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 $n \geq 1$. 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 ($q q_c$). 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.