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## A survey of max-type recursive distributional equations (2005)

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Venue: | Annals of Applied Probability 15 (2005 |

Citations: | 86 - 6 self |

### Citations

5195 |
Convergence of Probability Measures
- Billingsley
- 1968
(Show Context)
Citation Context ...x) + ∧ (Y + y) + . Observe that But, |Xn + x| d 0 ≤ Z x,y n ≤ (Xn + x) + ≤ |Xn + x| , ∀ n ≥ 1. (4.25) = |X + x| , ∀ n ≥ 1. So clearly {Z x,y n } ∞ n=1 we conclude ( using Theorem 25.12 of Billingsley =-=[11]-=- ) that E [Z x,y n ] −→ E [Z x,y ] . is uniformly integrable. Hence This completes the proof. □47 Proposition 34 The operator T maps F into F. Proof : First note that if f ∈ F, then by definition T (... |

1330 | Probability: theory and examples
- Durrett
- 1996
(Show Context)
Citation Context ...f the tail H is trivial then naturally X∅ is independent of it. For proving the converse we will need the following standard measure theoretic fact whose proof is easy using Dynkin’s π-λ Theorem (see =-=[13]-=-) and is omitted here. Lemma 14 Suppose (Ω, I, P) be a probability space and let F ∗ , G∗ and H∗ be three sub-σalgebras such that F ∗ is independent of H∗ ; G∗ is independent of H∗ ; and F ∗ and G∗ ar... |

1252 |
Interacting Particle Systems
- Liggett
- 1985
(Show Context)
Citation Context ...µ∞ implies T(µn) ↑ T(µ∞)], then µ∗ is a fixed point of T , and µ∗ � µ, for any other fixed point µ. This obvious result parallels the notion of lower invariant measure in interacting particle systems =-=[47]-=-. Returning to the case of general S, the Banach contraction theorem specializes to LEMMA 5 (The contraction method). Let P be a subset of P (S) such that T maps P into P . Let d be a complete metric ... |

902 | Applied Probability and Queues - Asmussen - 2003 |

748 | Branching Processes.
- Athreya, Ney
- 1972
(Show Context)
Citation Context ...Example Consider a sub-critical/critical Galton-Watson branching process with progeny distribution N. To exclude the trivial cases we assume that P (N = 1) = 1 and P (N = 0) = 1. A classical result =-=[8]-=- shows that the branching process starting from one individual goes extinct a.s. and hence the random family tree of the individual is a.s. finite. So the random variable { } H := min d ∣ no individua... |

355 |
Gibbs Measures and Phase Transitions (W. de Gruyter
- Georgii
- 1988
(Show Context)
Citation Context ...ult linking endogeny and bivariate uniqueness, without seeking minimal hypotheses. The result and proof are similar to standard results about Gibbs measures and Markov random fields (see Chapter 7 of =-=[34]-=-), but our emphasis is different, so it seems helpful to give a direct proof here, after a few remarks. THEOREM 11. Suppose S is a Polish space. Consider an invariant RTP with marginal distribution µ.... |

238 |
Étude de l’équation de la diffusion avec croissance de la quantité de la matière et son application a un problèm biologique, in:
- Kolmogorov, Petrovskii, et al.
- 1937
(Show Context)
Citation Context ...rovides an extensive bibliography of earlier work. The proof uses a mixture of analytic and probabilistic tools, for example, the “stretching” partial order (which goes back to the original KPP paper =-=[46]-=-), and multiplicative martingales.sRECURSIVE DISTRIBUTIONAL EQUATIONS 1083 THEOREM 44 ([14]). Consider an IBRW where E[N] < ∞, N ≥ 1, P(N > 1)>0, and where the offspring displacement has density f(x)=... |

232 | Iterated random functions
- Diaconis, Freedman
- 1999
(Show Context)
Citation Context ...the past (CFTP) technique for studying Markov chains [56]. That technique is part of a large circle of ideas (graphical representations in interacting particle systems [47]; iterated random functions =-=[27]-=-) for studying uniqueness of stationary distributions, and rates of convergence to stationarity, for Markov chains via sample path constructions. LEMMA 14. Consider an RTF and write T for the associat... |

147 | Steele,“The objective method: Probabilistic combinatorial optimization and local weak convergence,”
- Aldous, M
- 2003
(Show Context)
Citation Context ...is that a randomly chosen edge of that tree splits it into two subtrees, the smaller of which is distributed as a Galton–Watson tree with Poisson(1) offspring. For the detailed story see Section 3 of =-=[11]-=-, whose final result is: THEOREM 41. Suppose ν is nonatomic and has finite mean. Then lim n n −1 EWn = Eξ1(ξ>X+Z) where the r.v.’s on the right are independent, ξ has distribution ν, X is distributed ... |

129 |
Probability theory and combinatorial optimization.
- Steele
- 1997
(Show Context)
Citation Context ...n and consider a (complete) matching, that is, a collection of n/2 vertex-disjoint edges. Define Mn = minimum total length of a complete matching. This problem is often studied in the bipartite case (=-=[61]-=-, Chapter 4) but the two versions turn out to be equivalent in our asymptotic setting. The following limit behavior was argued nonrigorously in [53] and proved (in the bipartite setting) in [1, 6]. Th... |

104 |
Martingale convergence in the branching random walk,
- Biggins
- 1977
(Show Context)
Citation Context ...0. This in turn implies limm m−1Rm,1 = ess sup ξ1, contradicting Lemma 47. � p psRECURSIVE DISTRIBUTIONAL EQUATIONS 1087 REMARK 1. Some multiplicative martingales used in the study of BRW (see, e.g., =-=[20]-=-) are of the form Zn(θ) = m −n (θ) � i exp(θY n i ) where (Y n i ,i ≥ 1) are the positions of the generation-n individuals. The a.s. limit Z(θ) = limn Zn(θ) satisfies the RDE Z d = � i exp(θξi)Zi/m(θ)... |

