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

We study extremal statistics and return intervals in stationary long-range correlated sequences for which the underlying probability density function is bounded and uniform. The extremal statistics we consider (e.g., maximum relative to minimum) are such that the reference point from which the maximum is measured is itself a random quantity. We analytically calculate the limiting distributions for independent and identically distributed random variables, and use these as a reference point for correlated cases. The distributions are different from that of the maximum itself (i.e., a Weibull distribution), reflecting the fact that the distribution of the reference point either dominates over or convolves with the distribution of the maximum. The functional form of the limiting distributions is unaffected by correlations, although the convergence is slower. We show that our findings can be directly generalized to a wide class of stochastic processes. We also analyze return interval distributions, and compare them to recent conjectures of their functional form.

Figure 1

(Color online) Distribution of maximum relative to average for sequences of length $2^{11},2^{13},2^{15}$ (triangles, crosses, diamonds) and $\alpha =0.5$, with zero mean and unit standard deviation. A Gaussian is plotted alongside as a solid line.

Figure 2

(Color online) Distribution of maximum relative to initial value for sequences of length $2^{11},2^{13},2^{15}$ (triangles, crosses, diamonds) and $\alpha =0.5$, with zero mean and unit standard deviation. A uniform distribution is plotted alongside as a solid line.

Figure 3

(Color online) Decay in the fraction of zero maxima relative to initial value with increasing system size, for $\alpha =0.95,0.8,0.6,0.4,0.2$ (top to bottom). The decay $1/n$ for iid series is shown by the solid black line.

Figure 4

(Color online) Measurements of $C\left(\alpha \right)$ based on Eq. (16) for $\alpha =0.2,0.4,0.6,0.8,0.95$ (top to bottom) and $n=64,128,256,512,1024,2048$. For each $\alpha$, the lines are averages over the six system sizes. Error bars are estimates of statistical error (via standard error propagation) and represent one standard deviation: over $N$ measurements the number of instances of the maximum also being the initial value is binomially distributed with expectation $Np$ and variance $Np\left(1-p\right)$, where $p=1/n$ is the probability that the maximum lies at the initial value.

Figure 5

(Color online) Distribution of maximum relative to minimum for sequences of length $2^{11},2^{13},2^{15}$ (triangles, crosses, diamonds) and $\alpha =0.5$, with zero mean and unit standard deviation. The distribution in Eq. (13) is plotted alongside as a solid line.

Figure 6

(Color online) Top panels: distribution of return times above the (a) 97.9%, (b) 99.7% quantile for sequences of length $2^{19},2^{22}$ (triangles, crosses) and $\alpha =0.2$, with mean return times (a) $\tau \approx 48$, (b) $\tau \approx 287$. Stretched exponential ansatzes [see Eq. (17)] are plotted alongside for $\alpha =0.25,0.2,0.15$ from top to bottom. Bottom panels: distribution of return times above the (c) 97.9%, (d) 99.7% quantile for sequences of length $2^{19},2^{21}$ (triangles, crosses) and $\alpha =0.7$, with mean return times (c) $\tau \approx 48$, (d) $\tau \approx 287$. Stretched exponential ansatzes are plotted alongside for $\alpha =0.75,0.7,0.65$ from top to bottom.

Figure 7

(Color online) Main plot: distribution of return times above the 99.7% quantile for sequences of lengths $2^{19},2^{21}$ (triangles, crosses) for $\alpha =0.2$. Inset: distribution of return times above the 99.1%, 99.7% quantiles (triangles, crosses) for sequences of length $2^{21}$ and $\alpha =0.2$. A power law with slope $-\alpha$ is plotted alongside.

Figure 8

(Color online) Main plot: distribution of return times above the 99.7% quantile for sequences of lengths $2^{19},2^{21}$ (triangles, crosses) for $\alpha =0.7$. Inset: distribution of return times above the 99.1%, 99.7% quantiles (triangles, crosses) for sequences of length $2^{21}$ and $\alpha =0.7$. A power law with slope $-\alpha$ is plotted alongside.

Figure 9

(Color online) Distribution of step sizes for series of length $32768$ and $\alpha =2$. A Gaussian and a symmetric exponential, both with the same standard deviation as the data, have been plotted alongside as solid lines.

Figure 10

(Color online) Time evolution of the PDF for sequences initially starting within $0.475\le x\le 0.525$ after 25, 100, 500 steps (top to bottom), for $L=32768$ and $\alpha =2$. Gaussians with the same standard deviation as the data are plotted alongside. Inset: Time evolution of the standard deviation of the PDF (crosses) for sequences initially starting within $0.475\le x\le 0.525$. The standard deviations of the initial and final uniform distributions are $\sqrt{3}/120$ and $\sqrt{3}/6$ (horizontal lines), respectively. For a range of intermediate times the standard deviation grows with the square root of the number of steps (diagonal line).