Extreme value statistics and return intervals in long-range correlated uniform deviates

Area of Science:

• Statistical Physics
• Time Series Analysis
• Probability Theory

Background:

• Extremal statistics and return intervals are crucial for understanding complex systems.
• Previous studies often assumed independent and identically distributed (i.i.d.) data, limiting applicability to correlated processes.
• The influence of random reference points on extremal statistics in correlated sequences remains an open question.

Purpose of the Study:

• To investigate extremal statistics and return intervals in stationary, long-range correlated sequences with a bounded, uniform probability density function.
• To analytically derive limiting distributions for extremal statistics, considering random reference points.
• To compare these distributions with those of i.i.d. variables and analyze the impact of correlations.

Main Methods:

• Analytical calculation of limiting distributions for extremal statistics.
• Comparison of distributions derived for correlated sequences against those for i.i.d. random variables.
• Analysis of return interval distributions and comparison with existing conjectures.

Main Results:

• Limiting distributions for extremal statistics differ from the standard Weibull distribution due to the random reference point.
• The functional form of the limiting distributions is invariant to long-range correlations, though convergence rates are affected.
• Return interval distributions were analyzed and compared to recent theoretical predictions.

Conclusions:

• The presence of a random reference point significantly modifies extremal statistics compared to standard maximum distributions.
• Long-range correlations do not alter the fundamental shape of these extremal distributions but impact their convergence speed.
• The derived findings are generalizable to a broad range of stochastic processes, offering insights into their extreme value behavior.