Saddlepoint approximation for moments of random variables

In this paper, we introduce a saddlepoint approximation method for higher-order moments like E(S − a)₊ ^m , a>0, where the random variable S in these expectations could be a single random variable as well as the average or sum of some i.i.d random variables, and a > 0 is a constant. Numerical results are given to show the accuracy of this approximation method.     

An identity on the maximum of a set of random variables

It is shown that if X₁, X₂, …, X_n are symmetric random variables and max(X₁, …, X_n)⁺ = max(0, X₁, …, X_n), then E[max(X₁,…,X_n)⁺]=[max(X₁,X₁,+X₂,+X₁,+X₃,…X₁,+X_n)⁺], and in the case of independent identically distributed symmetric random variables, E[max(X₁, X₂)⁺] = E[(X₁)⁺] + (1/2)E[(X₁ + X₂)⁺], so that for independent standard normal random variables, E[max(X₁, X₂)⁺] = (1/√2π)[1 + (1/√2)].     

A universal result for consecutive random subdivision of polygons

We consider consecutive random subdivision of polygons described as follows. Given an initial convex polygon with edges, we choose a point at random on each edge, such that the proportions in which these points divide edges are i.i.d. copies of some random variable ξ. These new points form a new (smaller) polygon. By repeatedly implementing this procedure we obtain a sequence of random polygons. The aim of this paper is to show that under very mild non‐degenerateness conditions on ξ, the shapes of these polygons eventually become “flat” The convergence rate to flatness is also investigated; in particular, in the case of triangles (d = 3), we show how to calculate the exact value of the rate of convergence, connected to Lyapunov exponents. Using the theory of products of random matrices our paper greatly generalizes the results of [11] which are achieved mostly by using ad hoc methods. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 341–371, 2017     

A new approach to the giant component problem

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed on the size of the largest component in a random graph with a given degree sequence. We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n,p) with np = 1 + ω(n)n^?1/3, where ω(n) → ∞ arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Some calculations on the conditional densities of well-admissible measures on linear spaces

When a probability measurem on a topological vector spaceE is well-admissible in a directionk E, the conditional law in the directionk given the other directions is absolutely continuous with respect to the Lebesgue measure. We shall prove that its density function is differentiable (in the sense precised below) and we shall calculate their derivatives. We given then two applications of such calculations.     

Random walks which prefer unvisited edges: Exploring high girth even degree expanders in linear time

Let be a connected graph with vertices. A simple random walk on the vertex set of G is a process, which at each step moves from its current vertex position to a neighbouring vertex chosen uniformly at random. We consider a modified walk which, whenever possible, chooses an unvisited edge for the next transition; and makes a simple random walk otherwise. We call such a walk an edge‐process (or E‐process). The rule used to choose among unvisited edges at any step has no effect on our analysis. One possible method is to choose an unvisited edge uniformly at random, but we impose no such restriction. For the class of connected even degree graphs of constant maximum degree, we bound the vertex cover time of the E‐process in terms of the edge expansion rate of the graph G, as measured by eigenvalue gap of the transition matrix of a simple random walk on G. A vertex v is ?‐good, if any even degree subgraph containing all edges incident with v contains at least ? vertices. A graph G is ?‐good, if every vertex has the ?‐good property. Let G be an even degree ?‐good expander of bounded maximum degree. Any E‐process on G has vertex cover time This is to be compared with the lower bound on the cover time of any connected graph by a weighted random walk. Our result is independent of the rule used to select the order of the unvisited edges, which could, for example, be chosen on‐line by an adversary. As no walk based process can cover an n vertex graph in less than n – 1 steps, the cover time of the E‐process is of optimal order when . With high probability random r‐regular graphs, even, have . Thus the vertex cover time of the E‐process on such graphs is . © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 36–54, 2015     