93 |
On the value of a random minimum spanning tree problem.
- Frieze
- 1985
(Show Context)
Citation Context ... that is, a tree whose vertices are some subset of the n vertices, write |t|= number of edges in t, L(t) = � ξe = total length of t, e∈t a(t) = L(t)/|t|= average edge-length of t. A well-known result =-=[32]-=- on minimal spanning trees says that, if we insist on |t| =n − 1, then the smallest we can make a(t) is about ζ(3) := � i i−3 .Ifwe fix 0 <ε<1 and consider subtrees with around εn edges, then we guess... |

93 | A limit theorem for ”Quicksort”.
- Rosler
- 1991
(Show Context)
Citation Context ...(see Section 1.1), characterization of probability distributions [19], probabilistic analysis of random algorithms with suitable recursive structures, in particular the study of Quicksort algorithm2 =-=[26, 27, 16]-=- and the study of the find algorithm [12], in the study of branching random walk [9, 10], and in various statistical physics models on trees [4, 5], to name a few. Perhaps the most well studied case i... |

87 |
Fixed points of the smoothing transformation.
- Durret, Liggett
- 1983
(Show Context)
Citation Context ... be dependent. This and the extension (20) have been studied quite extensively; our discussion focuses on analogies with the maxtype cases later. Where the state space is R + , the key ideas are from =-=[28]-=- which assumed N is nonrandom; the extensions to random N (which is a frequent setting for our max-type examples) have been developed in [49, 50]. Here is a typical result (Corollaries 1.5 and 1.6 of ... |

71 | The contraction method for recursive algorithms.
- Rosler, Ruschendorf
- 2001
(Show Context)
Citation Context ...earance of the linear RDE (18) in the study of branching processes and branching random walks, invariant measures of infinite particle systems, and Hausdorff dimension of random Cantor-type sets. See =-=[26, 60]-=- for many references to linear RDEs arising in probabilistic analysis of algorithms which are analyzable by contraction. See [39, 40] for the specialization ∞� X d = h(ξi)Xi i=1 (S = R + ) where (ξi) ... |

70 |
Chernoff’s theorem in the branching random walk
- Biggins
- 1977
(Show Context)
Citation Context ...r Rn := position of rightmost individual in generation n with Rn =−∞if there is no such individual. Write nonextinction for the event that the process survives forever. Standard results going back to =-=[19]-=- show: PROPOSITION 23. If the BRW is supercritical, then there exists a constant −∞ <γ <∞ such that n−1Rn → γ a.s. on nonextinction. And γ is computable as the solution of � � inf log m(θ) − γθ = 0. θ... |

70 |
Postulates for subadditive processes
- Hammersley
- 1974
(Show Context)
Citation Context ...≥ 0 � (55) then (56) � Rn − median(Rn), n ≥ 1 � is bounded above, is tight. Harry Kesten (personal communication) attributes this type of argument to old work of Hammersley: it is perhaps implicit in =-=[36]-=-, page 662. PROOF OF LEMMA 43. Given ε>0 we can choose k<∞ and B>−∞ such that P � generation k has at least log 2 1/ε individuals in [B,∞) � ≥ 1 − ε. Then by conditioning on the positions of generatio... |

70 |
On the solution of the random link matching problems.
- Mézard, Parisi
- 1987
(Show Context)
Citation Context ...his problem is often studied in the bipartite case ([61], Chapter 4) but the two versions turn out to be equivalent in our asymptotic setting. The following limit behavior was argued nonrigorously in =-=[53]-=- and proved (in the bipartite setting) in [1, 6]. There are fascinating recent proofs [48, 55] of an underlying exact formula for EMn in the bipartite, exponential distribution setting, but it seems u... |

64 |
Interacting particle systems, volume 276 of Grundlehren der
- Liggett
- 1985
(Show Context)
Citation Context ... (µn) ↑ T (µ∞), then µ⋆ is a fixed point of T , and µ⋆ ≼ µ, for any other fixed point µ. This obvious result parallels the notion of lower invariant measure in attractive interacting particle systems =-=[20]-=-. Now considering the case of general S, the Banach contraction mapping principle specializes to yet another obvious result. Lemma 4 (Contraction Method) Let P be a subset of P(S) such that T maps P i... |

61 |
The cavity method at zero temperature
- Mézard, Parisi
- 2003
(Show Context)
Citation Context ...2]) 1/X d ∞� = e i=1 −θξiXi (S = R + ) which is somewhat in the spirit of the linear case. 7.5. The cavity method. The nonrigorous cavity method was developed in statistical physics in the 1980s; see =-=[54]-=- for a recent survey. Though typically applied to examples such as ground states of disordered Ising models, it can also be applied to the kind of “mean-field combinatorial optimization” examples of t... |

60 |
An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics, WileyInterscience,
- Khamsi, Kirk
- 2001
(Show Context)
Citation Context ...m 32). It is also worth mentioning that in several examples we have no rigorous proofs of existence of fixed points (see Sections 5 and 7.4) and so the use of other fixed point theorems from analysis =-=[45]-=- might be worth exploring.sRECURSIVE DISTRIBUTIONAL EQUATIONS 1053 2.3. Recursive tree processes. Consider again the setup from Section 2.1. Rather than considering only the induced map T , one can ma... |

60 |
Fixed points of a generalized smoothing transformation and applications to the branching random walk
- Liu
- 1998
(Show Context)
Citation Context ...ter. Where the state space is R + , the key ideas are from [28] which assumed N is nonrandom; the extensions to random N (which is a frequent setting for our max-type examples) have been developed in =-=[49, 50]-=-. Here is a typical result (Corollaries 1.5 and 1.6 of [49]; the case of nonrandom N is in [28]; minor nontriviality assumptions omitted). THEOREM 16. Suppose ξi ≥ 0, with ξi > 0 iff 1 ≤ i ≤ N, for so... |

