Search ResultsShowing 1-20 of 281
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arXiv:2505.10137 (Published 2025-05-15)
Small deviations for critical Galton-Watson processes with infinite variance
Comments: 30 pagesCategories: math.PRWe study the asymptotic behavior of small deviation probabilities for the critical Galton-Watson processes with infinite variance of the offspring sizes of particles and apply the obtained result to investigate the structure of a reduced critical Galton-Watson process.
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arXiv:2501.04047 (Published 2025-01-06)
New probabilistic methods for physics
We present how a probabilistic model can describe the asymptotic behavior of the iterations, with applications for ODE and approach of some problems in mechanics in R^d.
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arXiv:2412.15626 (Published 2024-12-20)
Stationary states for stable processes with partial resetting
Categories: math.PRWe study a $d$-dimensional stochastic process $\mathbf{X}$ which arises from a L\'evy process $\mathbf{Y}$ by partial resetting, that is the position of the process $\mathbf{X}$ at a Poisson moment equals $c$ times its position right before the moment, and it develops as $\mathbf{Y}$ between these two consecutive moments, $c \in (0, 1)$. We focus on $\mathbf{Y}$ being a strictly $\alpha$-stable process with $\alpha\in (0,2]$ having a transition density: We analyze properties of the transition density $p$ of the process $\mathbf{X}$. We establish a series representation of $p$. We prove its convergence as time goes to infinity (ergodicity), and we show that the limit $\rho_{\mathbf{Y}}$ (density of the ergodic measure) can be expressed by means of the transition density of the process $\mathbf{Y}$ starting from zero, which results in closed concise formulae for its moments. We show that the process $\mathbf{X}$ reaches a non-equilibrium stationary state. Furthermore, we check that $p$ satisfies the Fokker--Planck equation, and we confirm the harmonicity of $\rho_{\mathbf{Y}}$ with respect to the adjoint generator. In detail, we discuss the following cases: Brownian motion, isotropic and $d$-cylindrical $\alpha$-stable processes for $\alpha \in (0,2)$, and $\alpha$-stable subordinator for $\alpha\in (0,1)$. We find the asymptotic behavior of $p(t;x,y)$ as $t\to +\infty$ while $(t,y)$ stays in a certain space-time region. For Brownian motion, we discover a phase transition, that is a change of the asymptotic behavior of $p(t;0,y)$ with respect to $\rho_{\mathbf{Y}}(y)$.
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arXiv:2412.09667 (Published 2024-12-12)
Spatial preferential attachment with choice-based edge step
We study the asymptotic behavior of the maximum indegree in the spatial preferential attachment model with a choice-based edge step. We prove different types of behavior of maximal indegree based on the model's parameters.
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arXiv:2411.16913 (Published 2024-11-25)
Entropies of the Poisson distribution as functions of intensity: "normal" and "anomalous" behavior
Subjects: 94A17The paper extends the analysis of the entropies of the Poisson distribution with parameter $\lambda$. It demonstrates that the Tsallis and Sharma-Mittal entropies exhibit monotonic behavior with respect to $\lambda$, whereas two generalized forms of the R\'enyi entropy may exhibit "anomalous" (non-monotonic) behavior. Additionally, we examine the asymptotic behavior of the entropies as $\lambda \to \infty$ and provide both lower and upper bounds for them.
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arXiv:2411.07737 (Published 2024-11-12)
On Asymptotic Behavior of Extinction Moment of Critical Bisexual Branching Process in Random Environment
Categories: math.PRWe consider a critical bisexual branching process in a random environment generated by independent and identically distributed random variables. Assuming that the process starts with a large number of pairs $N$, we prove that its extinction time is of the order $\ln^2 N$. Interestingly, this result is valid for a general class of mating functions. Among them are the functions describing the monogamous and polygamous behavior of couples, as well as the function reducing the bisexual branching process to the simple one.
