Weighted Branching Process Research Articles

Probability Inequalities for Weighted Branching Processes in Random Environments

Probability Inequalities for Weighted Branching Processes in Random Environments

Small counts in nested Karlin’s occupancy scheme generated by discrete Weibull-like distributions

A nested Karlin’s occupancy scheme is a symbiosis of classical Karlin’s balls-in-boxes scheme and a weighted branching process. To define it, imagine a deterministic weighted branching process in which weights of the first generation individuals are given by the elements of a discrete probability distribution. For each positive integer j, identify the jth generation individuals with the jth generation boxes. The collection of balls is one and the same for all generations, and each ball starts at the root of the weighted branching process tree and moves along the tree according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights.Assume that there are n balls and that the discrete probability distribution (pk)k∈N responsible for the first generation is Weibull-like. This means that the counting function of 1/p1, 1/p2,… belongs to de Haan’s class Π, a subclass of the class of slowly varying functions. Denote by Kn(j)(l) and Kn∗(j)(l) the number of the jth generation boxes which contain at least l balls and exactly l balls, respectively. We prove functional limit theorems (FLTs) for the matrix-valued processes (K[eT+⋅](j)(l))j,l∈N and (K[eT+⋅]∗(j)(l))j,l∈N, properly normalized and centered, as T→∞. The present FLTs are an extension of a FLT proved by Iksanov et al. (2022) for the vector-valued process (K[eT+⋅](j)(1))j∈N. While the rows of each of the limit matrix-valued processes are independent and identically distributed, the entries within each row are stationary Gaussian processes with explicitly given covariances and cross-covariances. We provide an integral representation for each row. The results obtained are new even for Karlin’s occupancy scheme.

A functional limit theorem for nested Karlin's occupancy scheme generated by discrete Weibull-like distributions

Let (pk)k∈N be a discrete probability distribution for which the counting function x↦#{k∈N:pk≥1/x} belongs to the de Haan class Π. Consider a deterministic weighted branching process generated by (pk)k∈N. A nested Karlin's occupancy scheme is the sequence of Karlin balls-in-boxes schemes in which boxes of the jth level, j=1,2,… are identified with the jth generation individuals and the hitting probabilities of boxes are identified with the corresponding weights. The collection of balls is the same for all generations, and each ball starts at the root and moves along the tree of the deterministic weighted branching process according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights. Assuming there are n balls, denote by Kn(j) the number of occupied (ever hit) boxes in the jth level. For each j∈N, we prove a functional limit theorem for the vector-valued process (K⌊eT+u⌋(1),…,K⌊eT+u⌋(j))u∈R, properly normalized and centered, as T→∞. The limit is a vector-valued process whose components are independent stationary Gaussian processes. An integral representation of the limit process is obtained.

Late levels of nested occupancy scheme in random environment

Consider a weighted branching process generated by a point process on whose atoms sum up to one. Then the weights of all individuals in any given generation sum up to one, as well. We define a nested occupancy scheme in random environment as the sequence of balls-in-boxes schemes (with random probabilities) in which boxes of the jth level, are identified with the jth generation individuals and the hitting probabilities of boxes are identified with the corresponding weights. The collection of balls is the same for all generations, and each ball starts at the root and moves along the tree of the weighted branching process according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights. Assuming that there are n balls, we give a full classification of regimes of the a.s. convergence for the number of occupied (ever hit) boxes in the jth level, properly normalized, as n and j = jn grow to Here, is a sequence of positive numbers growing proportionally to We call such levels late, for the nested occupancy scheme gets extinct in the levels which grow super-logarithmically in n in the sense that each occupied box contains one ball. Also, in some regimes we prove the strong laws of large numbers (a) for the number of the jth level boxes which contain at least k balls, and (b) under the assumption that the mean number of the first level boxes is finite, for the number of empty boxes in the jth level.

