Yuri Kondratiev, José Luís da Silva
In this paper we investigate the long-time behavior of the subordination of the constant speed traveling waves by a general class of kernels. We use the Feller--Karamata Tauberian theorem in order to study the long-time behavior of the upper and lower wave. As a result we obtain the long-time behavior for the propagation of the front of the wave.
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The asymptotic behaviour of empirical measures has been studied extensively. In this paper, we consider empirical measures of given subordinated processes on complete (not necessarily compact) and connected Riemannian manifolds with possibly nonempty boundary. We obtain rates of convergence for empirical measures to the invariant measure of the subordinated process under the Wasserstein distance. The results, established for more general subordinated processes than [arXiv:2107.11568], generalize the recent ones in [Stoch. Proc. Appl. 144(2022), 271--287] and are shown to be sharp by a typical …
Long-Time Asymptotics for Subordinated Fractional Diffusion Equations
本文研究次化子驱动的分数阶扩散方程的长时渐近行为，结合概率论、渐近分析与PDE技术，刻画次化解动力学，扩展经典分数阶动力学并连接随机过程与确定性PDE。
Geometrical subordinated Poisson processes and its extensions
本文用几何计数过程对泊松型过程进行时间变换，定义了三种几何从属泊松过程，推导其分布性质与特例，并应用于冲击建模。
This paper considers the asymptotic behaviour of volumes of excursion sets of subordinated Gaussian random fields with (possibly) infinite variance. Actually, we consider integral functionals of such fields and obtain their limiting distribution using the Hermite expansion of the integrand. We consider the general non-stationary Gaussian random fields, including stationary and anisotropic special cases. The limiting random variables in our limit theorems have the form of multiple Wiener-Itô integrals. We illustrate most results with corresponding examples.
We study the asymptotic behavior of wavelet coefficients of random processes with long memory. These processes may be stationary or not and are obtained as the output of non--linear filter with Gaussian input. The wavelet coefficients that appear in the limit are random, typically non--Gaussian and belong to a Wiener chaos. They can be interpreted as wavelet coefficients of a generalized self-similar process.
Long-Term Behavior of Subordinated Branching Processes with Prevailing Emigration
本文研究了具有主导移民的从属临界分支过程的长期行为，通过更新机制引入时间随机性，揭示了初始人口规模和到达间隔平均时间对非灭绝概率渐近行为及Yaglom型极限定理的影响，并探讨了交替再生过程，补充了移民主导移出过程的研究。
We study the weak convergence (in the high-frequency limit) of the frequency components associated with Gaussian-subordinated, spherical and isotropic random fields. In particular, we provide conditions for asymptotic Gaussianity and we establish a new connection with random walks on the the dual of SO(3), which mirrors analogous results previously established for fields defined on Abelian groups (see Marinucci and Peccati (2007)). Our work is motivated by applications to cosmological data analysis, and specifically by the probabilistic modelling and the statistical analysis of the Cosmic Micr…
Let T* be a random field indexed by an Abelian compact group G, and suppose that T* has the form T* = F(T(g)), where T is Gaussian and isotropic. The aim of this paper is to establish high-frequency central limit theorems for the Fourier coefficients associated to T*. The proofs of our main results involve recently established criteria for the weak convergence of multiple Wiener-Itô integrals. Our research is motivated by physical applications, mainly related to the probabilistic modelization of the Cosmic Microwave Background radiation. In this connection, the case of the n-dimensional torus …
Asymptotics of Ruin Probabilities in a Subordinated Cramér-Lundberg Model
本文研究了在莱维子ordinator 时间变换下的非寿险投资组合模型，证明了该变换对 ruin 概率的负面影响及初始资本与 ruin 概率之间的缓慢依赖关系。
In this article, we show that a general class of weakly stationary time series can be modeled applying Gaussian subordinated processes. We show that, for any given weakly stationary time series $(z_t)_{z\in\mathbb{N}}$ with given equal one-dimensional marginal distribution, one can always construct a function $f$ and a Gaussian process $(X_t)_{t\in\mathbb{N}}$ such that $\left(f(X_t)\right)_{t\in\mathbb{N}}$ has the same marginal distributions and, asymptotically, the same autocovariance function as $(z_t)_{t\in\mathbb{N}}$. Consequently, we obtain asymptotic distributions for the mean and aut…
In this paper we characterize the limiting behavior of sums of extreme values of long range dependent sequences defined as functionals of linear processes with finite variance. The extremal sums behave completely different by compared to the i.i.d case. In particular, though we still have asymptotic normality, the scaling factor is relatively bigger than in the i.i.d case, meaning that the maximal terms have relatively smaller contribution to the whole sum. Also, the scaling need not depend on the tail index of the underlying marginal distribution, as it is well-known to be so in the i.i.d. si…
In this paper we study the effect of the subordination by a general random time-change to the solution of a model on spatial ecology in terms of its evolution density. In particular on traveling waves for a non-local spatial logistic equation. We study the Cesaro limit of the subordinated dynamics in a number of particular cases related to the considered fractional derivative making use of the Karamata-Tauberian theorem.