58 |
The ζ(2) limit in the random assignment problem. Random Structures and Algorithms
- Aldous
- 2001
(Show Context)
Citation Context ...case ([61], Chapter 4) but the two versions turn out to be equivalent in our asymptotic setting. The following limit behavior was argued nonrigorously in [53] and proved (in the bipartite setting) in =-=[1, 6]-=-. There are fascinating recent proofs [48, 55] of an underlying exact formula for EMn in the bipartite, exponential distribution setting, but it seems unlikely that the applicability of exact methods ... |

50 |
Gibbs measures and phase transitions, volume 9 of de Gruyter
- Georgii
- 1988
(Show Context)
Citation Context ... much richer probabilistic model which we call recursive tree process (RTP) (see Section 2.4 for a formal definition). This treatment is some what similar to the general theory of Markov random field =-=[17]-=- but our emphasis is quite different. 2.1 The General Setting Here we record a careful setup for RDEs. Let (S, S) be a measurable space, and P(S) the set of probability measures on (S, S). Let (Θ, T) ... |

49 | A proof of Parisi’s conjecture on the random assignment problem.
- Linusson, Wästlund
- 2004
(Show Context)
Citation Context ...turn out to be equivalent in our asymptotic setting. The following limit behavior was argued nonrigorously in [53] and proved (in the bipartite setting) in [1, 6]. There are fascinating recent proofs =-=[48, 55]-=- of an underlying exact formula for EMn in the bipartite, exponential distribution setting, but it seems unlikely that the applicability of exact methods extends far into the broad realm of problems a... |

48 | Martingales and profile of binary search trees
- Chauvin, Klein, et al.
(Show Context)
Citation Context ... AND A. BANDYOPADHYAY 8.3. Dependence on parameters. When an RDE depends on a parameter (e.g., our (77) and (96); see also examples involving multiplicative martingales for branching processes, e.g., =-=[23]-=-, Theorem 3.3), it is natural to ask whether the solution depends continuously on the parameter. This has apparently not been studied in any generality. 8.4. Continuous-time analogs. We set up recursi... |

48 |
Differential equations : with applications and historical notes
- Simmons
- 1972
(Show Context)
Citation Context ... will prove that the operator associated with (4.10) ( defined on an appropriate space ) is monotone and has unique fixed-point as H. The techniques we will use here are similar to Eulerian recursion =-=[28]-=-, and are based heavily on analytic arguments. Let F be the set of all functions f : R → [0, 1] such that • H 2 (x) ≤ f(x) ≤ H(x), ∀ x ∈ R, • f is a tail of a distribution, that is, ∃ random variable ... |

47 |
Characterization Problems of Mathematical Statistics.
- Kagan, Linnik, et al.
- 1973
(Show Context)
Citation Context ...n (RDE). RDEs have arisen in a variety of settings, for example, in the study of the Galton-Watson branching process and related trees (see Section 1.1), characterization of probability distributions =-=[19]-=-, probabilistic analysis of random algorithms with suitable recursive structures, in particular the study of Quicksort algorithm2 [26, 27, 16] and the study of the find algorithm [12], in the study o... |

42 |
Asymptotic properties and absolute continuity of laws stable by random weighted mean
- Liu
- 2001
(Show Context)
Citation Context ...ter. Where the state space is R + , the key ideas are from [28] which assumed N is nonrandom; the extensions to random N (which is a frequent setting for our max-type examples) have been developed in =-=[49, 50]-=-. Here is a typical result (Corollaries 1.5 and 1.6 of [49]; the case of nonrandom N is in [28]; minor nontriviality assumptions omitted). THEOREM 16. Suppose ξi ≥ 0, with ξi > 0 iff 1 ≤ i ≤ N, for so... |

42 |
Replicas and optimization
- Mézard, Parisi
- 1985
(Show Context)
Citation Context ... position Rn of the rightmost point in BRW; its spread (Lemma 43) stays bounded with n. 7.4. TSP and other matching problems. Here we follow Sections 6.1 and 6.2 of [6], which showed how earlier work =-=[51, 52]-=- fits into the current framework. As suggested at the start of Section 7, one can define a mean-field model of distance with any real pseudo-dimension 0 <d<∞ to mimic distances betweensRECURSIVE DISTR... |

42 |
A replica analysis of the traveling salesman problem.
- Mezard, Parisi
- 1986
(Show Context)
Citation Context ... position Rn of the rightmost point in BRW; its spread (Lemma 43) stays bounded with n. 7.4. TSP and other matching problems. Here we follow Sections 6.1 and 6.2 of [6], which showed how earlier work =-=[51, 52]-=- fits into the current framework. As suggested at the start of Section 7, one can define a mean-field model of distance with any real pseudo-dimension 0 <d<∞ to mimic distances betweensRECURSIVE DISTR... |

41 |
Asymptotics in the Random Assignment,” Problem.
- Aldous
- 1992
(Show Context)
Citation Context ... U)X2 + C(U) (22) = , Y UY1 + (1 − U)Y2 + C(U)sRECURSIVE DISTRIBUTIONAL EQUATIONS 1065 where (X1,Y1) and (X2,Y2) are i.i.d. having the same distribution as (X, Y ) and are independent of U d = Uniform=-=[0, 1]-=-. First consider the case σ = 0. In this case both X and Y have finite second moment and hence so does D = X − Y . Naturally the distribution of D satisfies the RDE D d = UD1 + (1 − U)D2 (on R), where... |

41 |
A fixed point theorem for distributions. Stochastic Process
- Rösler
- 1992
(Show Context)
Citation Context ...d = UX1 + (1 − U)X2 + C(U) (S = R) where C(x) := 2x log x + 2(1 − x)log(1 − x) + 1, and U d = U(0, 1). Thereis a unique solution with E[X 2 ] < ∞ because T is a contraction under the metric d2 at (7) =-=[59]-=-. But there are also other solutions. THEOREM 20 ([31]). Let ν be the solution of the RDE (21) with zero mean and finite variance. Then the set of all solutions is the set of distributions of the form... |

