Disaggregation of Long Memory Processes on C^{\infty} Class

On estimating the memory for finitarily Markovian processes

Finitarily Markovian processes are those processes $\{X_n\}_{n=-\infty}^{\infty}$ for which there is a finite $K$ ($K = K(\{X_n\}_{n=-\infty}^0$) such that the conditional distribution of $X_1$ given the entire past is equal to the conditional distribution of $X_1$ given only $\{X_n\}_{n=1-K}^0$. The least such value of $K$ is called the memory length. We give a rather complete analysis of the problems of universally estimating the least such value of $K$, both in the backward sense that we have just described and in the forward sense, where one observes successive values of $\{X_n\}$ for $n \…

Uniform asymptotic normality of weighted sums of short-memory linear processes

Let $X_1, X_2,\dots$ be a short-memory linear process of random variables. For $1\leq q<2$, let $\cF$ be a bounded set of real-valued functions on $[0,1]$ with finite $q$-variation. It is proved that $\{n^{-1/2}\sum_{i=1}^nX_if(i/n)\colon\,f\in\cF\}$ converges in outer distribution in the Banach space of bounded functions on $\cF$ as $n\to\infty$. Several applications to a regression model and a multiple change point model are given.

Long run behaviour of the autocovariance function of ARCH($\infty$) models

The asymptotic properties of the memory structure of ARCH($\infty$) equations are investigated. This asymptotic analysis is achieved by expressing the autocovariance function of ARCH($\infty$) equations as the solution of a linear Volterra summation equation and analysing the properties of an associated resolvent equation via the admissibility theory of linear Volterra operators. It is shown that the autocovariance function decays subexponentially (or geometrically) if and only if the kernel of the resolvent equation has the same decay property. It is also shown that upper subexponential bound…

A Black--Scholes Model with Long Memory

This note develops a stochastic model of asset volatility. The volatility obeys a continuous-time autoregressive equation. Conditions under which the process is asymptotically stationary and possesses long memory are characterised. Connections with the class of ARCH($\infty$) processes are sketched.

重尾近不稳定累积INAR(∞)过程的标度极限及粗糙分数扩散

Scaling limit of heavy tailed nearly unstable cumulative INAR($\infty$) processes and rough fractional diffusions

本文研究了重尾近不稳定累积INAR(∞)过程的标度极限，揭示其与集成分数Cox-Ingersoll-Ross过程的关系，并提出高效模拟方法。

Maxima of long memory stationary symmetric $α$-stable processes, and self-similar processes with stationary max-increments

We derive a functional limit theorem for the partial maxima process based on a long memory stationary $α$-stable process. The length of memory in the stable process is parameterized by a certain ergodic-theoretical parameter in an integral representation of the process. The limiting process is no longer a classical extremal Fréchet process. It is a self-similar process with $α$-Fréchet marginals, and it has stationary max-increments, a property which we introduce in this paper. The functional limit theorem is established in the space $D[0,\infty)$ equipped with the Skorohod $M_1$-topology; in …