Additive tree functionals represent the cost of many divide-and-conquer algorithms. We derive the limiting distribution of the additive functionals induced by toll functions of the form (a) n^αwhen α> 0 and (b) log n (the so-called shape functional) on uniformly distributed binary search trees, sometimes called Catalan trees. The Gaussian law obtained in the latter case complements the central limit theorem for the shape functional under the random permutation model. Our results give rise to an apparently new family of distributions containing the Airy distribution (α= 1) and the normal distribution [case (b), and case (a) as $α\downarrow 0$]. The main theoretical tools employed are recent results relating asymptotics of the generating functions of sequences to those of their Hadamard product, and the method of moments.
Suppose Vν is the pseudo-variance function of the Cauchy-Stieltjes Kernel (CSK) family K+(ν) generated by a non degenerate probability measure ν with support bounded from above. We determine the formula for pseudo-variance function (or variance function Vν in case of existence) under boolean additive convolution power. This formulas is used to identify the relation between variance functions under Boolean Bercovici-Bata Bijection between probability measures. We also gives the connection between boolean cumulants and variance function and we relate boolean cumulants of some probability measure…
When some eigenvalues of a spiked multiplicative resp. additive deformation model of a Hermitian Wigner matrix resp. a sample covariance matrix separate from the bulk, we study how the corresponding eigenvectors project onto those of the perturbation. We point out that the inverse of the subordination function relative to the free additive resp. multiplicative convolution plays an important part in the asymptotic behavior.
In this paper, we first introduce the explicit analytical formulas for the expected values of the Gutman and Schultz indices for a random cyclooctane chain COCn. Meanwhile, the explicit formulas of the variances of the Gutman and Schultz indices for a random cyclooctane chain are determined and we prove these two indices are asymptotically subject to normal distribution. Furthermore, we are surprised to find the variances of Kf*(COCn) and Kf+(COCn) for a random cyclooctane chain based on the known results of others' paper and they are asymptotically subject to normal distribution.
We study the shape of the normalized stable Lévy tree $\mathcal{T}$ near its root. We show that, when zooming in at the root at the proper speed with a scaling depending on the index of stability, we get the unnormalized Kesten tree. In particular the limit is described by a tree-valued Poisson point process which does not depend on the initial normalization. We apply this to study the asymptotic behavior of additive functionals of the form \[\mathbf{Z}_{α,β}=\int_{\mathcal{T}} μ(\mathrm{d} x) \int_0^{H(x)} σ_{r,x}^α\mathfrak{h}_{r,x}^β\,\mathrm{d} r\]as $\max(α,β) \to \infty$, where $μ$ is th…
For two decades, the Colless index has been the most frequently used statistic for assessing the balance of phylogenetic trees. In this article, this statistic is studied under the Yule and uniform model of phylogenetic trees. The main tool of analysis is a coupling argument with another well-known index called the Sackin statistic. Asymptotics for the mean, variance and covariance of these two statistics are obtained, as well as their limiting joint distribution for large phylogenies. Under the Yule model, the limiting distribution arises as a solution of a functional fixed point equation. Un…
Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on $m$-ary search trees on $n$ keys with toll sequence (i) $n^α$ with $α\geq 0$ ($α=0$ and $α=1$ correspond roughly to the space requirement and total path length, respectively); (ii) $\ln \binom{n}{m-1}$, which corresponds to the so-called shape functional; and (iii) $\mathbf{1}_{n=m-1}$, which corresponds to the number of leaves.
We develop a unified approach to the problem of clustering in the three different fields of applications, as indicated in the title the paper. The approach is based on Khintchine's probabilistic method that grew out of the Darwin-Fawler method in statistical physics. Our main result is the derivation of asymptotic formulae for probabilities of clusters (= groups) of certain sizes as the number of particles goes to infinity. Based on these formulae we prove the zero-one law for the distribution of the largest cluster and establish the threshold function in the phase transition from 0 to 1 in th…
We study SDEs arising from limiting fluctuations in a $(2+1)$-dimensional surface growth model called the Whittaker driven particle system, which is believed to be in the anisotropic Kardar--Parisi--Zhang class. The main result of this paper proves an irrelevance of nonlinearity in the surface growth model in the continuum by weak convergence in a path space; the first instance of this irrelevance is obtained recently for this model in terms of the covariance functions along certain diverging characteristics. With the same limiting scheme, we prove that the derived SDEs converge in distributio…
Recent works have shown that the family of probability distributions with moments given by the Fuss-Catalan numbers permit a simple parameterized form for their density. We extend this result to the Raney distribution which by definition has its moments given by a generalization of the Fuss-Catalan numbers. Such computations begin with an algebraic equation satisfied by the Stieltjes transform, which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the Raney distribution. For the Fuss-Catalan distribution, …
Using the renewal approach we prove Bernstein-like inequalities for additive functionals of geometrically ergodic Markov chains, thus obtaining counterparts of inequalities for sums of independent random variables. The coefficient in the sub-Gaussian part of our estimate is the asymptotic variance of the additive functional, i.e. the variance of the limiting Gaussian variable in the Central Limit Theorem for Markov chains. This refines earlier results by R. Adamczak and W. Bednorz, which were obtained under the additional assumption of strong aperiodicity of the chain.