And/or trees: A local limit point of view

We present here a new and universal approach for the study of random and/or trees, unifying in one framework many different models, including some novel ones not yet understood in the literature. An and/or tree is a Boolean expression represented in (one of) its tree shapes. Fix an integer k, take a sequence of random (rooted) trees of increasing size, say , and label each of these random trees uniformly at random in order to get a random Boolean expression on k variables. We prove that, under rather weak local conditions on the sequence of random trees , the distribution induced on Boolean functions by this procedure converges as n tends to infinity. In particular, we characterize two different behaviors of this limit distribution depending on the shape of the local limit of : a degenerate case when the local limit has no leaves; and a non‐degenerate case, which we are able to describe in more details under stronger conditions. In this latter case, we provide a relationship between the probability of a given Boolean function and its complexity. The examples covered by this unified framework include trees that interpolate between models with logarithmic typical distances (such as random binary search trees) and other ones with square root typical distances (such as conditioned Galton–Watson trees).     

An Iterative Approach to Determining the Length of the Longest Common Subsequence of Two Strings

This paper concerns the longest common subsequence (LCS) shared by two sequences (or strings) of length N, whose elements are chosen at random from a finite alphabet. The exact distribution and the expected value of the length of the LCS, k say, between two random sequences is still an open problem in applied probability. While the expected value E(N) of the length of the LCS of two random strings is known to lie within certain limits, the exact value of E(N) and the exact distribution are unknown. In this paper, we calculate the length of the LCS for all possible pairs of binary sequences from N=1 to 14. The length of the LCS and the Hamming distance are represented in color on two all-against-all arrays. An iterative approach is then introduced in which we determine the pairs of sequences whose LCS lengths increased by one upon the addition of one letter to each sequence. The pairs whose score did increase are shown in black and white on an array, which has an interesting fractal-like structure. As the sequence length increases, R(N) (the proportion of sequences whose score increased) approaches the Chvátal–Sankoff constant a _c (the proportionality constant for the linear growth of the expected length of the LCS with sequence length). We show that R(N) is converging more rapidly to a _c than E(N)/N.     

Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues

We consider a class of random matrix ensembles which can be constructed from the random permutation matrices by replacing the nonzero entries of the n×n permutation matrix matrix with M×M diagonal matrices whose entries are random Kth roots of unity or random points on the unit circle. Let X be the number of eigenvalues lying in a specified arc I of the unit circle, and consider the standardized random variable (X–E[X])/(Var(X))^1/2. We show that for a fixed set of arcs I ₁,...,I _N , the corresponding standardized random variables are jointly normal in the large n limit, and compare the covariance structures which arise with results for other random matrix ensembles.     

Delaying satisfiability for random 2SAT

Let (C₁,C′_(*)),(C₂,C′_(*)),…,(C _m,C′_(*)) be a sequence of ordered pairs of 2CNF clauses chosen uniformly at random (with replacement) from the set of all 4 \begin{align*}\binom{n}{2}\end{align*} clauses on n variables. Choosing exactly one clause from each pair defines a probability distribution over 2CNF formulas. The choice at each step must be made on‐line, without backtracking, but may depend on the clauses chosen previously. We show that there exists an on‐line choice algorithm in the above process which results whp in a satisfiable 2CNF formula as long as m/n ≤ (1000/999)^1/4. This contrasts with the well‐known fact that a random m ‐clause formula constructed without the choice of two clauses at each step is unsatisfiable whp whenever m/n > 1. Thus the choice algorithm is able to delay satisfiability of a random 2CNF formula beyond the classical satisfiability threshold. Choice processes of this kind in random structures are known as “Achlioptas processes.” This paper joins a series of previous results studying Achlioptas processes in different settings, such as delaying the appearance of a giant component or a Hamilton cycle in a random graph. In addition to the on‐line setting above, we also consider an off‐line version in which all m clause‐pairs are presented in advance, and the algorithm chooses one clause from each pair with knowledge of all pairs. For the off‐line setting, we show that the two‐choice satisfiability threshold for k ‐SAT for any fixed k coincides with the standard satisfiability threshold for random 2k ‐SAT.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

On calculation of moments of ladder heights

Let X,X ₁,X ₂, … be independent identically distributed random variables, F(x) = P{X < x}, S ₀ = 0, and S _n =Σ _i=1 ⁿ X _i . We consider the random variables, ladder heights Z ₊ and Z ₋ that are respectively the first positive sum and the first negative sum in the random walk {S _n }, n = 0, 1, 2, …. We calculate the first three (four in the case EX = 0) moments of random variables Z ₊ and Z ₋ in the qualitatively different cases EX > 0, EX < 0, and EX = 0. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 159–179, April–June, 2006.     

