longmemory

Long memory, or long-range dependence, is a property of certain stochastic processes in which observations separated by long time lags remain significantly correlated. Unlike short-memory processes, whose autocorrelations decay quickly, long-memory processes exhibit autocorrelations that decay slowly, typically at a hyperbolic rate, so that the sum of autocorrelations can diverge.

A common mathematical characterization uses the autocovariance function gamma(k) behaving as gamma(k) ~ L(k) k^{-(1-2d)} for large k, with parameter 0 < d < 0.5, where L is a slowly varying function. Equivalently, the process can be described by a Hurst exponent H = d + 0.5 in the interval (0.5, 1). In this framework, H > 0.5 indicates persistent behavior.

Canonical models for long memory include ARFIMA(p,d,q) processes, where the fractional differencing parameter d controls the memory. Stationary long memory corresponds to 0 < d < 0.5. Continuous-time counterparts include fractional Brownian motion (fBm) and its increments, fractional Gaussian noise (fGn), where long-range dependence occurs for H > 0.5.

Estimation and testing focus on the memory parameter or the Hurst exponent, using methods such as rescaled range analysis, time- and frequency-domain estimators (e.g., local Whittle), wavelet-based techniques, and detrended fluctuation analysis. These methods face challenges from finite samples and model misspecification.

Applications of long-memory models appear in hydrology, finance and economics (notably in volatility and trading activity), telecommunications and network traffic, and climatology. Debate continues regarding the prevalence of genuine long memory in some domains, due in part to potential confounding factors like structural breaks and nonstationarities.