In this paper we study the central limit theorem for additive functionals of stationary Markov chains with general state space by using a new idea involving conditioning with respect to both the past and future of the chain. Practically, we show that any additive functionals of a stationary and totally ergodic Markov chain with the variance of S_n divided by n uniformly bounded, satisfies a square root central limit theorem with a random centering. We do not assume that the Markov chain is irreducible and aperiodic. However, the random centering is not needed if the Markov chain satisfies stro…
We consider linear preferential attachment random trees with additive fitness, where fitness is defined as the random initial vertex attractiveness. We show that when the fitness distribution has positive bounded support, the weak local limit of this family can be constructed using a sequence of mixed Poisson point processes. We also provide a rate of convergence of the total variation distance between the r- neighbourhood of the uniformly chosen vertex in the preferential attachment tree and that of the root vertex of its weak local limit. We apply the theorem to obtain the limiting degree di…
Search trees are fundamental data structures in computer science. We study functionals on random search trees that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the so-called shape functional fall under this framework. Our goal is to derive asymptotics of moments and identify limiting distributions of these functionals under two commonly studied probability models -- the random permutation model and the uniform model. For the random permutation model, our approach is based on establishing transfer t…
Error Distribution of the Local Linearization Method for Stochastic Differential Equations with Additive Brownian Noise
Graduate School of Engineering Science, The University of Osaka（大阪大学工学研究科）；Department of Statistics, London School of Economics and Political Science（伦敦政治经济学院统计学系）
We consider sequences of additive functionals of difference approximations for uniformly non-degenerate multidimensional diffusions. The conditions are given, sufficient for such a sequence to converge weakly to a W-functional of the limiting process. The class of the W-functionals, that can be obtained as the limiting ones, is completely described in the terms of the associated W-measures, and coincides with the class of the functionals that are regular w.r.t. the phase variable.
We study the additive functional $X_n(α)$ on conditioned Galton-Watson trees given, for arbitrary complex $α$, by summing the $α$th power of all subtree sizes. Allowing complex $α$ is advantageous, even for the study of real $α$, since it allows us to use powerful results from the theory of analytic functions in the proofs. For $\Reα< 0$, we prove that $X_n(α)$, suitably normalized, has a complex normal limiting distribution; moreover, as processes in $α$, the weak convergence holds in the space of analytic functions in the left half-plane. We establish, and prove similar process-convergence e…
In this paper we study the diameter of the random graph $G(n,p)$, i.e., the the largest finite distance between two vertices, for a wide range of functions $p=p(n)$. For $p=\la/n$ with $\la>1$ constant, we give a simple proof of an essentially best possible result, with an $O_p(1)$ additive correction term. Using similar techniques, we establish 2-point concentration in the case that $np\to\infty$. For $p=(1+ε)/n$ with $ε\to 0$, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an $O_p(1/ε)$ additive correction term whose (approp…
We study the asymptotic behavior of mixed functionals of the form $I_T(t)=F_T(ξ_T(t))+\int_0^tg_T(ξ_T(s))\,dξ_T(s)$, $t\ge0$, as $T\to\infty$. Here $ξ_T(t)$ is a strong solution of the stochastic differential equation $dξ_T(t)=a_T(ξ_T(t))\,dt+dW_T(t)$, $T>0$ is a parameter, $a_T=a_T(x)$ are measurable functions such that $\left|a_T(x)\right|\leq C_T$ for all $x\in \mathbb {R}$, $W_T(t)$ are standard Wiener processes, $F_T=F_T(x)$, $x\in \mathbb {R}$, are continuous functions, $g_T=g_T(x)$, $x\in \mathbb {R}$, are locally bounded functions, and everything is real-valued. The explicit form of th…
The limiting distribution of eigenvalues of N x N random matrices has many applications. One of the most studied ensembles are real symmetric matrices with independent entries iidrv; the limiting rescaled spectral measure (LRSM) $\widetildeμ$ is the semi-circle. Studies have determined the LRSMs for many structured ensembles, such as Toeplitz and circulant matrices. These have very different behavior; the LRSM for both have unbounded support. Given a structured ensemble such that (i) each random variable occurs o(N) times in each row and (ii) the LRSM exists, we introduce a parameter to contin…
We investigate the asymptotic behavior of the eigenvalues of spiked perturbations of Wigner matrices when the dimension goes to infinity. The entries of the Hermitian Wigner matrix have a distribution which is symmetric and satisfies a Poincaré inequality. The perturbation matrix is a deterministic Hermitian matrix whose spectral measure converges to some probability measure with compact support. We assume that this perturbation matrix has a fixed number of fixed eigenvalues (spikes) outside the support of its limiting spectral measure whereas the distance between the other eigenvalues and thi…
We consider very general "random integers" and (attempt to) prove that many multiplicative and additive functions of such integers have limiting distributions. These integers include, for instance, the curvatures of Apollonian circle packings, trace of Frobenius elements for elliptic curves, the Ramanujan tau-function, Mersenne numbers, and many others.
In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of selfadjoint commuting random variables in an infinitesimal triangular array are determined. These distributions are characterized by their bi-freely infinite divisibility, and moreover, a transfer principle is established for limit theorems in classical probability theory and Voiculescu's bi-free probability theory. Complete descriptions of bi-free stability and fullness of planar probability distributions are also set down. All these results…