A characterization of probability distributions by moments of sums of independent random variables

In this paper we investigate the conditions under which the distribution of i.i.d. random variables{X _k } _k=1 is determined by the sequence of momentsa _n =E| _k=1 ⁿ X _k | ^p (n=1, 2,...), where positivep is fixed.     

Efficacy of cones on topological vector spaces and application to common fixed points of multifunctions

Let (E, τ) be a topological vector space and P a cone in E. We shall define a topology τ _P on E so that (E, τ _P ) is a normable topological vector space and P is a normal cone with normal constant M = 1. Then by using the norm, we shall give some results about common fixed points of two multifunctions on cone metric spaces.     

Scaling limits of random bipartite planar maps with a prescribed degree sequence

We study the asymptotic behavior of uniform random maps with a prescribed face‐degree sequence, in the bipartite case, as the number of faces tends to infinity. Under mild assumptions, we show that, properly rescaled, such maps converge in distribution toward the Brownian map in the Gromov–Hausdorff sense. This result encompasses a previous one of Le Gall for uniform random q‐angulations where q is an even integer. It applies also to random maps sampled from a Boltzmann distribution, under a second moment assumption only, conditioned to be large in either of the sense of the number of edges, vertices, or faces. The proof relies on the convergence of so‐called “discrete snakes” obtained by adding spatial positions to the nodes of uniform random plane trees with a prescribed child sequence recently studied by Broutin and Marckert. This paper can alternatively be seen as a contribution to the study of the geometry of such trees.     

Almost sure central limit theorems for random functions

Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn Rn, where supn E|Rn| <∞and Rn = o(n~(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc.     

Random Logistic Maps. I

Let {C _i} ₀ be a sequence of independent and identically distributed random variables with vales in [0, 4]. Let {X _n} ₀ be a sequence of random variables with values in [0, 1] defined recursively by X _n+1=C _n+1 X _n(1–X _n). It is shown here that: (i) E ln C ₁<0X _n0 w.p.1. (ii) E ln C ₁=0X _n0 in probability (iii) E ln C ₁>0, E |ln(4–C ₁)| such that (0, 1)=1 and is invariant for {X _n}. (iv) If there exits an invariant probability measure such that {0}=0, then E ln C ₁>0 and – ln(1–x) (dx)=E ln C ₁. (v) E ln C ₁>0, E |ln(4–C ₁)|< and {X _n} is Harris irreducible implies that the probability distribution of X _n converges in the Cesaro sense to a unique probability distribution on (0, 1) for all X ₀0.     

A concentration result with application to subgraph count

Let H = (V,E) be a k ‐uniform hypergraph with a vertex set V and an edge set E. Let V _p be constructed by taking every vertex in V independently with probability p. Let X be the number of edges in E that are contained in V _p. We give a condition that guarantees the concentration of X within a small interval around its mean. The applicability of this result is demonstrated by deriving new sub‐Gaussian tail bounds for the number of copies of small complete and complete bipartite graphs in the binomial random graph. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

What is the furthest graph from a hereditary property?

For a graph property P, the edit distance of a graph G from P, denoted E_P(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n,P). This question is motivated by algorithmic edge‐modification problems, in which one wishes to find or approximate the value of E_P(G) given an input graph G. A monotone graph property is closed under removal of edges and vertices. Trivially, for any monotone property, the largest edit distance is attained by a complete graph. We show that this is a simple instance of a much broader phenomenon. A hereditary graph property is closed under removal of vertices. We prove that for any hereditary graph property P, a random graph with an edge density that depends on P essentially achieves the maximal distance from P, that is: ed(n,P) = E_P(G(n,p(P))) + o(n²) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi regularity lemma, properties of random graphs and probabilistic arguments. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008