Using the regenerative scheme of [8], we establish a functional central limit theorem (FCLT) for discrete time stochastic processes (chains) with summable memory decay. Furthermore, under stronger assumptions on the memory decay, we identify the limiting variance in terms of the process only. As applications, we define classes of binary autoregressive processes and power-law Ising chains for wh...

The continuous-time random walk (CTRW) formalism can be adapted to encompass stochastic processes with memory. In this paper we will show how the random combination of two different unbiased CTRWs can give rise to a process with clear drift, if one of them is a CTRW with memory. If one identifies the other one as noise, the effect can be thought of as a kind of stochastic resonance. The ultimat...

Identifying and quantifying memory are often critical steps in developing a mechanistic understanding of stochastic processes. These are particularly challenging and necessary when exploring processes that exhibit long-range correlations. The most common signatures employed rely on second-order temporal statistics and lead, for example, to identifying long memory in processes with power-law aut...

We consider semiparametric estimation of memory parameter in long memory stochastic volatility models. It is known that log periodogram regression estimator by Geweke and Porter-Hudak (1983) results in significant negative bias due to the existence of the spectrum of non-Gaussian noise process. Through wavelet transform of the squared process, we effectively remove the noise spectrum around zer...

We study the dynamics of a point particle in a periodic array of spherical scatterers and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (BoltzmannGrad limit). A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two.

We study the dynamics of a point particle in a periodic array of spherical scatterers, and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (Boltzmann-Grad limit). A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two.

Stochastic volatility (SV) models play an important role in finance. Under these models, the volatility of an asset follows an individual stochastic process. In contrast to the GARCH model, the volatility process in the SV model is autonomous with no need to refer to the asset price. It is often assumed that the log-volatility process follows a standard ARMA process in an SV model. However, emp...

Quantifying ecological memory could be done at several levels from the rate of physiological changes in an ecosystem all way down to responses genetic level. One unlocking information encoded a collective environmental is examine recorded time-series data generated by different components ecosystem. In this paper, we probe into case Great Barrier Reef (GBR) which threatened elevated sea surface...