Power-Law Generalized Covariances

• Power-law generalized covariances are canonical functions defined for spatial intrinsic random functions that capture local regularity through a power-law singularity at the origin.
• They extend to space–time settings via an anisotropic spectral density, enabling independent tuning of spatial and temporal smoothness with parameters such as α₁, α₂, and ν.
• Explicit convergent series and asymptotic expansions provide practical tools to model realistic global behavior and facilitate efficient numerical approximations in geostatistics.

Power-law generalized covariances are canonical generalized covariance functions for isotropic spatial intrinsic random functions, and they provide a simple model for describing local behavior through a power law in the inter-point distance. In Stein’s formulation, the classical spatial class is embedded in a space–time family of generalized covariance functions whose spatial and temporal smoothness can be chosen independently while retaining explicit convergent and asymptotic series expansions (Stein, 2013). Closely related literatures study power-law covariance decay in time series by different mechanisms, notably monotone transformations of Gaussian processes with prescribed marginals and time changes of Pearson diffusions by inverse subordinators (Carpena et al., 2019, Mijena et al., 2014).

1. Classical spatial power-law generalized covariances

Let $Z(x)$, $x\in\mathbb R^d$, be an intrinsic random function of order $\lfloor \zeta \rfloor$. A canonical generalized covariance for such a process is, for inter-point distance $r=\|h\|\ge 0$ and $\zeta>0$,

$\gamma_{\zeta}(r)= \begin{cases} \Gamma(-\zeta)\,r^{2\zeta}, & \zeta\notin\mathbb N,\[6pt] 2(-1)^{\zeta+1}\,\zeta!\;r^{2\zeta}\log r, & \zeta\in\mathbb N. \end{cases}$

This form is valid in any dimension $d$ (Stein, 2013).

Locally, $\gamma_{\zeta}(r)\sim r^{2\zeta}$. Accordingly, $Z$ is $m$-times mean-square differentiable in every direction if and only if $x\in\mathbb R^d$0 (Stein, 2013). The central role of $x\in\mathbb R^d$1 is therefore not merely algebraic: it directly encodes local regularity through a power-law singularity at the origin.

The logarithmic branch for integer $x\in\mathbb R^d$2 is part of the same canonical class rather than an exceptional construction. This preserves the local power-law interpretation while maintaining the generalized covariance structure required for intrinsic random functions. A plausible implication is that the class is best viewed as a regularity model indexed by $x\in\mathbb R^d$3, with the noninteger and integer cases differing only in analytic representation.

2. Spectral extension to space–time intrinsic random functions

To extend $x\in\mathbb R^d$4 to a space–time intrinsic random function on $x\in\mathbb R^d$5, Stein starts from the stationary spectral density family

$x\in\mathbb R^d$6

with $x\in\mathbb R^d$7, $x\in\mathbb R^d$8, $x\in\mathbb R^d$9, and

$\lfloor \zeta \rfloor$0

Here $\lfloor \zeta \rfloor$1 and $\lfloor \zeta \rfloor$2 separately control the high-frequency decay, and hence smoothness, in the spatial frequency $\lfloor \zeta \rfloor$3 and temporal frequency $\lfloor \zeta \rfloor$4 (Stein, 2013).

Since range parameters $\lfloor \zeta \rfloor$5 do not affect the local power-law behavior, one may set $\lfloor \zeta \rfloor$6 and $\lfloor \zeta \rfloor$7, yielding

$\lfloor \zeta \rfloor$8

This non-integrable density corresponds to a GC-$\lfloor \zeta \rfloor$9, with

$r=\|h\|\ge 0$0

that has exactly the desired power-law behavior in space and in time, and whose smoothness in each coordinate direction can be set by choosing $r=\|h\|\ge 0$1, and $r=\|h\|\ge 0$2 (Stein, 2013).

This construction generalizes the isotropic spatial power-law model without collapsing space and time into a single regularity scale. The decisive structural feature is anisotropy in the spectral domain, implemented through separate exponents for $r=\|h\|\ge 0$3 and $r=\|h\|\ge 0$4.

3. Convergent series and asymptotic representations

For $r=\|h\|\ge 0$5, $r=\|h\|\ge 0$6, and $r=\|h\|\ge 0$7, the resulting generalized covariance of order $r=\|h\|\ge 0$8 can be written in isotropic form as

$r=\|h\|\ge 0$9

with the convergent power series, valid for $\zeta>0$0,

$\zeta>0$1

where

$\zeta>0$2

and

$\zeta>0$3

The function $\zeta>0$4 is the classical power-law generalized covariance defined above (Stein, 2013).

