InterEventTimeVerteilungen

InterEventTimeVerteilungen refer to the statistical distributions of waiting times between consecutive events in a temporal sequence. Given event times t1, t2, ..., the inter-event times are Δt_i = t_{i+1} - t_i. The distribution of Δt_i describes the temporal dynamics of the underlying process and is studied across domains such as human communication, neuronal activity, seismology, finance, and network traffic.

In the simplest case, a homogeneous Poisson process, Δt_i follow an exponential distribution with rate λ, implying memoryless intervals and a constant event rate. More general models use renewal processes with an arbitrary inter-event time distribution f(τ), corresponding survival function S(τ) = P(Δt > τ) and mean μ = E[Δt]. Heavy tails may occur when f is Pareto, lognormal, or similar, indicating bursts of rapid events followed by long quiescent periods.

Non-stationary and dependent models capture more complex behavior. A non-homogeneous Poisson process has a time-varying rate λ(t). Hawkes processes and related self-exciting models produce clusters of events: after an event, the instantaneous rate temporarily increases, shaping the shape of the inter-event-time distribution and often generating power-law-like tails.

Estimation and model selection typically proceed from observed event times. One computes Δt from data, fits candidate distributions using maximum likelihood or Bayesian methods, and assesses goodness of fit with tools such as Kolmogorov–Smirnov tests or information criteria. Burstiness measures compare dispersion to a Poisson benchmark.

Applications include modeling user activity on online platforms, communication patterns, neural spike trains, earthquake sequences, and queuing systems. InterEventTimeVerteilungen thus provide a compact summary of temporal structure and aid in predicting future events and understanding underlying mechanisms.