Emergence of Correlations Between Records and Maximum in Random Time Series
Abstract
We compute exactly the covariance between the maximum M and the number of records K in a discrete time series of n entries as a quantitative probe of the history leading to an extreme event. For continuous IID series, with a common probability distribution function (PDF) p(x), we find the striking result that M and K are uncorrelated for any n ≥ 1. We show that this independence is rather fragile and the covariance Cq(n) between M and K becomes nonzero when p(x) becomes discontinuous due to a delta-component at x=0 with an amplitude 0< q < 1. This time series models the occurrence of dry days with probability q in a precipitation time series when there is no rainfall. We show that there is an interesting shape transition at a critical value qc where the covariance Cq(n), as a function of n, changes from monotonic (q<qc) to non monotonic behavior (for q>qc). We compute the critical strength qc exactly. Furthermore, for large n, we show that Cq(n) decays as a power law, with an exponent that depends on the tail of p(x). We then compare our theoretical predictions with real precipitation data from several weather stations across Germany, finding a very good agreement.
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BibTeX
@misc{majumdar2026emergence,
title = {{Emergence of Correlations Between Records and Maximum in Random Time Series}},
author = {Satya N. Majumdar and Sanjib Sabhapandit and Gregory Schehr},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23217},
eprint = {2610.23217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23217},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Emergence of Correlations Between Records and Maximum in Random Time Series
AU - Satya N. Majumdar
AU - Sanjib Sabhapandit
AU - Gregory Schehr
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23217
UR - https://arxiv.org/abs/2610.23217
AB - We compute exactly the covariance between the maximum M and the number of records K in a discrete time series of n entries as a quantitative probe of the history leading to an extreme event. For continuous IID series, with a common probability distribution function (PDF) p(x), we find the striking result that M and K are uncorrelated for any n ≥ 1. We show that this independence is rather fragile and the covariance C_q(n) between M and K becomes nonzero when p(x) becomes discontinuous due to a delta-component at x=0 with an amplitude 0< q < 1. This time series models the occurrence of dry days with probability q in a precipitation time series when there is no rainfall. We show that there is an interesting shape transition at a critical value q_c where the covariance C_q(n), as a function of n, changes from monotonic (q<q_c) to non monotonic behavior (for q>q_c). We compute the critical strength q_c exactly. Furthermore, for large n, we show that C_q(n) decays as a power law, with an exponent that depends on the tail of p(x). We then compare our theoretical predictions with real precipitation data from several weather stations across Germany, finding a very good agreement.
ER -