On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I

Consider a weighted branching process generated by the lengths of intervals obtained by stick-breaking of unit length (a.k.a. the residual allocation model) and associate with each weight a `box'. Given the weights `balls' are thrown independently into the boxes of the first generation with probability of hitting a box being equal to its weight. Each ball located in a box of the $j$th generation, independently of the others, hits a daughter box in the $(j+1)$th generation with probability being equal the ratio of the daughter weight and the mother weight. This is what we call nested occupancy scheme in random environment. Restricting attention to a particular generation one obtains the classical Karlin occupancy scheme in random environment. Assuming that the stick-breaking factor has a uniform distribution on $[0,1]$ and that the number of balls is $n$ we investigate occupancy of intermediate generations, that is, those with indices $\lfloor j_n u\rfloor$ for $u>0$, where $j_n$ diverges to infinity at a sublogarithmic rate as $n$ becomes large. Denote by $K_n(j)$ the number of occupied (ever hit) boxes in the $j$th generation. It is shown that the finite-dimensional distributions of the process $(K_n(\lfloor j_n u\rfloor))_{u>0}$, properly normalized and centered, converge weakly to those of an integral functional of a Brownian motion. The case of a more general stick-breaking is also analyzed.

Stochastic fixed-point equations

The weighted branching process is a natural extension of branching processes. Branching processes count only individuals, weighted branching processes give every individual an abstract weight. Mathematically, we face a (random) dynamical system indexed by a tree. We give an overview of research into this directions, including and unifying many examples of this structure, like Biggins branching random walk and Mandelbrot cascades. WBP are closely connected to stochastic fixed-point equations, fractals and have many applications in genetics, computer science and algorithms. The most advanced example we present is the Quicksort process.

Performance Analysis of Sorting Process with Different Sampling Strategies

Sorting data is one of the most important problems that play an important rule in many applications in operations research, computer science and many other applications. Many sorting algorithms are well studied but the problem is not to find a way or algorithm to sort elements, but to find an efficiently way to sort elements and do the job. The output is a stream of data in time and it is a sorted data array. We are interested in this flow of data to estaplish a smart technique to sort elements as well as efficient complexity. For the performance of such algorithms, there has been little research on their stochastic behavior and mathematical properties such existance and convergence properties. In this paper we study the mathematical behavior of some different versions sorting algorithms in the case when the size of the input is very large. This work also discuss the corresponding running time using some different strategies in terms of number of comparisons and swaps. Here, we use a nice approach to show the existence of partial sorting process via the weighted branching process. This approach was inspired by the methods used for the analysis of Quickselect and Quichsort in the standard cases, where fixed point equations on the Cadlag space were considered for the first time.

Analysis of Clusters in Network Graphs for Personalized Web Search

PageRank is one of the most popular measures of the importance of Web pages. The dual nature of PageRank is not yet completely understood. From its definition the PageRank of a Web page (node) is the probability to find a random walk at this node when the process has reached the steady state. From another side, PageRank of a randomly selected page is a hidden Markov process due to the random number of its in- and out-degrees. Considering the stochastic nature of PageRank we aim to study its extremal index that represents the dependence measure of extremes and plays a fundamental role in the theory of extreme values. Using the representation of PageRank as a weighted branching process and the corresponding Thorny Branching Tree, we propose a nonparametric estimation of the extremal index of PageRank by samples of moderate sizes. It is based on the representation of the reciprocal of the extremal index by a mean cluster size. The cluster implies a block of node ranks with at least one exceedance over a sufficiently high threshold. It is proposed to consider generations of successors of a root node of the PageRank branching process as blocks. Among practical advantages the extremal index determines the mean first hitting time (or the mean minimal time) to reach an influential node with a large PageRank. We consider a Max-linear model as an alternative to PageRank and compare its distribution with that one of PageRank.

Coupling on weighted branching trees

Abstract In this paper we consider linear functions constructed on two different weighted branching processes and provide explicit bounds for their Kantorovich–Rubinstein distance in terms of couplings of their corresponding generic branching vectors. Motivated by applications to the analysis of random graphs, we also consider a variation of the weighted branching process where the generic branching vector has a different dependence structure from the usual one. By applying the bounds to sequences of weighted branching processes, we derive sufficient conditions for the convergence in the Kantorovich–Rubinstein distance of linear functions. We focus on the case where the limits are endogenous fixed points of suitable smoothing transformations.

Weighted moments for Mandelbrot's martingales

Let $(Y_n)_{n\ge0}$ be a Mandelbrot's martingale defined as sums of products of random weights indexed by nodes of a Galton-Watson tree, and let $Y$ be its limit. We show a necessary and sufficient condition for the existence of weighted moments of $Y$ of the forms $\mathbb{E}Y^{\alpha}\ell(Y)$, where $\alpha>1$ and $\ell$ is a positive function slowly varying at $\infty$. We also show a sufficient condition in the case of $\alpha=1$. Our results complete those of Alsmeyer and Kuhlbusch (2010) for weighted branching processes by removing their extra conditions on $\ell$.