Let $(G(X_j))_{j\geq1}$ be a multivariate subordinated Gaussian process, which exhibits long-range dependence. We study the asymptotic behaviour of the corresponding sequential empirical process under two different types of subordination. The limiting process is either a product of a deterministic function and a Hermite process as in the one-dimensional case or a sum of various processes of this kind.
In this paper we address the question of finding the best $L^p$-norm constant for martingale transforms with one-sided orthogonality. We consider two martingales on a probability space with filtration $\mathcal{B}$ generated by a two-dimensional Brownian motion $B_t$. One is differentially subordinated to the other. Here we find the sharp estimate for subordinate martingales if the subordinated martingale is orthogonal and $1<p<2$, and we find the best constant if $p>2$, but the orthogonal martingale is a subordinator. The answers are given in terms of zeros of Laguerre polynomials. As an appl…
Analysing Models for Volatility Clustering with Subordinated Processes: VGSA and Beyond
本文研究由 Lévy 从属与 CIR/CKLS 随机到达机制构建的时间变换过程，证明 VG、VGSA 及 CGMY 模型的一致性与渐近正态性，并通过 Monte Carlo 模拟验证。
In this paper we introduce the concept of random time changes in dynamical systems. The subordination principle may be applied to study the long time behavior of the random time systems. We show, under certain assumptions on the class of random time, that the subordinated system exhibits a slower time decay which is determined by the random time characteristics. In the path asymptotic a random time change is reflected in the new velocity of the resulting dynamics.
We study a class of directed random graphs. In these graphs, the interval [0,x] is the vertex set, and from each y\in [0,x], directed links are drawn to points in the interval (y,x] which are chosen uniformly with density one. We analyze the length of the longest directed path starting from the origin. In the large x limit, we employ traveling wave techniques to extract the asymptotic behavior of this quantity. We also study the size of a cascade tree composed of vertices which can be reached via directed paths starting at the origin.
The food web is a directed graph in which nodes label species and directed links represent the predation between species. The cascade model generates random food webs. The continuum cascade model is a Poisson approximation of the cascade model. The recursion to obtain the probability distribution of the longest chain length generated by the continuum cascade model has the solution with traveling wave. We consider a branching random walk to study the asymptotic probability on the position of wave front.
KPP traveling waves in the half-space
本文研究半空间中带Dirichlet边界条件的KPP方程行波，证明最小速度波在平移和旋转下唯一，并利用分支布朗运动的鞅表示揭示其渐近行为及与一维波的对数平移收敛关系。
We formulate the notion of the classical Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) reaction diffusion equation associated with a homogeneous conservative fragmentation process and study its traveling waves. Specifically, we establish existence, uniqueness and asymptotics. In the spirit of classical works such as McKean [Comm. Pure Appl. Math. 28 (1975) 323-331] and [Comm. Pure Appl. Math. 29 (1976) 553-554], Neveu [In Seminar on Stochastic Processes (1988) 223-242 Birkhäuser] and Chauvin [Ann. Probab. 19 (1991) 1195-1205], our analysis exposes the relation between traveling waves and certa…
Central limit theory for Peaks-over-Threshold partial sums of long memory linear time series
本文研究长记忆线性时间序列在阈值依赖于样本量的子ordination机制下的部分和渐进行为，提出定制化减少原理，扩展了传统方法，适用于无限方差创新过程，并在重尾和轻尾情况下得到新结果。
The aim of the paper is to address the long time behavior of the Kuramoto model of mean-field coupled phase rotators, subject to white noise and quenched frequencies. We analyse the influence of the fluctuations of both thermal noise and frequencies (seen as a disorder) on a large but finite population of $N$ rotators, in the case where the law of the disorder is symmetric. On a finite time scale $[0, T]$, the system is known to be self-averaging: the empirical measure of the system converges as $N\to \infty$ to the deterministic solution of a nonlinear Fokker-Planck equation which exhibits a …
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