Conditional Limit Theorem Research Articles

A Conditioned Local Limit Theorem for Nonnegative Random Matrices

A Conditioned Local Limit Theorem for Nonnegative Random Matrices

Conditional Functional Limit Theorem for a Random Recurrence Sequence Conditioned on a Large Deviation Event

Conditional Functional Limit Theorem for a Random Recurrence Sequence Conditioned on a Large Deviation Event

The intransitive dice kernel: 1x≥y-1x≤y4-3(x-y)(1+xy)8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{\\mathbbm {1}_{x\\ge y}-\\mathbbm {1}_{x\\le y}}{4} - \\frac{3(x-y)(1+xy)}{8}$$\\end{document}

Answering a pair of questions of Conrey, Gabbard, Grant, Liu, and Morrison, we prove that a triplet of dice drawn from the multiset model are intransitive with probability 1/4+o(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/4+o(1)$$\\end{document} and the probability a random pair of dice tie tends toward αn-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha n^{-1}$$\\end{document} for an explicitly defined constant α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. This extends and sharpens the recent results of Polymath regarding the balanced sequence model. We further show the distribution of larger tournaments converges to a universal tournamenton in both models. This limit naturally arises from the discrete spectrum of a certain skew-symmetric operator (given by the kernel in the title acting on L2([-1,1])\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2([-1,1])$$\\end{document}). The limit exhibits a degree of symmetry and can be used to prove that, for instance, the limiting probability that Ai\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_i$$\\end{document} beats Ai+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_{i+1}$$\\end{document} for 1≤i≤4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le i\\le 4$$\\end{document} and that A5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_5$$\\end{document} beats A1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_1$$\\end{document} is 1/32+o(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/32+o(1)$$\\end{document}. Furthermore, the limiting tournamenton has range contained in the discrete set {0,1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{0,1\\}$$\\end{document}. This proves that the associated tournamenton is non-quasirandom in a dramatic fashion, vastly extending work of Cornacchia and Hązła regarding the continuous analogue of the balanced sequence model. The proof is based on a reduction to conditional central limit theorems (related to work of Polymath), the use of a “Poissonization” style method to reduce to computations with independent random variables, and the systematic use of switching-based arguments to extract cancellations in Fourier estimates when establishing local limit-type estimates.

Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls

In this paper, we prove a Sanov-type large deviation principle for the sequence of empirical measures of vectors chosen uniformly at random from an Orlicz ball. From this level-2 large deviation result, in a combination with Gibbs conditioning, entropy maximization and an Orlicz version of the Poincaré-Maxwell-Borel lemma, we deduce a conditional limit theorem for high-dimensional Orlicz balls. In more geometric parlance, the latter shows that if V1 and V2 are Orlicz functions, then random points in the V1-Orlicz ball, conditioned on having a small V2-Orlicz radius, look like an appropriately scaled V2-Orlicz ball. In fact, we show that the limiting distribution in our Poincaré-Maxwell-Borel lemma, and thus the geometric interpretation, undergoes a phase transition depending on the magnitude of the V2-Orlicz radius.

Conditional central limit theorem for critical branching random walk

Conditional central limit theorem for critical branching random walk

Conditional functional limit theorem for random reccurence sequence conditioned on large deviation event

Пусть $Z_n, n\geqslant 0\}$ - ветвящийся процесс в случайной среде, представляющей собой последовательность независимых одинаково распределенных случайных величин, $\{S_n, n\geqslant 1\}$ - его сопровождающее блуждание, $\xi_i$ - шаги сопровождающего блуждания. В предположении, что $\xi_1$ удовлетворяет условию Крамера и выполнены моментные условия на количество потомков одной частицы, для $Z_n$ известна асимптотика вероятностей больших уклонений $\mathbf{P}(\ln Z_n > x)$, где $x/n>\mu^*$, $\mu^*$ - некоторый параметр, зависящий от типа процесса. В работе исследуется поведение траектории процесса, совершающего такого рода уклонение. В частности, получена условная функциональная предельная теорема для траектории $(Z_{[nt]}, t\in [0,1])$, рассматриваемой при условии совершения ею большого уклонения $\ln Z_n>x$. Результат получен для более общей модели случайной рекуррентной последовательности $Y_{n+1}=A_n Y_n + B_n$, $n\geqslant 0$, где последовательность $\{A_i\}$ является последовательностью независимых одинаково распределенных величин, а $Y_0$, $B_i$, $i\geqslant 0$, вообще говоря, зависимы и имеют различное распределение, но удовлетворяют некоторым моментным условиям.

