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Series expansion calculation of persistence exponents
George C M A Ehrhardt1, Alan J Bray
1Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom.
Physical Review Letters
|February 28, 2002
Summary
The probability of consecutive same-signed values in Gaussian stationary processes decays exponentially. This study calculates the discrete persistence exponent and extrapolates to find the continuum persistence exponent, showing agreement with diffusion equation estimates.
Area of Science:
- Statistical Physics
- Time Series Analysis
- Stochastic Processes
Background:
- Gaussian stationary processes are fundamental in signal processing and physics.
- Understanding the persistence of same-signed values is crucial for analyzing complex systems.
- Previous methods for calculating persistence exponents had limitations.
Purpose of the Study:
- To calculate the discrete persistence exponent for Gaussian stationary processes.
- To determine the continuum persistence exponent by extrapolating discrete results.
- To validate the findings against known models like the diffusion equation.
Main Methods:
- Analyzing an arbitrary Gaussian stationary process X(T) with a known correlator C(T).
- Sampling the process at discrete times Tn = nΔT.
- Calculating the discrete persistence exponent θ(D) using a series expansion up to the 14th order.
- Extrapolating to the continuum limit (ΔT → 0) using constrained Padé approximants.
Main Results:
- Derived a series expansion for the discrete persistence exponent θ(D) in terms of the correlator C(ΔT).
- Obtained the continuum persistence exponent θs via extrapolation.
- Demonstrated exceptional agreement between the calculated results and numerical estimates for the diffusion equation.
Conclusions:
- The developed method provides an accurate way to determine persistence exponents for Gaussian stationary processes.
- The results align well with theoretical predictions and numerical simulations for related physical systems.
- This work offers a robust analytical approach for studying sign persistence in time series data.