38 | Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method.
- Gamarnik, Nowicki, et al.
- 2006
(Show Context)
Citation Context ... in the sense of local weak convergence, is the infinite r-regular tree Tr. Thus one can seek to carry out the general program formalizing the cavity method (Section 7.5) in this setting. Recent work =-=[33]-=- provides interesting positive and negative results. The relevant RDE is [cf. (51)] (99) X d = max 1≤i≤r−1 (0,ξi − Xi) (S = R + ). THEOREM 66 ([33]). Let Tr−1 be the map associated with the RDE (99). ... |

32 | Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stochastic Process
- Iksanov
(Show Context)
Citation Context ...efficient is ρ(α) = 1 and hence the contraction argument cannot be used directly. The proof of Theorem 16 instead involves somewhat intricate analysis to find the moment generating function of X. See =-=[38]-=- for the case where N may be infinite. See [49] for many references to the appearance of the linear RDE (18) in the study of branching processes and branching random walks, invariant measures of infin... |

28 |
Travelling-waves for the F-K-P-P equation via probabilistic arguments,
- Harris
- 1999
(Show Context)
Citation Context ...me Markov chains. Let us mention two contexts where continuous-parameter analogs of RTPs arise. The first is the classical KPP equation, which is the analog of (54) for branching Brownian motion; see =-=[37]-=- for a recent probabilistic account. The second concerns the maximum X of standard Brownian excursion of length 1. By scaling, the maximum Xt for Brownian excursion of length t satisfies Xt d = t1/2X.... |

27 | Coupling from the past: a user’s guide
- Propp, Wilson
- 1998
(Show Context)
Citation Context ... . �sRECURSIVE DISTRIBUTIONAL EQUATIONS 1061 2.6. Tree-structured coupling from the past. The next lemma is clearly analogous to the coupling from the past (CFTP) technique for studying Markov chains =-=[56]-=-. That technique is part of a large circle of ideas (graphical representations in interacting particle systems [47]; iterated random functions [27]) for studying uniqueness of stationary distributions... |

24 | The percolation process on a tree where infinite clusters are frozen
- Aldous
(Show Context)
Citation Context ...zen percolation process on infinite binary tree. A different setting where a particular “max-type” RDE plays the crucial role is the frozen percolation process on the infinite binary tree, studied in =-=[5]-=-. Let T3 = (V, E) be the infinite binary tree, where each vertex has degree 3; V is the set of vertices and E is the set of undirected edges. Let (Ue)e∈E be independent edge weights with Uniform(0, 1)... |

22 |
volume 321 of Grundlehren der Mathematischen Wissenschaften
- Percolation
- 1999
(Show Context)
Citation Context ...se set At = At−. Here formally a cluster is a connected subgraph of T3. A more familiar process of similar ∣ := {e ∈ E ≤ t } for kind is the standard percolation process on T3 defined as Bt 0 ≤ t ≤ 1 =-=[18]-=-. From definition we see that At ⊆ Bt for all 0 ≤ t ≤ 1. We note that ∣ Ue53 Bt has no infinite cluster for t ≤ 1/2, since all the connected components are then subcritical or critical (when t = 1/2)... |

21 |
Limit theorems for the minimal position in a branching random walk with independent logconcave displacements,
- Bachmann
- 2000
(Show Context)
Citation Context ...f course, because if (57) and (58) hold, then the limit X must satisfy the RDE (54). 5.2. Limit theorems. This topic has been studied carefully only in the IBRW setting. We quote a recent result from =-=[14]-=-, which provides an extensive bibliography of earlier work. The proof uses a mixture of analytic and probabilistic tools, for example, the “stretching” partial order (which goes back to the original K... |

16 | Lindley-type equations in the branching random walk
- Biggins
- 1998
(Show Context)
Citation Context ...see Open Problem 30 later. The lemma also leaves open the question of whether there may be other invariant measures in the supercritical case. A thorough treatment of the latter question was given in =-=[21]-=- within a slightly more general setting, including the next result showing that nonuniqueness is typical. PROPOSITION 25 ([21], Theorem 1). Under technical hypotheses (omitted here) the RDE (25) has a... |

16 | The zeta (2) Limit - Aldous - 2001 |

16 |
Rachev and Ludger Rüschendorf. Mass transportation problems. Vol
- Svetlozar
- 1998
(Show Context)
Citation Context ...emma 3 the RDE (3.3) will have a solution if an only if T n (δ0) is tight. The following theorem which is a generalization of Rachev and Rüschendorf’s result (see the comments on equation (9.1.18) of =-=[25]-=-) which only deals with non-random fixed N, gives a sufficient condition for existence of a solution. Theorem 19 Suppose 0 ≤ ξi < 1 for all i ≥ 1 and η ≥ 0 has all moments finite. For p ≥ 1 write c(p)... |

15 | A characterization of the set of fixed points of the Quicksort transformation.
- Fill, Janson
- 2000
(Show Context)
Citation Context ...g x + 2(1 − x)log(1 − x) + 1, and U d = U(0, 1). Thereis a unique solution with E[X 2 ] < ∞ because T is a contraction under the metric d2 at (7) [59]. But there are also other solutions. THEOREM 20 (=-=[31]-=-). Let ν be the solution of the RDE (21) with zero mean and finite variance. Then the set of all solutions is the set of distributions of the form ν ∗ Cauchy(m, σ 2 ) where m ∈ R and σ 2 ≥ 0, and ∗ de... |

