. Appl. Probab , [6] Bandyopadhyay, A., A necessary and sufficient condition for the tail-triviality of a recursive tree process. Sankhyā , 23. [7] Bandyopadhyay, A. and Gamarnik, D., Counting ithout sampling. Ne algorithms for enumeration problems using statistical physics. In: Proc. Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia PA: SIAM 2006, pp

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