frecven

Frecven is a theoretical metric used in the analysis of recurrent events to describe the long-run density of occurrences in time. It aims to capture how frequently events recur on average, independent of short-term timing details. In formal terms, frecven is defined for a counting process N(t) that counts events up to time t by the limit f = lim as t → ∞ of N(t)/t, when this limit exists. When inter-arrival times T_i have a finite mean μ, frecven equals 1/μ. In renewal processes, this relationship is fundamental, and in a Poisson process with rate λ, frecven coincides with λ.

Interpretation and use cases: Frecven represents the average event rate over the long run. It is distinct from instantaneous or local frequency, which may vary with time or context. As an estimate, frecven can be approximated by N(T)/T for large observation windows, or by more sophisticated methods that account for non-stationarity or censoring in the data.

Applications: The concept is used in theoretical discussions across fields such as signal processing, reliability engineering, seismology, and neuroscience to quantify how often a recurring phenomenon occurs per unit time. It provides a simple scalar summary of recurrence intensity that can be compared across systems or conditions.

Limitations: Frecven does not capture temporal structure within recurrence intervals, such as clustering, periodic modulation, or bursts. If the underlying process has infinite mean inter-arrival time, frecven is not defined. Etymology: the term frecven blends frequency and recurrence to emphasize long-run density rather than precise timing.