Quadratic transportation cost in the conditional central limit theorem for dependent sequences

Quadratic transportation cost in the conditional central limit theorem for dependent sequences

A large deviation theorem for a supercritical super-Brownian motion with absorption

AbstractWe consider a one-dimensional superprocess with a supercritical local branching mechanism $\psi$ , where particles move as a Brownian motion with drift $-\rho$ and are killed when they reach the origin. It is known that the process survives with positive probability if and only if $\rho<\sqrt{2\alpha}$ , where $\alpha=-\psi'(0)$ . When $\rho<\sqrt{2 \alpha}$ , Kyprianou et al. [18] proved that $\lim_{t\to \infty}R_t/t =\sqrt{2\alpha}-\rho$ almost surely on the survival set, where $R_t$ is the rightmost position of the support at time t. Motivated by this work, we investigate its large deviation, in other words, the convergence rate of $\mathbb{P}_{\delta_x} (R_t >\gamma t+\theta)$ as $t \to \infty$ , where $\gamma >\sqrt{2 \alpha} -\rho$ , $\theta \ge 0$ . As a by-product, a related Yaglom-type conditional limit theorem is obtained. Analogous results for branching Brownian motion can be found in Harris et al. [13].

Conditioned limit theorems for hyperbolic dynamical systems

AbstractLet $({\mathbb X}, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu $ and let f be a real-valued Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$ . Consider the Birkhoff sums $S_n f = \sum _{k=0}^{n-1} f \circ T^{k}$ , $n\geqslant 1$ . For any $t \in {\mathbb R}$ , denote by $\tau _t^f$ the first time when the sum $t+ S_n f$ leaves the positive half-line for some $n\geqslant 1$ . By analogy with the case of random walks with independent and identically distributed increments, we study the asymptotic as $ n\to \infty $ of the probabilities $ \nu (x\in {\mathbb X}: \tau _t^f(x)>n) $ and $ {\nu (x\in {\mathbb X}: \tau _t^f(x)=n) }$ . We also establish integral and local-type limit theorems for the sum $t+ S_n f(x)$ conditioned on the set $\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$

Stein’s method for conditional central limit theorem

In the seventies, Charles Stein revolutionized the way of proving the central limit theorem by introducing a method that utilizes a characterization equation for Gaussian distribution. In the last 50 years, much research has been done to adapt and strengthen this method to a variety of different settings and other limiting distributions. However, it has not been yet extended to study conditional convergences. In this article we develop a novel approach, using Stein’s method for exchangeable pairs, to find a rate of convergence in the conditional central limit theorem of the form (Xn∣Yn=k), where (Xn,Yn) are asymptotically jointly Gaussian, and extend this result to a multivariate version. We apply our general result to several concrete examples, including pattern count in a random binary sequence and subgraph count in Erdős–Rényi random graph.

Conditional central limit theorem for subcritical branching random walk

Conditional central limit theorem for subcritical branching random walk

Random walks conditioned to stay nonnegative and branching processes in an unfavourable environment

Let $\{S_n, n\geqslant 0\}$ be a random walk with increments that belong (without centering) to the domain of attraction of an $alpha$-stable law, that is, there exists a process $\{Y_t, t\geqslant 0\}$ such that $S_{nt}/a_{n}$ $\Rightarrow$ $Y_t$, $t\geqslant 0$, as $n\to\infty$ for some scaling constants $a_n$. Assuming that $S_{0}=o(a_n)$ and $S_n\leqslant \varphi (n)=o(a_n)$, we prove several conditional limit theorems for the distribution of the random variable $S_{n-m}$ given that $m=o(n)$ and $\min_{0\leqslant k\leqslant n}S_k\geqslant 0$. These theorems supplement the assertions established by Caravenna and Chaumont in 2013. Our results are used to study the population size of a critical branching process evolving in an unfavourable environment. Bibliography: 28 titles.

