Fractionally Integrated Moving Average Stable Processes With Long-Range Dependence

Stochastic differential equations with a fractionally filtered delay: a semimartingale model for long-range dependent processes

In this paper we introduce a model, the stochastic fractional delay differential equation (SFDDE), which is based on the linear stochastic delay differential equation and produces stationary processes with hyperbolically decaying autocovariance functions. The model departs from the usual way of incorporating this type of long-range dependence into a short-memory model as it is obtained by applying a fractional filter to the drift term rather than to the noise term. The advantages of this approach are that the corresponding long-range dependent solutions are semimartingales and the local behavi…

分数积分分数布朗运动的持久概率

Persistence probabilities for fractionally integrated fractional Brownian noise

本文研究了分数积分分数布朗运动I(t/a,H)的持久指数e(a,H)，通过分析其在不同参数下的行为，推翻了长期假设e(2,H)=H(1-H)，并揭示了分数布朗运动之间的分数积分关系。

具有增响应函数的更新散粒噪声过程的泛函极限定理

Functional limit theorems for renewal shot noise processes with increasing response functions

我们考虑更新散粒噪声过程的响应函数最终是非减的，并定期在无穷大变化。本文证明了更新散粒噪声过程在$J_1$或$M_1$拓扑下，在空间$D[0,\infty)$中适当地归一化和中心化后的弱收敛性.极限过程要么是谱非正稳定的Lévy过程，包括布朗运动，要么是逆稳定从属（当响应函数缓慢变化时），要么是分数积分稳定过程或分数积分逆稳定从属（当正则变差指数为正时）。证明利用了更新过程的优良性质、稳定Lévy过程的分布性质和连续映射定理。

Cramér's estimate for stable processes with power drift

We investigate the upper tail probabilities of the all-time maximum of a stable Lévy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Lévy process. We also study the lower tail probabilities of the integrated stable Lévy process in the presence of a power positive drift.

带移民的随机过程的渐近性Ⅰ：标度极限

Asymptotics of random processes with immigration I: scaling limits

设$(X_1, ξ_1), (X_2,ξ_2),\ldots$为i.i.d.~一对$(X,ξ)$的副本，其中$X$是具有Skorokhod空间中的路径的随机过程$D[0,\infty)$，并且$ξ$是正随机变量。定义$S_k := ξ_1+\ldots+ξ_k$、$k \in \mathbb{N}_0$和$Y(t) := \sum_{k\geq 0} X_{k+1}(t-S_k) 1_{\{S_k \leq t\}}$、$t\geq 0$。我们称这个过程为在更新过程的历元上有移民的$(Y(t))_{t \geq 0}$随机过程。我们研究了$(Y(ut))_{u>0}$的有限维分布与$t\to\infty$的弱收敛性。在假设$X$的协方差函数在$(0,\infty)\times (0,\infty)$中均匀变化的条件下，当$ξ$的律属于一个有限均值稳定律的吸引域时，极限过程类是相当丰富的，包括协方差函数显式给定的高斯过程、分数积分稳定的Lévy运动及其和，当$ξ$律属于无穷均值稳定律的吸引域时，给出了协方差函数、分数积分逆稳定从属子及其和的条件高斯过程.

积分超均匀无穷可分移动平均的Fubini型极限定理

A Fubini-type limit theorem for the integrated hyperuniform infinitely divisible moving averages

本文研究高维无穷可分移动平均积分的极限行为，证明其弱极限为带积分核的同类移动平均，无需矩假设，适用于时间序列与随机场，平方可积时满足超均匀性。

具有递减响应函数的更新散粒噪声过程的极限定理

Limit theorems for renewal shot noise processes with decreasing response functions

本文考虑具有确定性响应函数$h$的散粒噪声过程$(X(t))_{t \geq 0}$和在零延迟更新过程的更新时期$0= S_0 < S_1 < S_2 ...$发生的散粒噪声。我们证明了收敛的有限维分布的$(X(ut))_{u \geq 0}$作为$t \to \infty$在不同的制度。若响应函数$h$直接Riemann可积，则$(X(ut))_{u \geq 0}$的有限维分布弱收敛为$t \to \infty$ .在这种情况下，既不需要缩放也不需要居中。如果响应函数最终是递减的，不可积的可积幂，然后，经过适当的移动，过程的有限维分布收敛。同样，不需要缩放。在这两种情况下，极限都是确定的。如果$S_1$的分布在$α$稳定律的吸引域中，并且响应函数在$\infty$处以指数$β$（具有$β< 1/α$或$β\leq 1/α$，取决于是$\mathbb{E} S_1 < \infty$还是$\mathbb{E} S_1 = \infty$）规则地变化，则需要缩放以获得$(X(ut))_{u \geq 0}$的有限维分布的弱收敛。极限过程是分数积分稳定的Lévy运动，如果$\mathbb{E} S_1 < \infty$和分数积分逆稳定从属，如果$\mathbb{E} S_1 = \infty$。

Minimax-robust forecasting of sequences with periodically stationary long memory multiple seasonal increments

We introduce stochastic sequences $ζ(k)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the problem of optimal estimation of linear functionals constructed from unobserved values of stochastic sequences $ζ(k)$ based on their observations at points $ k<0$. For sequences with known spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of functionals. Formulas that det…

平稳增量Lévy驱动移动平均线的极限定理

Limit theorems for stationary increments Lévy driven moving averages

本文给出了Lévy驱动的平稳增量的$k$阶增量的幂变差的几个新的极限定理.在这种填充抽样设置中，渐近理论给出了非常令人惊讶的结果，这（部分）在离散移动平均线理论中没有对应的结果。更具体地说，我们将表明，一阶极限定理和模式的收敛强烈依赖于之间的相互作用，所考虑的电源$p>0$，Blumenthal-Getoor指数$β\in (0,2)$的驱动纯跳Lévy过程$L$和行为的核函数$g$在$0$的权力$α$。一阶渐近理论基本上包括三种情况：稳定收敛于某一无穷可分分布，遍历型极限定理和依概率收敛于一个完整的随机过程。我们还证明了与遍历型结果相联系的二阶极限定理。当驱动Lévy过程$L$是对称$β$ -稳定过程时，我们得到了两个不同的极限：中心极限定理和分布收敛于$(1-α)β$ -稳定随机变量.

Limit theorems for a class of stationary increments Levy driven moving averages

In this paper we present some new limit theorems for power variation of $k$th order increments of stationary increments Lévy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives very surprising results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we will show that the first order limit theorems and the mode of convergence strongly depend on the interplay between the given order of the increments, the considered power $p>0$, t…

A functional limit theorem for random processes with immigration in the case of heavy tails

Let $(X_k,ξ_k)_{k\in \mathbb {N}}$ be a sequence of independent copies of a pair $(X,ξ)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\infty)$ and $ξ$ is a positive random variable. The random process with immigration $(Y(u))_{u\in \mathbb {R}}$ is defined as the a.s. finite sum $Y(u)=\sum_{k\geq0}X_{k+1}(u- ξ_1-\cdots-ξ_k)1\mkern-4.5mu\mathrm{l}_{\{ξ_1+\cdots+ξ_k\leq u\}}$. We obtain a functional limit theorem for the process $(Y(ut))_{u\geq 0}$, as $t\to\infty$, when the law of $ξ$ belongs to the domain of attraction of an $α$-stable law with $α\in(0,1)$, and the proc…