Asymptotics for Weighted Random Sums

Let {X i} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C 1, C 2,…) be a nonnegative random vector independent of the {X i } with N∈ℕ∪ {∞}. We study the weighted random sum S N =∑{i=1} N C i X i , and its maximum, M N =sup{1≤k N+1∑ i=1 k C i X i . This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P(M N > x)∼ P(S N > x)∼ E[∑ i=1 N F̄(x/C i )] as x→∞. When E[X 1]>0 and the distribution of Z N =∑ i=1 N C i is also intermediate regularly varying, we obtain the asymptotics P(M N > x)∼ P(S N > x)∼ E[∑ i=1 N F̄}(x/C i )] +P(Z N > x/E[X 1]). For completeness, when the distribution of Z N is intermediate regularly varying and heavier than F̄, we also obtain conditions under which the asymptotic relations P(M N > x) ∼ P(S N > x)∼ P(Z N > x / E[X 1 ] hold.

Asymptotics for Weighted Random Sums

Let {Xi} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C1, C2,…) be a nonnegative random vector independent of the {Xi} with N∈ℕ∪ {∞}. We study the weighted random sum SN=∑{i=1}NCiXi, and its maximum, MN=sup{1≤kN+1∑i=1kCiXi. This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P(MN > x)∼ P(SN > x)∼ E[∑i=1NF̄(x/Ci)] as x→∞. When E[X1]>0 and the distribution of ZN=∑ i=1NCi is also intermediate regularly varying, we obtain the asymptotics P(MN > x)∼ P(SN > x)∼ E[∑i=1NF̄}(x/Ci)] +P(ZN > x/E[X1]). For completeness, when the distribution of ZN is intermediate regularly varying and heavier than F̄, we also obtain conditions under which the asymptotic relations P(MN > x) ∼ P(SN > x)∼ P(ZN > x / E[X1] hold.

The analysis of find and versions of it

Analysis of Algorithms In the running time analysis of the algorithm Find and versions of it appear as limiting distributions solutions of stochastic fixed points equation of the form X D = Sigma(i) AiXi o Bi + C on the space D of cadlag functions. The distribution of the D-valued process X is invariant by some random linear affine transformation of space and random time change. We show the existence of solutions in some generality via the Weighted Branching Process. Finite exponential moments are connected to stochastic fixed point of supremum type X D = sup(i) (A(i)X(i) + C-i) on the positive reals. Specifically we present a running time analysis of m-median and adapted versions of Find. The finite dimensional distributions converge in L-1 and are continuous in the cylinder coordinates. We present the optimal adapted version in the sense of low asymptotic average number of comparisons. The limit distribution of the optimal adapted version of Find is a point measure on the function [0, 1] there exists t -> 1 + mint, 1 - t.

Information ranking and power laws on trees

In this paper we consider the stochastic analysis of information ranking algorithms of large interconnected data sets, e.g. Google's PageRank algorithm for ranking pages on the World Wide Web. The stochastic formulation of the problem results in an equation of the formwhereN,Q, {Ri}i≥1, and {C,Ci}i≥1are independent nonnegative random variables, the {C,Ci}i≥1are identically distributed, and the {Ri}i≥1are independent copies ofstands for equality in distribution. We study the asymptotic properties of the distribution ofRthat, in the context of PageRank, represents the frequencies of highly ranked pages. The preceding equation is interesting in its own right since it belongs to a more general class of weighted branching processes that have been found to be useful in the analysis of many other algorithms. Our first main result shows that if ENE[Cα] = 1, α > 0, andQ,Nsatisfy additional moment conditions, thenRhas a power law distribution of index α. This result is obtained using a new approach based on an extension of Goldie's (1991) implicit renewal theorem. Furthermore, whenNis regularly varying of index α > 1, ENE[Cα] < 1, andQ,Chave higher moments than α, then the distributions ofRandNare tail equivalent. The latter result is derived via a novel sample path large deviation method for recursive random sums. Similarly, we characterize the situation when the distribution ofRis determined by the tail ofQ. The preceding approaches may be of independent interest, as they can be used for analyzing other functionals on trees. We also briefly discuss the engineering implications of our results.

