Long-range dependence, also known as long memory or long-range persistence, refers to the phenomenon where correlations between elements of a time series decay more slowly than an exponential decay, typically following a power law. This characteristic implies that past values have a significant influence on future values over long time intervals, impacting fields such as finance, hydrology, and network traffic analysis.

Long-range Dependence

Long-range dependence

long memory

long-range persistence

correlations between elements of a time series

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value, leading to a rapid decline that slows over time. This mathematical model is crucial for understanding phenomena such as radioactive decay, population decline, and depreciation of assets.

exponential decay

A power law is a functional relationship between two quantities, where one quantity varies as a power of another. It is characterized by the property that a small number of occurrences are very common, while larger occurrences are rare, often visualized as a straight line on a log-log graph.

power law

past values

future values

long time intervals

Finance is the science of managing money, including the processes of acquiring, investing, and utilizing funds to achieve personal, corporate, or governmental goals. It encompasses various activities such as budgeting, saving, lending, investing, and forecasting to optimize financial resources and ensure stability and growth.

finance

Hydrology is the scientific study of the movement, distribution, and quality of water on Earth, encompassing both the water cycle and water resources. It plays a critical role in understanding environmental processes, managing water resources, and mitigating the impacts of natural disasters like floods and droughts.

hydrology

Network Traffic Analysis involves monitoring and examining data packets on a network to ensure security, optimize performance, and identify potential threats. It is essential for detecting anomalies, understanding network behavior, and maintaining the integrity and confidentiality of data transmission.

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