Two boundary reductions recover the intended marginal power laws. For $\zeta>0$5,

$\zeta>0$6

matching $\zeta>0$7 with $\zeta>0$8. For $\zeta>0$9,

$\gamma_{\zeta}(r)= \begin{cases} \Gamma(-\zeta)\,r^{2\zeta}, & \zeta\notin\mathbb N,\[6pt] 2(-1)^{\zeta+1}\,\zeta!\;r^{2\zeta}\log r, & \zeta\in\mathbb N. \end{cases}$0

matching $\gamma_{\zeta}(r)= \begin{cases} \Gamma(-\zeta)\,r^{2\zeta}, & \zeta\notin\mathbb N,\[6pt] 2(-1)^{\zeta+1}\,\zeta!\;r^{2\zeta}\log r, & \zeta\in\mathbb N. \end{cases}$1 with $\gamma_{\zeta}(r)= \begin{cases} \Gamma(-\zeta)\,r^{2\zeta}, & \zeta\notin\mathbb N,\[6pt] 2(-1)^{\zeta+1}\,\zeta!\;r^{2\zeta}\log r, & \zeta\in\mathbb N. \end{cases}$2 (Stein, 2013).

In most cases $\gamma_{\zeta}(r)= \begin{cases} \Gamma(-\zeta)\,r^{2\zeta}, & \zeta\notin\mathbb N,\[6pt] 2(-1)^{\zeta+1}\,\zeta!\;r^{2\zeta}\log r, & \zeta\in\mathbb N. \end{cases}$3, $\gamma_{\zeta}(r)= \begin{cases} \Gamma(-\zeta)\,r^{2\zeta}, & \zeta\notin\mathbb N,\[6pt] 2(-1)^{\zeta+1}\,\zeta!\;r^{2\zeta}\log r, & \zeta\in\mathbb N. \end{cases}$4 also admits a Fox–$\gamma_{\zeta}(r)= \begin{cases} \Gamma(-\zeta)\,r^{2\zeta}, & \zeta\notin\mathbb N,\[6pt] 2(-1)^{\zeta+1}\,\zeta!\;r^{2\zeta}\log r, & \zeta\in\mathbb N. \end{cases}$5 representation, and from known asymptotics for small argument of $\gamma_{\zeta}(r)= \begin{cases} \Gamma(-\zeta)\,r^{2\zeta}, & \zeta\notin\mathbb N,\[6pt] 2(-1)^{\zeta+1}\,\zeta!\;r^{2\zeta}\log r, & \zeta\in\mathbb N. \end{cases}$6, one obtains, for fixed $\gamma_{\zeta}(r)= \begin{cases} \Gamma(-\zeta)\,r^{2\zeta}, & \zeta\notin\mathbb N,\[6pt] 2(-1)^{\zeta+1}\,\zeta!\;r^{2\zeta}\log r, & \zeta\in\mathbb N. \end{cases}$7,

$\gamma_{\zeta}(r)= \begin{cases} \Gamma(-\zeta)\,r^{2\zeta}, & \zeta\notin\mathbb N,\[6pt] 2(-1)^{\zeta+1}\,\zeta!\;r^{2\zeta}\log r, & \zeta\in\mathbb N. \end{cases}$8

with explicit constants $\gamma_{\zeta}(r)= \begin{cases} \Gamma(-\zeta)\,r^{2\zeta}, & \zeta\notin\mathbb N,\[6pt] 2(-1)^{\zeta+1}\,\zeta!\;r^{2\zeta}\log r, & \zeta\in\mathbb N. \end{cases}$9, $d$0 (Stein, 2013). In particular the leading non-polynomial term is the power-law $d$1, showing that the temporal smoothness exponent is $d$2.

The convergent series and asymptotic expansions are analytically complementary. The former provides an explicit constructive representation, while the latter isolates the non-polynomial term that governs local regularity.

4. Smoothness, parameter interpretation, and validity conditions

The parameters $d$3 and $d$4 control the decay of the spectral density in spatial and temporal frequencies, hence the mean-square differentiability in those directions. One shows that $d$5 is $d$6-times differentiable in space if and only if

$d$7

and likewise in time if and only if

$d$8

The parameter $d$9, together with the $\gamma_{\zeta}(r)\sim r^{2\zeta}$0, sets the overall smoothness level $\gamma_{\zeta}(r)\sim r^{2\zeta}$1, interpreted as the local Hölder exponent in space or time (Stein, 2013).

Any $\gamma_{\zeta}(r)\sim r^{2\zeta}$2 can be matched by choosing

$\gamma_{\zeta}(r)\sim r^{2\zeta}$3

By contrast, $\gamma_{\zeta}(r)\sim r^{2\zeta}$4, when re-introduced, are range parameters, controlling large-scale correlation without affecting local smoothness (Stein, 2013).