Large Deviations of Weighted Sums of Independent Identically Distributed Random Variables with Functionally-Defined Weights

Let \( {S}_n=\sum \limits_{j=1}^n{a}_{j,n}{X}_{j,n} \) be a weighted sum with independent, identically distributed steps Xj,n, j ≤ n, where aj,n = f(j/n) for some f ∈ C2[0, 1]. Under Cramer’s condition, we prove an integro-local limit theorem for P(Sn ∈ [x, x + Δn)) as x/n ∈ [m−, m+] for some m−, m+ and any sequence Δn tending to zero slowly enough. This result covers the whole scope of normal, moderate, and large deviations. For the stochastic process Yn(t) corresponding to S0, . . . , Sn we obtain a conditional functional limit theorem concerning convergence Yn(t) to the Brownian bridge given the condition Sn ∈ [x, x + Δn).

Asymptotic results for heavy-tailed Lévy processes and their exponential functionals

In this paper, we first provide several conditional limit theorems for Lévy processes with negative drift and regularly varying tail. Then we apply them to study the asymptotic behavior of expectations of some exponential functionals of heavy-tailed Lévy processes. As the key point, we observe that the asymptotic mainly depends on the sample paths with early arrival of large jump. Both the polynomial decay rate and the exact expression of the limit coefficients are given. As an application, we give an exact description for the extinction speed of continuous-state branching processes in heavy-tailed Lévy random environment with stable branching mechanism.

Statistical properties of sketching algorithms.

SummarySketching is a probabilistic data compression technique that has been largely developed by the computer science community. Numerical operations on big datasets can be intolerably slow; sketching algorithms address this issue by generating a smaller surrogate dataset. Typically, inference proceeds on the compressed dataset. Sketching algorithms generally use random projections to compress the original dataset, and this stochastic generation process makes them amenable to statistical analysis. We argue that the sketched data can be modelled as a random sample, thus placing this family of data compression methods firmly within an inferential framework. In particular, we focus on the Gaussian, Hadamard and Clarkson–Woodruff sketches and their use in single-pass sketching algorithms for linear regression with huge samples. We explore the statistical properties of sketched regression algorithms and derive new distributional results for a large class of sketching estimators. A key result is a conditional central limit theorem for data-oblivious sketches. An important finding is that the best choice of sketching algorithm in terms of mean squared error is related to the signal-to-noise ratio in the source dataset. Finally, we demonstrate the theory and the limits of its applicability on two datasets.

Critical Galton-Watson branching processes with a countable set of types and infinite second moments

We consider an indecomposable Galton-Watson branching process with a countable set of types. Assuming that the process is critical and may have infinite variance of the offspring sizes of some (or all) types of particles we describe the asymptotic behaviour of the survival probability of the process and establish a Yaglom-type conditional limit theorem for the infinite-dimensional vector of the number of particles of all types. Bibliography: 20 titles.

Rates of Convergence in Laplace’s Integrals and Sums and Conditional Central Limit Theorems

We obtained the exact estimates for the error terms in Laplace’s integrals and sums implying the corresponding estimates for the related laws of large number and central limit theorems including the large deviations approximation.

Subcritical Branching Processes in Random Environment with Immigration Stopped at Zero

We consider the subcritical branching processes with immigration which evolve under the influence of a random environment and study the tail distribution of life periods of such processes defined as the length of the time interval between the moment when first invader (or invaders) came to an empty site until the moment when the site becomes empty again. We prove that the tail distribution decays with exponential rate. The main tools are change of measure and some conditional limit theorems for random walks.

The Initial Evolution Stage of a Weakly Subcritical Branching Process in a Random Environment

A conditional limit theorem is proved describing the distribution of the population size in the initial evolution stage of a weakly subcritical branching process in a random environment given its s...

Sample path large deviations for Lévy processes and random walks with regularly varying increments

Let $X$ be a Lévy process with regularly varying Lévy measure $\nu$. We obtain sample-path large deviations for scaled processes $\bar{X}_{n}(t)\triangleq X(nt)/n$ and obtain a similar result for random walks with regularly varying increments. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the increment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connections with the classical large deviations framework. In that setting, we show that a weak large deviation principle (with logarithmic speed) holds, but a full large deviation principle does not hold.