Information ranking and power laws on trees

In this paper we consider the stochastic analysis of information ranking algorithms of large interconnected data sets, e.g. Google's PageRank algorithm for ranking pages on the World Wide Web. The stochastic formulation of the problem results in an equation of the form where N, Q, {Ri}i≥1, and {C, Ci}i≥1 are independent nonnegative random variables, the {C, Ci}i≥1 are identically distributed, and the {Ri}i≥1 are independent copies of stands for equality in distribution. We study the asymptotic properties of the distribution of R that, in the context of PageRank, represents the frequencies of highly ranked pages. The preceding equation is interesting in its own right since it belongs to a more general class of weighted branching processes that have been found to be useful in the analysis of many other algorithms. Our first main result shows that if ENE[Cα] = 1, α > 0, and Q, N satisfy additional moment conditions, then R has a power law distribution of index α. This result is obtained using a new approach based on an extension of Goldie's (1991) implicit renewal theorem. Furthermore, when N is regularly varying of index α > 1, ENE[Cα] < 1, and Q, C have higher moments than α, then the distributions of R and N are tail equivalent. The latter result is derived via a novel sample path large deviation method for recursive random sums. Similarly, we characterize the situation when the distribution of R is determined by the tail of Q. The preceding approaches may be of independent interest, as they can be used for analyzing other functionals on trees. We also briefly discuss the engineering implications of our results.

A Note on the Transience of Critical Branching Random Walks on the Line

Gantert and Müller (2006) proved that a critical branching random walk (BRW) on the integer lattice is transient by analyzing this problem within the more general framework of branching Markov chains and making use of Lyapunov functions. The main purpose of this note is to show how the same result can be derived quite elegantly and even extended to the nonlattice case within the theory of weighted branching processes. This is done by an analysis of certain associated random weighted location measures which, upon taking expectations, provide a useful connection to the well established theory of ordinary random walks with i.i.d. increments. A brief discussion of the asymptotic behavior of the left- and rightmost particles in a critical BRW as time goes to infinity is provided in the final section by drawing on recent work by Hu and Shi (2008).

Probability Theory on Trees and Analysis of Algorithms

The analysis of algorithms and data structures, started by D. Knuth, is a rapidly growing area at the interface of Mathematics and Theoretical Computer Science. Probability theory enters the subject in a natural way when studying an algorithm as to its performance over randomized inputs and/or if an algorithm itself takes randomization steps. The latter holds true for so-called divide and conquer algorithms, which were a central topic of this workshop. The sorting algorithm is presumably the most prominent example. While early work was mostly based on the generating function approach, the last two decades have seen an increasing number of contributions based on probabilistic methods involving martingales, random trees and other stochastic processes. The first article of this type was written in the eighties by L. Devroye, a Humboldt award winner of 2004 and also one of the organizers. Later another pioneering contribution came by U. Rösler who, by introducing weighted branching processes and the contraction method, determined the asymptotic distributional behavior of . One can say that much of the work presented at this workshop is to some extent spawn by these articles. The aim and scope of the mini-workshop was to bring together leading junior and senior experts in the field with a strong probabilistic background and a focus on divide an conquer algorithms and related data structures. There were 16 one hour talks presented by 12 of the 14 participants from 7 countries. Main topics were branching processes, the contraction method, the asymptotic analysis of random trees, stochastic fixed point equations and randomized algorithms.

On weighted branching processes in random environment

In this paper we investigate the nonnegative martingale W n=Z n/μ n( U ), n⩾0 and its a.s. limit W, when ( Z n ) n⩾0 is a weighted branching process in random environment with stationary ergodic environmental sequence U=( U n ) n⩾0 and μ n ( U) denotes the conditional expectation of Z n given U for n⩾0. We find necessary and sufficient conditions for W to be nondegenerate, generalizing earlier results in the literature on ordinary branching processes in random environment and also weighted branching processes. In the important special case of i.i.d. random environment, a Z log Z -condition turns out to be crucial. Deterministic and nonvarying environments are treated as special cases. Our arguments adapt the probabilistic proof of Biggins’ theorem for branching random walks given by Lyons (1997) to our situation.

Convergence conditions for weighted branching processes

Convergence conditions for weighted branching processes

A fixed point theorem for distributions

We study in a systematic form the contractive behavior of the map S of distributions to distributions S(F) = D ∑ iT iX i+C,(C,T=(T 1,T 2…)) , X i are independent r.v., L( X i ) = F. Further we show higher and exponential moments of the fixed point. Applications of this structure are given for (a) weighted branching processes, (b) the Hausdorff dimension of random Cantor sets and (c) the sorting algorithm Quicksort.