The spectral density

$\gamma_{\zeta}(r)\sim r^{2\zeta}$5

is nonnegative and, for $\gamma_{\zeta}(r)\sim r^{2\zeta}$6, yields a well-defined IRF-$\gamma_{\zeta}(r)\sim r^{2\zeta}$7 generalized covariance in $\gamma_{\zeta}(r)\sim r^{2\zeta}$8. Equivalently, in the stationary sense it satisfies the slow-variation condition at high frequencies

$\gamma_{\zeta}(r)\sim r^{2\zeta}$9

Under this condition, $Z$0 is smoother away from the origin than at the origin and avoids separable anomalies such as the discontinuous incremental correlation, or “dimple” (Stein, 2013).

This suggests that the family is designed not only to realize prescribed local exponents, but also to enforce a specific global regularity profile in which singularity is localized at the origin rather than propagated through separable structure.

5. Special cases and modeling implications

Several special cases clarify the scope of the class. In the purely spatial case, one may set $Z$1, or equivalently consider $Z$2, and the construction reduces to $Z$3. The purely temporal case is obtained similarly from $Z$4 (Stein, 2013).

The Matérn-type isotropic case $Z$5, $Z$6 yields, for $Z$7,

$Z$8

recovering the classical isotropic power-law variogram in $Z$9 dimensions (Stein, 2013).

Separable models, such as $m$0, correspond to $m$1, $m$2, or to product spectral densities; these are excluded by the high-frequency slow-variation condition (Stein, 2013). The exclusion is methodological rather than incidental: it distinguishes the power-law generalized covariance framework from product constructions that may fail to reproduce the desired local singular structure.

The family provides a tool to model environmental or geostatistical phenomena with independently tunable space and time smoothness. Its explicit series and asymptotic formulas permit fast numerical approximation. By avoiding separability and ensuring “smoother away from the origin” behavior, these models yield more realistic kriging predictors in space–time applications such as climate data interpolation, pollutant transport, or soil-moisture dynamics (Stein, 2013).

6. Related temporal power-law covariance constructions

A distinct but related line of work studies how Gaussian correlations are transformed when Gaussian variables or Gaussian time series are mapped to non-Gaussian marginals. Starting from two standard Gaussian variables $m$3 with $m$4, and defining $m$5, $m$6, the Pearson correlation of $m$7 is

$m$8

with

$m$9

All $x\in\mathbb R^d$00, $x\in\mathbb R^d$01, and for symmetric $x\in\mathbb R^d$02 only odd $x\in\mathbb R^d$03 appear. The map is sign preserving and monotonic in $x\in\mathbb R^d$04. For a stationary Gaussian series $x\in\mathbb R^d$05 with autocorrelation $x\in\mathbb R^d$06, setting

$x\in\mathbb R^d$07

gives

$x\in\mathbb R^d$08

If $x\in\mathbb R^d$09 and the first-order term dominates, then

$x\in\mathbb R^d$10

Hence the tail exponent is preserved and only the amplitude is rescaled by $x\in\mathbb R^d$11. A practical recipe is to generate a zero-mean, unit-variance Gaussian series with autocorrelation exponent $x\in\mathbb R^d$12, for example by Fourier Filtering or Davies–Harte, and then transform pointwise through $x\in\mathbb R^d$13. In the IBM stock example, daily absolute returns $x\in\mathbb R^d$14 with $x\in\mathbb R^d$15 yielded a measured $x\in\mathbb R^d$16, corresponding to $x\in\mathbb R^d$17; the synthesized series and the data collapse onto the same power law $x\in\mathbb R^d$18 for $x\in\mathbb R^d$19 up to several hundred days (Carpena et al., 2019).

Another distinct construction is the fractional Pearson diffusion

$x\in\mathbb R^d$20

obtained by time-changing a classical Pearson diffusion $x\in\mathbb R^d$21 by the inverse of a suitable subordinator. Replacing the ordinary derivative by a distributed fractional derivative

$x\in\mathbb R^d$22

leads, in steady state, to a covariance of the form

$x\in\mathbb R^d$23

a finite sum of generalized Mittag-Leffler kernels. As $x\in\mathbb R^d$24,

$x\in\mathbb R^d$25

so the covariance decays like a power law with exponent equal to the smallest fractional order $x\in\mathbb R^d$26. In particular, whenever $x\in\mathbb R^d$27, $x\in\mathbb R^d$28, and the process has long-range dependence. Setting $x\in\mathbb R^d$29 recovers the classical Pearson diffusion with exponentially decaying covariance $x\in\mathbb R^d$30 (Mijena et al., 2014).

These temporal constructions do not define generalized covariances in the intrinsic-random-function sense of Stein’s space–time framework. They are, however, directly relevant to the broader study of power-law covariance behavior: one realizes prescribed local regularity in space–time through generalized covariance functions, while the others realize long-range temporal dependence through marginal transformation or fractional time change.