13 |
Scaling and universality in continuous length combinatorial optimization
- Aldous, Percus
- 2003
(Show Context)
Citation Context ... measure of how different an almost-optimal solution can be from the optimal solution. Remarkably, it is not so hard to study this question by an extension of the methods of Section 7.3. It turns out =-=[10]-=- that the key is the extension of the RDE (86) to the following RDE on S = R3 : ⎛ ⎛ ⎞ min(ξi − Xi) X ⎜ � i ⎝ Y ⎠ ⎜ = ⎜min ξi − (Zi + λ)1(i = i ⎝ i Z ∗ ) − Yi1(i �= i ∗ ) � ⎞ ⎟ (96) ⎟ ⎠ min(ξi − Yi) i ... |

12 |
Fixed points with finite variance of a smoothing transformation. Stochastic Processes.
- Caliebe, Rosler
- 2003
(Show Context)
Citation Context ...ointing out that there is no very complete “general theory” for S = R: OPEN PROBLEM 19. Study analogs of Theorem 16 for S = R. Of course the contraction method remains useful in particular cases. See =-=[22]-=- for results on smoothness of solutions in the case of finite second moment. (20) 3.1. The Quicksort RDE. A slight extension of the linear case is the case g((ξi,Xi)) d = ξ0 + � i≥1 ξiXi. As a well-kn... |

12 | On the probabilistic worst-case time of FIND.
- Devroye
- 2001
(Show Context)
Citation Context ...aysRECURSIVE DISTRIBUTIONAL EQUATIONS 1077 4.5. Examples of discounted tree sums. EXAMPLE 36. Take U d = Uniform(0, 1) and consider the RDE X d = 1 + max � � UX1,(1− U)X2 (S = R + (42) ). This arises =-=[25]-=- in the context of the probabilistic worst-case analysis of Hoare’s FIND algorithm. Theorem 32 implies existence of a fixed point with all moments finite, unique amongst possible fixed points with fin... |

11 |
Stochastic fixed points for the maximum
- Jagers, Rösler
- 2004
(Show Context)
Citation Context ...8) is equivalent to ˆX d = max(ˆξi + ˆXi) i and this RDE, to be studied in Section 5, is the fundamental example of a max-type RDE which cannot be solved by any simple probabilistic construction. See =-=[41]-=- for further discussion of (38). saysRECURSIVE DISTRIBUTIONAL EQUATIONS 1077 4.5. Examples of discounted tree sums. EXAMPLE 36. Take U d = Uniform(0, 1) and consider the RDE X d = 1 + max � � UX1,(1− ... |

11 |
Diffusion-limited aggregation on a tree, Probab
- Barlow, Pemantle, et al.
- 1997
(Show Context)
Citation Context ...rpreted as nonhomogeneous percolation, in which case the time X taken to percolate to infinity satisfies the RDE (49) X d =η + cmin(X1,X2). This setting has been studied from a different viewpoint in =-=[17]-=-. 4.6. Matchings on Galton–Watson trees. Amongst many possible examples involving Galton–Watson trees, the following rather subtle example provides a warm-up to the harder example in Section 7.3. Cons... |

11 |
Probability Theory and Combinatorial Optimization, NSFCBMS Volume 69
- Steele
- 1997
(Show Context)
Citation Context ...en and consider a (complete) matching, that is a collection of n/2 vertex-disjoint edges. Define Mn = minimum total length of a complete matching. This problem is often studied in the bipartite case (=-=[56]-=- Chapter 4) but the two versions turn out to be equivalent in our asymptotic setting. The following limit behavior was argued non-rigorously in [48] and proved (in the bipartite setting) in [2, 7]. Th... |

11 |
Branching processes, volume 3
- Asmussen, Hering
- 1983
(Show Context)
Citation Context ...ike N = 0 in (1.6)) as zero. The RDE (1.6) is a natural prototype for “max-type” RDEs and one of our main motivating examples. So although the following result is implicit in the classical literature =-=[8, 6]-=- we provide a proof which is a simple consequence of the above RDE. Proposition 1 Let N be a non-negative integer valued random variable with E [N] ≤ 1 and assume that 0 < P (N = 0) < 1. Let φ be the ... |

10 |
Greedy search on the binary tree with random edge-weights
- Aldous
- 1992
(Show Context)
Citation Context ...sage percolation Yes Conj. Y Determines critical c Mean-field scaling analysis 5 c + maxi(Xi + ξi) Extremes in BRW No c specified by dist(ξi) 7.3 7.4 mini(ξi − Xi) min Mean-field minimal matching Yes =-=[2]-=- i (ξi − Xi) Mean-field TSP Conj. Y min [2] Other S denotes second smallest 6 7.6 �(min(X1,X2), ξ0) Frozen percolation on tree See (96), (97), (98) Mean-field scaling Yes Conj. Y � defined in Section ... |

10 | A Metropolis-type optimization algorithm on the infinite tree
- Aldous
- 1998
(Show Context)
Citation Context ... such that γ(p) > 0 iff p > p crit, we will typically have linear scaling for γ: γ(p) ∼ c(p − p crit) as p ↓ p crit. But how does speed(p) scale? A special case permits explicit analysis. Theorem 29 (=-=[4]-=- Theorem 6) Consider IBRW where each individual has exactly two children whose displacements ξ satisfy P(ξ = 1) = p, P(ξ = −1) = 1 − p. The critical point p crit is the smaller solution of 16p crit(1 ... |

10 | Percolation-like scaling exponents for minimal paths and trees in the stochastic mean field
- Aldous
(Show Context)
Citation Context ...complete graph with random edge-lengths. Over all cycles πn,u in Kn containing un vertices, let Cn,u be the minimum average edge-length of πn,u. We anticipate a limit lim n ECn,u = p(u). It turns out =-=[8]-=- that what is relevant is the following RDE for a distribution (X, Z) on R2 : ⎛ � � max(λ − ξi + Xi − Z X d= ⎜ i ⎝ Z + i ) max(λ − ξi + Xi − Z i + [2] i ) + max (λ − ξi + Xi − Z i + i ) ⎞ ⎟ (97) ⎠ . H... |