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arXiv:2408.16585 (Published 2024-08-29)
ASEP via Mallows coloring
Comments: 13 pagesIn this paper we study the asymptotic behavior of the Asymmetric Simple Exclusion Process (=ASEP) with finitely many particles. It turns out that a certain randomized initial condition is the most amenable to such an analysis. Our main result is the behavior of such an ASEP in the KPZ limit regime. A key technical tool introduced in the paper -- the coloring of ASEP particles with the use of random Mallows permutations -- may be of independent interest.
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arXiv:2407.19741 (Published 2024-07-29)
Scaling limits for supercritical nearly unstable Hawkes processes
Categories: math.PRIn this paper,we investigate the asymptotic behavior of nearly unstable Hawkes processes whose regression kernel has $L^1$ norm strictly greater than one and close to one as time goes to infinity. We find that,the scaling size determines the scaling behavior of the processes like in \cite{MR3313750}.Specifically,after suitable rescaling,the limit of the sequence of Hawkes processes is deterministic.And also with another appropriate rescaling,the sequence converges in law to an integrated Cox Ingersoll Ross like process.This theoretical result may apply to model the recent COVID19 in epidemiology and in social network.
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arXiv:2407.05514 (Published 2024-07-07)
Exact convergence rates to derivatives of local time for some self-similar Gaussian processes
Comments: 21 pagesCategories: math.PRIn this article, for some $d-$dimensional Gaussian processes \[X=\big\{X_t=(X^1_t,\cdots,X^d_t):t\ge0\big\},\] whose components are i.i.d. $1-$dimensional self-similar Gaussian process with Hurst index $H\in(0,1)$, we consider the asymptotic behavior of approximation of its $\boldsymbol{k}-$th derivatives of local time under certain mild conditions, where $\boldsymbol{k}=(k_1,\cdots,k_d)$ and $k_\ell$'s are non-negative real numbers. We will give a derivative version of the limit theorems for functional of Gaussian processes and use this result to get the asymptotic behaviors.
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arXiv:2406.15150 (Published 2024-06-21)
Asymptotic behavior for the quenched survival probability of a supercritical branching random walk in random environment with a barrier
We introduce a random barrier to a supercritical branching random walk in an i.i.d. random environment $\{\mathcal{L}_n\}$ indexed by time $n,$ i.e., in each generation, only the individuals born below the barrier can survive and reproduce. At generation $n$ ($n\in\mathbb{N}$), the barrier is set as $\chi_n+\varepsilon n,$ where $\{\chi_n\}$ is a random walk determined by the random environment. Lv \& Hong (2024) showed that for almost every $\mathcal{L}:=\{\mathcal{L}_n\},$ the quenched survival probability (denoted by $\varrho_{\mathcal{L}}(\varepsilon)$) of the particles system will be 0 (resp., positive) when $\varepsilon\leq 0$ (resp., $\varepsilon>0$). In the present paper, we prove that $\sqrt{\varepsilon}\log\varrho_\mathcal{L}(\varepsilon)$ will converge in Probability/ almost surely/ in $L^p$ to an explicit negative constant (depending on the environment) as $\varepsilon\downarrow 0$ under some integrability conditions respectively. This result extends the scope of the result of Gantert et al. (2011) to the random environment case.
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arXiv:2406.09852 (Published 2024-06-14)
Asymptotic behavior of some strongly critical decomposable 3-type Galton-Watson processes with immigration
Comments: 55 pagesCategories: math.PRWe study the asymptotic behaviour of a critical decomposable 3-type Galton-Watson process with immigration when its offspring mean matrix is triangular with diagonal entries 1. It is proved that, under second or fourth order moment assumptions on the offspring and immigration distributions, a sequence of appropriately scaled random step processes formed from such a Galton-Watson process converges weakly. The limit process can be described using independent squared Bessel processes $(\mathcal{X}_{t,1})_{t\geq0}$, $(\mathcal{X}_{t,2})_{t\geq0}$, and $(\mathcal{X}_{t,3})_{t\geq0}$, the linear combinations of the integral processes of $(\mathcal{X}_{t,1})_{t\geq0}$ and $(\mathcal{X}_{t,2})_{t\geq0}$, and possibly the 2-fold iterated integral process of $(\mathcal{X}_{t,1})_{t\geq0}$. The presence of the 2-fold iterated integral process in the limit distribution is a new phenomenon in the description of asymptotic behavior of critical multi-type Galton-Watson processes with immigration. Our results complete and extend some results of Foster and Ney (1978) for some strongly critical decomposable 3-type Galton-Watson processes with immigration.
