On the Detection of Long-Term Memory in Short Records

Long-term memory is ubiquitous in nature and has important consequences for the occurrence of natural hazards, but its detection often is complicated by the short length of the considered records. Here we study synthetic long-term correlated records of length N that are characterized by a correlation exponent γ, 0<γ<1. We show that the autocorrelation function CN(s) has the general form CN(s)=(C∞(s)-E)/(1-E), where C∞(0)=1, C∞(s>0)=B· s-γ and E=E(B,γ,N)= 2B/((2-γ)(1-γ))· N-γ+O(N-1). Due to the finite-size correction E, a direct determination of γ is difficult to achieve and generally leads to an enhanced value of γ. The parameter E also occurs in related quantities, that are characterized by the Hurst-exponent, which describe how the fluctuations of the records in time windows of length s decay with s, for example the variance of the local mean in time windows of length s. We show how to estimate E from a given data set which then allows a more accurate determination of γ. This approach can be applied also to long-term correlated records in the presence of additional white noise, where the determination of γ is particularly difficult (which is the case, e.g. in the records of return intervals between consecutive events above a certain threshold).