9 | Density approximation and exact simulation of random variables that are solutions of fixed-point equations
- Devroye, Neininger
- 2002
(Show Context)
Citation Context ...earance of the linear RDE (18) in the study of branching processes and branching random walks, invariant measures of infinite particle systems, and Hausdorff dimension of random Cantor-type sets. See =-=[26, 60]-=- for many references to linear RDEs arising in probabilistic analysis of algorithms which are analyzable by contraction. See [39, 40] for the specialization ∞� X d = h(ξi)Xi i=1 (S = R + ) where (ξi) ... |

8 | Cost-volume relationships for flows through a disordered network”,
- Aldous
- 2008
(Show Context)
Citation Context ...τ}, 0 <τ<1. We have δ ∗ (τ) > 0iffτ>τFPP, whereτFPP is the time constant in first passage percolation on T. As above, to study scaling exponents the key is a certain RDEsfor S = R + , which turns out =-=[7]-=- to be RECURSIVE DISTRIBUTIONAL EQUATIONS 1103 Z d � 3� � = min Z2 + ξ2 − a,Z3 + ξ3 − a, (Zi + ξi − a) i=1 (98) � − min 0, � (Zi + ξi − a), i=1,2 � � (Zi + ξi − a) . i=1,3 Here a is a parameter ∈ (τFP... |

7 | The ‘birth-andassassination’ process
- Aldous, WB
- 1990
(Show Context)
Citation Context ...amples where the distribution arising in an RDE is the distribution of a stochastic process, rather than a single real-valued random variable. Here is an illustration. Birth and assassination process =-=[9]-=-. Start with one individual at time 0. During each individual’s lifetime, children are born at the times of a Poisson (rate λ) process. An individual cannot die before the time of its parent’s death (... |

7 |
The asymptotic behaviour of fragmentation processes
- Bertoin
(Show Context)
Citation Context ...conditional on their lengths), we can write (for infinitesimal δ) X = δ + max t i 1/2 i (δ) Xi where (ti(δ), i ≥ 1) are the lengths of excursions above level δ within standard Brownian excursion. See =-=[18]-=- for this kind of decomposition. 8.5. Process-valued analogs. There are examples where the distribution arising in an RDE is the distribution of a stochastic process, rather than a single real-valued ... |

5 |
A surprising Poisson process arising from a species competition model
- Durrett, Limic
- 2001
(Show Context)
Citation Context ... where (ξi,i ≥ 1) are the points of a Poisson rate 1 process on (0, ∞) and where η has Exponential(1) distribution independent of (ξi). This is a new example, arising from a species competition model =-=[29]-=-. Time reversal of the process in [29], together with a transformation of (0, 1) to (0, ∞) by x →−log(1 − x), yields a branching Markov process taking values in the space of countable subsets of (0, ∞... |

5 | On fixed points of Poisson shot noise transforms
- Iksanov, Jurek
- 2002
(Show Context)
Citation Context ...e systems, and Hausdorff dimension of random Cantor-type sets. See [26, 60] for many references to linear RDEs arising in probabilistic analysis of algorithms which are analyzable by contraction. See =-=[39, 40]-=- for the specialization ∞� X d = h(ξi)Xi i=1 (S = R + ) where (ξi) are the points of a Poisson process on (0, ∞). Often, within one model there are different questions which lead to both linear and ma... |

5 |
C.S.: On a Pitman-Yor problem
- Iksanov, Kim
- 2004
(Show Context)
Citation Context ...e systems, and Hausdorff dimension of random Cantor-type sets. See [26, 60] for many references to linear RDEs arising in probabilistic analysis of algorithms which are analyzable by contraction. See =-=[39, 40]-=- for the specialization ∞� X d = h(ξi)Xi i=1 (S = R + ) where (ξi) are the points of a Poisson process on (0, ∞). Often, within one model there are different questions which lead to both linear and ma... |

5 |
A proof of Parisi’s conjecture for the finite random assignment problem. Unpublished manuscript
- NAIR, PRABHAKAR, et al.
- 2003
(Show Context)
Citation Context ...turn out to be equivalent in our asymptotic setting. The following limit behavior was argued nonrigorously in [53] and proved (in the bipartite setting) in [1, 6]. There are fascinating recent proofs =-=[48, 55]-=- of an underlying exact formula for EMn in the bipartite, exponential distribution setting, but it seems unlikely that the applicability of exact methods extends far into the broad realm of problems a... |

4 | On the critical value for “percolation” of minimum-weight trees in the mean-field distance model
- Aldous
- 1998
(Show Context)
Citation Context ...f a subtree t subject to the constraint that a(t) ≤ c. Forfixed 0 <c<∞ consider the RDE on S =[0, ∞) Y d ∞� = (c − ξi + Yi) + (77) ; (ξi) a Poisson rate 1 point process on (0, ∞). i=1 PROPOSITION 56 (=-=[4]-=-). Define M(n,c) = max{|t| : t a subtree of Kn, a(t) ≤ c}. Then there exists a critical point c(0) ∈[e −2 ,e −1 ] such that (78) (79) and if c<c(0) then n −1 M(n,c) d → 0, if c>c(0) then ∃ η(c) > 0 su... |

4 |
Discounted branching random walks
- Athreya
- 1985
(Show Context)
Citation Context ...ifferent way it can be proved [25] that any fixed point has all moments finite, and hence the fixed point is unique. (43) EXAMPLE 37. Consider the RDE where 0 <c<1. X d = η + c max(X1,X2) This arises =-=[13]-=- as a “discounted branching random walk.” One interpretation is as nonhomogeneous percolation on the planted binary tree (the root has degree 1), where an edge at depth d has traversal time distribute... |