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Hitting probability for Reflected Brownian Motion at Small Target
Comments: 23 pages, 7 figuresCategories: math.PRWe derive the asymptotic behavior of hitting probability at small target of size $O(\epsilon)$ for reflected Brownian motion in domains with suitable smooth boundary conditions, where the boundary of domain contains both reflecting part, absorbing part and target. In this case the domain could be localized near the target and explicit computations are possible. The asymptotic behavior is only related to $\epsilon$ up to some multiplicative constants that depends on the domain and boundary conditions.
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arXiv:2401.12605 (Published 2024-01-23)
The asymptotic behavior of fraudulent algorithms
Categories: math.PRLet $U$ be a Morse function on a compact connected $m$-dimensional Riemannian manifold, $m \geq 2,$ satisfying $\min U=0$ and let $\mathcal{U} = \{x \in M \: : U(x) = 0\}$ be the set of global minimizers. Consider the stochastic algorithm $X^{(\beta)}:=(X^{(\beta)}(t))_{t\geq 0}$ defined on $N = M \setminus \mathcal{U},$ whose generator is$U \Delta \cdot-\beta\langle \nabla U,\nabla \cdot\rangle$, where $\beta\in\RR$ is a real parameter.We show that for $\beta>\frac{m}{2}-1,$ $X^{(\beta)}(t)$ converges a.s.\ as $t \rightarrow \infty$, toward a point $p \in \mathcal{U}$ and that each $p \in \mathcal{U}$ has a positive probability to be selected. On the other hand, for $\beta < \frac{m}{2}-1,$ the law of $(X^{(\beta)}(t))$ converges in total variation (at an exponential rate) toward the probability measure $\pi_{\beta}$ having density proportional to $U(x)^{-1-\beta}$ with respect to the Riemannian measure.
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arXiv:2401.07876 (Published 2024-01-15)
Characterization of the asymptotic behavior of $U$-statistics on row-column exchangeable matrices
Categories: math.PRWe consider $U$-statistics on row-column exchangeable matrices. We derive a decomposition for them, based on orthogonal projections on probability spaces generated by sets of Aldous-Hoover-Kallenberg variables. The specificity of these sets is that they are indexed by bipartite graphs, which allows for the use of concepts from graph theory to describe this decomposition. The decomposition is used to investigate the asymptotic behavior of $U$-statistics of row-column exchangeable matrices, including in degenerate cases. In particular, it depends only on a few terms of the decomposition, corresponding to the non-zero elements that are indexed by the smallest graphs, named principal support graphs, after an analogous concept suggested by Janson and Nowicki (1991). Hence, we show that the asymptotic behavior of a $U$-statistic and its degeneracy are characterized by the properties of their principal support graphs. Indeed, their number of nodes gives the convergence rate of a $U$-statistic to its limit distribution. Specifically, the latter is degenerate if and only if this number is strictly greater than 1. Also, when the principal support graphs are connected, we find that the limit distribution is Gaussian, even in degenerate cases.
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arXiv:2312.14696 (Published 2023-12-22)
On short Edgeworth expansions for weighted sums of random vectors
Comments: arXiv admin note: text overlap with arXiv:2112.05815Categories: math.PRThe "typical" asymptotic behavior of the weighted sums of independent, identically distibuted random vectors in k-dimensional space is considered. It is shown that under finitnes of fifth absolute moment of an individual term the rate of convergence by Edgeworth correction in the multivariate central limit theorem is of order O(1/n^3/2 ). This extends the one-dimensional Bobkov(2020) result.