4 |
Bivariate uniqueness in the logistic fixed point equation
- Bandyopadhyay
- 2002
(Show Context)
Citation Context ...= 1 2E� (X1 + X2) +�2 by a general formula = 1 4E(X1 + X2) 2 by symmetry (93) = 1 2 EX2 1 = π 2 /6, the last step using a standard fact that the logistic distribution has variance π 2 /3. THEOREM 61 (=-=[15]-=-). The invariant RTP associated with the RDE (86) is endogenous. The significance of this result is pointed out in Section 7.5. The proof involves somewhat intricate analytic study of the iterates T (... |

4 |
An assignment problem at high temperature
- TALAGRAND
- 2003
(Show Context)
Citation Context ...he RDE (95), and that the associated invariant RTP is endogenous. Instead of studying minimal matchings one could study Gibbs distributions on matchings; this leads to a different RDE ([6], (46), and =-=[62]-=-) 1/X d ∞� = e i=1 −θξiXi (S = R + ) which is somewhat in the spirit of the linear case. 7.5. The cavity method. The nonrigorous cavity method was developed in statistical physics in the 1980s; see [5... |

4 | Diffusion-limited aggregation for trees
- Barlow, Pemantle, et al.
- 1992
(Show Context)
Citation Context ...ted as non-homogeneous percolation, in which case the time X taken to percolate to infinity satisfies the RDE X d = η + c min(X1, X2). (49) This setting has been studied from a different viewpoint in =-=[16]-=-. 4.6 Matchings on Galton-Watson trees Amongst many possible examples involving Galton-Watson trees, the following rather subtle example provides a warm-up to the harder example in section 7.3. Consid... |

3 |
Higher-order Lindley equations. Stochastic Process
- Kelbert, Suhov
- 1994
(Show Context)
Citation Context ... of the height of a Galton–Watson tree (take ξi = 1), as well as the study of the rightmost position of a random walk [take N = 1 and compare with (5)]. Applications to queueing networks are given in =-=[44]-=-. Conditioning on the first-generation offspring leads to the RDE below, and Lemma 15 establishes the other assertions. Lemma 24. Suppose extinction is certain, or suppose supercritical with γ < 0. Th... |

2 | Bivariate Uniqueness and Endogeny for Recursive Distributional Equations : Two Examples
- Bandyopadhyay
(Show Context)
Citation Context ...hat the behavior near the root was affected by the behavior at infinity. For several years we conjectured in seminar talks that the RDE is nonendogenous, but recently proved the opposite. THEOREM 55 (=-=[16]-=-). The invariant RTP associated with (67), (70) is endogenous. 7. Combinatorial optimization within the mean-field model of distance. In problems involving n random points in d-dimensional space, expl... |

2 |
The efficient construction of an unbiased random source
- ELIAS
- 1972
(Show Context)
Citation Context ... induced T is as above. In the associated RTP where Xi has Bern(1/2) marginals, observe that X∅ = ξ∅ and so the endogenous property holds. Now consider the well-known von Neumann random bit extractor =-=[30]-=-, which is a certain function ¯g : {0, 1} ∞ →{0, 1} which, applied to an independent Bern(p) input sequence for any 0 <p<1, gives a Bern(1/2) output. Set � a, if x1 = x2 = x3 ···, g(a,x1,x2,...)= ¯g(x... |

2 |
Renormalization of random hierarchical systems
- JORDAN
- 2003
(Show Context)
Citation Context ... the sparse random graph model above a certain critical value. So this setting provides an important test bed for the range of applicability of the method. 8.7. Random fractal graphs. A recent thesis =-=[42]-=- studies RDEs arising in the context of constructing random fractal graphs, and discusses examples such as the following: X d = g(X1,X2,ξ) (S = R + )s1106 D. J. ALDOUS AND A. BANDYOPADHYAY where ξ d =... |

2 | and Antar Bandyopadhyay. Probability Distributions Arising as Fixed-points of Max-type Operators - Aldous - 2002 |

1 |
Diffusion-limited aggregation on atree.Probab. Theory Related Fields 107
- BARLOW, PEMANTLE, et al.
- 1997
(Show Context)
Citation Context ...reted as nonhomogeneous percolation, in which case the time X taken to percolate to infinity satisfies the RDE (49) X d = η + c min(X1,X2). This setting has been studied from a different viewpoint in =-=[17]-=-. 4.6. Matchings on Galton–Watson trees. Amongst many possible examples involving Galton–Watson trees, the following rather subtle example provides a warm-up to the harder example in Section 7.3. Cons... |

1 |
Limit distributions for minima displacements of branching random walks. Probab. Theory Related Fields 90
- DEKKING, HOST
- 1991
(Show Context)
Citation Context ...e and we are studying the position of the leftmost particle) it is clear that Rn is decreasing and so (55) holds automatically, and then the lemma implies (56). A slicker argument for this case is in =-=[24]-=-, Proposition 2. The same holds (by translation) if there is a constant upper bound on displacement, that is, if P(ξ1 ≤ x0) = 1 for some constant x0 < ∞. From these tightness results it is natural to ... |

1 |
Higher-order Lindley equations. Stochastic Process
- andSUHOV
- 1994
(Show Context)
Citation Context ... of the height of a Galton–Watson tree (take ξi = 1), as well as the study of the rightmost position of a random walk [take N = 1 and compare with (5)]. Applications to queueing networks are given in =-=[44]-=-. Conditioning on the first-generation offspring leads to the RDE below, and Lemma 15 establishes the other assertions. is1068 D. J. ALDOUS AND A. BANDYOPADHYAY LEMMA 24. γ<0. Then Suppose extinction ... |

1 |
Probability Metrics and the Theory of Stochastic Models
- RACHEV
- 1991
(Show Context)
Citation Context ...traction, that is, d(T(µ1), T (µ2)) sup < 1. µ1�=µ2∈P d(µ1,µ2) Then T has a unique fixed point µ in P , whose domain of attraction is all of P . A thorough account of specific metrics can be found in =-=[57]-=-. Most commonly used is the Wasserstein metric on distributions on R with finite pth moment, 1 ≤ p<∞: dp(µ, ν) := inf{(E[|Z − W| p ]) 1/p |Z d = µ and W d (7) = ν}. Contraction is a powerful tool in t... |

1 |
Mass Transportation Problems 2: Applications
- RACHEV, L
- 1998
(Show Context)
Citation Context ...on makes sense when we allow P(ξi > 1)>0, though the concrete examples we know involve only the case ξi < 1 a.s. We content ourselves with recording a simple contraction argument (essentially that of =-=[58]-=-, (9.1.18), in the setting of finite nonrandom N) designed to handle the case ξi < 1a.s. THEOREM 32. Suppose 0 ≤ ξi < 1 and η ≥ 0 with E[ηp ] < ∞∀p≥ 1. For 1 ≤ p<∞ write c(p) := �∞ i=0 E[ξ p i ]≤∞. Su... |

1 |
Chance and Stability. VSP,Utrecht
- UCHAIKIN, M
- 1999
(Show Context)
Citation Context ...Perhaps the best known fixed point equation is (2) X d = 2 −1/2 (X1 + X2) (S = R) whose solutions are the Normal(0,σ 2 ) family. This example extends to give characterizations of stable distributions =-=[63]-=-. Moreover, there is a classical topic “characterization of probability distributions” [43] which considers the named families of distributions in mathematical statistics and studies many different ty... |

1 |
A non-rigorous scaling law in mean-field first passage percolation. Unpublished
- Aldous
- 2003
(Show Context)
Citation Context ...τ < 1. We have δ ∗ (τ) > 0 iff τ > τFPP, where τFPP is the time constant in first passage percolation on T. As above, to study scaling exponents the key is a certain RDE for S = R + , which turns out =-=[8]-=- to be Z d = min − ( Z2 + ξ2 − a, Z3 + ξ3 − a, 3∑ ) (Zi + ξi − a) ⎛ min ⎝0, ∑ (Zi + ξi − a), ∑ ⎞ (Zi + ξi − a) ⎠ . (104) i=1,2 i=1,3 Here a is a parameter ∈ (τFPP,1). In terms of the solution of this ... |

1 |
The asymptotic behaviour of self-similar fragmentations. Prépublication du Laboratoire de Probabilités et modèles aléatoires, Universite Paris VI. Available via http://www.proba.jussieu.fr
- Bertoin
- 2001
(Show Context)
Citation Context ...(conditional on their lengths) we can write (for infinitesimal δ) X = δ + max i t 1/2 i (δ) Xi where (ti(δ), i ≥ 1) are the lengths of excursions above level δ within standard Brownian excursion. See =-=[17]-=- for this kind of decomposition. 8.3 Process-valued analogs There are examples where the distribution arising in a RDE is the distribution of a stochastic process, rather than a single real-valued ran... |

1 |
Mass Transportation Problems. Volume 2: Applications
- Rachev, Rüschendorf
- 1998
(Show Context)
Citation Context ...makes sense when we allow P(ξi > 1) > 0, though the concrete examples we know involve only the case ξi < 1 a.s.. We content ourselves with recording a simple contraction argument (essentially that of =-=[53]-=- equation 9.1.18 in the setting of finite non-random N) designed to handle the case ξi < 1 a.s. Theorem 32 Suppose 0 ≤ ξi < 1 and η ≥ 0 with E[ηp ] < ∞ ∀ p ≥ 1. For 1 ≤ p < ∞ write c(p) := ∑∞ i=0 E[ξp... |

1 |
Stable distributions and their applications, With a foreword by V. Yu. Korolev and Zolotarev
- Chance, VSP, et al.
- 1999
(Show Context)
Citation Context ...erhaps the best known fixed point equation is X d = 2 −1/2 (X1 + X2) (S = R) (2) whose solutions are the Normal(0, σ 2 ) family. This example extends to give characterizations of stable distributions =-=[58]-=-. Moreover there is a classical topic “characterization of probability distributions” [38] which considers the named families of distributions in mathematical statistics and studies many different typ... |

1 |
Cost-volume relationship for flows in a disordered network. Unpublished
- Aldous
- 2004
(Show Context)
Citation Context ...τ < 1. We have δ ∗ (τ) > 0 iff τ > τFPP, where τFPP is the time constant in first passage percolation on T. As above, to study scaling exponents the key is a certain RDE for S = R + , which turns out =-=[8]-=- to be Z d = min − ( Z2 + ξ2 − a, Z3 + ξ3 − a, 3∑ ) (Zi + ξi − a) ⎛ min ⎝0, ∑ (Zi + ξi − a), ∑ ⎞ (Zi + ξi − a) ⎠ . (98) i=1,2 i=1,3 Here a is a parameter ∈ (τFPP,1). In terms of the solution of this R... |

1 |
Iksanov and Che-Soong Kim. On a Pitman-Yor problem
- Aleksander
(Show Context)
Citation Context ...e systems, and Hausdorff dimension of random Cantor-type sets. See [26, 60] for many references to linear RDEs arising in probabilistic analysis of algorithms which are analyzable by contraction. See =-=[40, 39]-=- for the specialization X d ∞∑ = h(ξi)Xi, (S = R + ) i=1 where (ξi) are the points of a Poisson process on (0, ∞). Often, within one model there are different questions which lead to both linear and m... |

1 |
and Uwe Rösler. Stochastic fixed points for the maximum. Unpublished
- Jagers
- 2004
(Show Context)
Citation Context ... is equivalent to ˆX d = max( i ˆ ξi + ˆ Xi) and this RDE, to be studied in section 5, is the fundamental example of a maxtype RDE which cannot be solved by any simple probabilistic construction. See =-=[41]-=- for further discussion of (38). 4.5 Examples of discounted tree sums Example 36 Take U d = Uniform(0, 1) and consider the RDE X d = 1 + max(UX1, (1 − U)X2), (S = R + ). (42) 30This arises [25] in th... |