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arXiv:2311.11217 (Published 2023-11-19)
On the association of the Airy$_1$ process
Categories: math.PRWe first show that the Airy$_1$ process is associated using the association property of solution to stochastic heat equation and convergence of the KPZ equation to the KPZ fixed point. Then we apply Newman's inequality to establish the ergodicity and central limit theorem for the Airy$_1$ process. Combining with the asymptotic behavior of the GOE Tracy-Widom distribution, we derive a Poisson limit theorem for the Airy$_1$ process and give some estimate on the asymptotic behavior of the maximum of the the Airy$_1$ process over an interval. Analogous results for the Airy$_2$ process are also presented.
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arXiv:2311.09804 (Published 2023-11-16)
Average Jaccard Index of Random Graphs
The asymptotic behavior of the Jaccard index in $G(n,p)$, the classical Erd\"{o}s-R\'{e}nyi random graphs model, is studied in this paper, as $n$ goes to infinity. We first derive the asymptotic distribution of the Jaccard index of any pair of distinct vertices, as well as the first two moments of this index. Then the average of the Jaccard indices over all vertex pairs in $G(n,p)$ is shown to be asymptotically normal under an additional mild condition that $np\to\infty$ and $n^2(1-p)\to\infty$.
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arXiv:2311.06652 (Published 2023-11-11)
A Classification of Asymptotic Behaviors of Green Functions of Random Walks in the Quadrant
This paper investigates the asymptotic behavior of Green functions associated to partially homogeneous random walks in the quadrant $Z_+^2$. There are four possible distributions for the jumps of these processes, depending on the location of the starting point: in the interior, on the two positive axes of the boundary, and at the origin $(0,0)$. With mild conditions on the positive jumps of the random walk, which can be unbounded, a complete analysis of the asymptotic behavior of the Green function of the random walk killed at $(0,0)$ is achieved. The main result is that {\em eight} regions of the set of parameters determine completely the possible limiting behaviors of Green functions of these Markov chains. These regions are defined by a set of relations for several characteristics of the distributions of the jumps. In the transient case, a description of the Martin boundary is obtained and in the positive recurrent case, our results give the exact limiting behavior of the invariant distribution of a state whose norm goes to infinity along some asymptotic direction in the quadrant. These limit theorems extend results of the literature obtained, up to now, essentially for random walks whose jump sizes are either $0$ or $1$ on each coordinate. Our approach relies on a combination of several methods: probabilistic representations of solutions of analytical equations, Lyapounov functions, convex analysis, methods of homogeneous random walks, and complex analysis arguments.
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arXiv:2310.16657 (Published 2023-10-25)
The asymptotic behavior of rarely visited edges of the simple random walk
Categories: math.PRIn this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\mathbb{Z}$. Let $\alpha(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\mathbb{E}(\alpha(n))$, show that $n\to \mathbb{E}(\alpha(n))$ is non-decreasing in $n$ and that $\lim\limits_{n\to+\infty}\mathbb{E}(\alpha(n))=2$. Then we study the asymptotic behavior of $\mathbb{P} (\alpha(n)>a(\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\in(0,+\infty)$ such that $\limsup\limits_{n\to+\infty}\frac{\alpha(n)}{(\log n)^2}=C$ almost surely.
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arXiv:2310.01670 (Published 2023-10-02)
Asymptotic behavior of Wasserstein distance for weighted empirical measures of diffusion processes on compact Riemannian manifolds
Categories: math.PRLet $(X_t)_{t \geq 0}$ be a diffusion process defined on a compact Riemannian manifold, and for $\alpha > 0$, let $$ \mu_t^{(\alpha)} = \frac{\alpha}{t^\alpha} \int_{0}^{t} \delta_{X_s} \, s^{\alpha - 1} \mathrm{d} s $$ be the associated weighted empirical measure. We investigate asymptotic behavior of $\mathbb{E}^\nu \big[ \mathrm{W}_2^2(\mu_t^{(\alpha)}, \mu) \big]$ for sufficient large $t$, where $\mathrm{W}_2$ is the quadratic Wasserstein distance and $\mu$ is the invariant measure of the process. In the particular case $\alpha = 1$, our result sharpens the limit theorem achieved in [26]. The proof is based on the PDE and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan.