Spatio-Temporal Expanding Distance Asymptotic Framework for Locally Stationary Processes

Abstract

Spatio-temporal data indexed by sampling locations and sampling time points are encountered in many scientific disciplines such as climatology, environmental sciences, and public health. Here, we propose a novel spatio-temporal expanding distance (STED) asymptotic framework for studying the properties of statistical inference for nonstationary spatio-temporal models. In particular, to model spatio-temporal dependence, we develop a new class of locally stationary spatio-temporal covariance functions. The STED asymptotic framework has a fixed spatio-temporal domain for spatio-temporal processes that are globally nonstationary in a rescaled fixed domain and locally stationary in a distance expanding domain. The utility of STED is illustrated by establishing the asymptotic properties of the maximum likelihood estimation for a general class of spatio-temporal covariance functions. A simulation study suggests sound finite-sample properties and the method is applied to a sea-surface temperature dataset.

Appendices

Appendix A.1: Proof of Proposition 1

Proof.

First, it can be seen that the generalized spatio-temporal Matérn covariance function in Eq. 1 is bounded and twice continuously differentiable with respect to 𝜃; thus, (LS.4) is satisfied. Next, we will show that the generalized spatio-temporal Matérn covariance function satisfies (LS.1) and (LS.2).

For any u₁ and u₂, define

$$ \begin{array}{@{}rcl@{}} \begin{array}{ll} h(\boldsymbol{u}_{1},u_{2}) =\left\{ \begin{array}{ll} \frac{ \theta_{3}^{d/2}2^{1-\nu}}{({\theta_{1}^{2}}{u_{2}^{2}}+1)^{\nu}({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})^{d/2}{\Gamma}(\nu)}m(\boldsymbol{u}_{1},u_{2};\boldsymbol{\theta})^{\nu}K_{\nu}\left\{m(\boldsymbol{u}_{1},u_{2};\boldsymbol{\theta})\right\}, & \text{if} ~ \Vert \boldsymbol{u}_{1}\Vert>0, \\ \frac{ \theta_{3}^{d/2}}{({\theta_{1}^{2}}{u_{2}^{2}}+1)^{\nu}({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})^{d/2}}, & \text{if}~\Vert \boldsymbol{u}_{1}\Vert=0. \end{array} \right. \end{array} \end{array} $$

For any s and t, let

$$ \begin{array}{@{}rcl@{}} \begin{array}{ll} &g_{n}(\boldsymbol{s}^{\prime}-\boldsymbol{s},t^{\prime}-t,\boldsymbol{s},t) = g(\boldsymbol{u}_{1},u_{2},\boldsymbol{s},t)\\ &=\left\{ \begin{array}{ll} D(\boldsymbol{s},t)^{2}\sigma^{2} h(\boldsymbol{u}_{1},u_{2}), & \text{if} ~ \Vert \boldsymbol{u}_{1}\Vert>0 ~\text{or}~ |u_{2}|>0, \\ D(\boldsymbol{s},t)^{2}\sigma^{2} + \tau^{2}, & \text{otherwise}. \end{array} \right. \end{array} \end{array} $$

Then, γ_n((s, t),(s, t)) = g(0,0,s, t). For all $(\boldsymbol {s},t), (\boldsymbol {s}+{\boldsymbol {u}_{1}}/{\varrho _{1,n}},t+{u_{2}}/{\varrho _{2,n}}) \in \mathcal {R}\times \mathcal {T}$ with ∥u₁∥ > 0 or |u₂| > 0, we have

$$ \begin{array}{@{}rcl@{}} &&|\gamma_{n}((\boldsymbol{s},t), (\boldsymbol{s}^{\prime},t^{\prime})) - g(\boldsymbol{u}_{1},u_{2},\boldsymbol{s},t)| = D(\boldsymbol{s},t)h(\boldsymbol{u}_{1},u_{2})\sigma^{2}|D(\boldsymbol{s}^{\prime},t^{\prime}) - D(\boldsymbol{s},t)|\\ &&\leq D(\boldsymbol{s},t)h(\boldsymbol{u}_{1},u_{2})\sigma^{2} (\widetilde{C}_{1}\|\boldsymbol{s}-\boldsymbol{s}^{\prime}\| + \widetilde{C}_{2}|t-t^{\prime}| ) = \mathcal{O}(\|\boldsymbol{s}-\boldsymbol{s}^{\prime}\|+|t-t^{\prime}|) \end{array} $$

uniformly since D(s, t) is bounded on $\mathcal {R}\times \mathcal {T}$ and |h(u₁,u₂)|≤ 1. Thus, (LS.1) is satisfied.

For g(s, t) defined in (LS.2), we have g(s, t) = g(0,0,s, t) = D(s, t)²σ² + τ². Note that $\boldsymbol {s}^{\prime }=\boldsymbol {s} + \boldsymbol {u}_{1}/\varrho _{1,n}$ and $t^{\prime } = t+u_{2}/\varrho _{2,n}$, and we have $|g(\boldsymbol {s},t) - g(\boldsymbol {s}^{\prime },t^{\prime })| = |D(\boldsymbol {s},t)^{2}-D(\boldsymbol {s}^{\prime },t^{\prime })^{2}|\sigma ^{2} = |D(\boldsymbol {s},t)+D(\boldsymbol {s}^{\prime },t^{\prime })||D(\boldsymbol {s},t)-D(\boldsymbol {s}^{\prime },t^{\prime })|\sigma ^{2} \leq |D(\boldsymbol {s},t)+D(\boldsymbol {s}^{\prime },t^{\prime })| (\widetilde {C}_{1}\Vert \boldsymbol {s} - \boldsymbol {s}^{\prime }\Vert + \widetilde {C}_{2} |t - t^{\prime }|)\sigma ^{2}$. Thus, (LS.2) holds by adjusting the constants.

Further, we will show that the generalized spatio-temporal Matérn covariance function (1) satisfies (LS.3) and (LS.5). For all $(\boldsymbol {s},t), (\boldsymbol {s}^{\prime },t^{\prime }) \in \mathcal {R}\times \mathcal {T}$, we have

$$ \gamma_{n}((\boldsymbol{s},t),(\boldsymbol{s}^{\prime},t^{\prime}))\leq (\max D(\boldsymbol{s},t))^{2}(\sigma^{2} +\tau^{2})h(\boldsymbol{u}_{1},u_{2}). $$

Thus, to show (LS.3), it suffices to find γ₀(∥u₁∥) and γ₁(|u₂|) to bound h(u₁,u₂). Moreover, straightforward calculation yields that all first- and second-order partial derivatives of γ_n, denoted by γ_{n, k} and $\gamma _{n,kk^{\prime }}$, can be bounded by the partial derivatives of h(u₁,u₂) up to some constant scales, which enables us to obtain γ₂(∥u₁∥) and γ₃(|u₂|) in (LS.5).

In addition, we have

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial h(\boldsymbol{u}_{1},u_{2})}{\partial \theta_{1}} = \frac{2^{1-\nu}}{\Gamma(\nu)} \left( \frac{\partial h(0,u_{2})}{\partial \theta_{1}}m^{\nu}K_{\nu}(m) - h(0,u_{2})m^{\nu}K_{\nu-1}(m)\frac{\partial m}{\partial \theta_{1}}\right) ,\\ &&\frac{\partial h(\boldsymbol{u}_{1},u_{2})}{\partial \theta_{2}} = -\frac{2^{1-\nu}}{\Gamma(\nu)} h(0,u_{2})m^{\nu}K_{\nu-1}(m)\frac{\partial m}{\partial \theta_{2}},\\ &&\frac{\partial h(\boldsymbol{u}_{1},u_{2})}{\partial \theta_{3}} = \frac{2^{1-\nu}}{\Gamma(\nu)} \left( \frac{\partial h(0,u_{2})}{\partial \theta_{3}}m^{\nu}K_{\nu}(m) - h(0,u_{2})m^{\nu}K_{\nu-1}(m)\frac{\partial m}{\partial \theta_{3}}\right), \end{array} $$

where m = m(u₁,u₂;𝜃) and

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial m(\boldsymbol{u}_{1},u_{2};\boldsymbol{\theta})}{\partial \theta_{1}} = \frac{m(\boldsymbol{u}_{1},u_{2};\boldsymbol{\theta})\theta_{1}{u_{2}^{2}}(\theta_{3}-1)}{({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})({\theta_{1}^{2}}{u_{2}^{2}}+1)},\\ &&\frac{\partial m(\boldsymbol{u}_{1},u_{2};\boldsymbol{\theta})}{\partial \theta_{2}} = \frac{m(\boldsymbol{u}_{1},u_{2};\boldsymbol{\theta})}{\theta_{2}}, \frac{\partial m(\boldsymbol{u}_{1},u_{2};\boldsymbol{\theta})}{\partial \theta_{3}} = -\frac{m(\boldsymbol{u}_{1},u_{2};\boldsymbol{\theta})}{2({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})},\\ &&\frac{\partial h(0,u_{2})}{\partial \theta_{1}} = -\frac{\theta_{1}{u_{2}^{2}}(2\nu({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})+d({\theta_{1}^{2}}{u_{2}^{2}}+1))} {({\theta_{1}^{2}}{u_{2}^{2}}+1)({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})}h(0,u_{2}),\\ &&\frac{\partial h(0,u_{2})}{\partial \theta_{3}} = \frac{d\theta_{3}^{d/2-1}{\theta_{1}^{2}}{u_{2}^{2}}}{2({\theta_{1}^{2}}{u_{2}^{2}}+1)^{\nu}({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})^{d/2+1}} = \frac{d{\theta_{1}^{2}}{u_{2}^{2}}}{2\theta_{3}({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})}h(0,u_{2}). \end{array} $$

For (LS.3), it can be seen that $m(\boldsymbol {u}_{1},u_{2};\boldsymbol {\theta }) \leq \max \limits \left \{\theta _{2} \theta _{3}^{-1/2},\theta _{2}\right \}\Vert \boldsymbol {u}_{1}\Vert$. Thus, we have

$$ h(\boldsymbol{u}_{1},u_{2})\leq \frac{ \theta_{3}^{d/2}2^{1-\nu}\widetilde{m}(\boldsymbol{u}_{1})^{\nu}K_{\nu}\left\{\widetilde{m}(\boldsymbol{u}_{1})\right\}}{({\theta_{1}^{2}}{u_{2}^{2}}+1)^{\nu}({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})^{d/2}{\Gamma}(\nu)}\leq 1, $$

where $\widetilde {m}(\boldsymbol {u}_{1}) = \max \limits \left \{\theta _{2} \theta _{3}^{-1/2},\theta _{2}\right \}\Vert \boldsymbol {u}_{1}\Vert$. We can see that, up to some constant scales,

$$ h(\boldsymbol{u}_{1},u_{2})\leq \left( \widetilde{m}(\boldsymbol{u}_{1})^{\nu}K_{\nu}\left\{\widetilde{m}(\boldsymbol{u}_{1})\right\}\right)\left( |u_{2}|^{-2\nu-d}\right) \equiv \gamma_{0}(\Vert\boldsymbol{u}_{1}\Vert)\gamma_{1}(|u_{2}|) $$

Here, γ₀(∥u₁∥) is a linear combination of a polynomial of ∥u₁∥ with degree ν and a modified Bessel function of the second kind and γ₁(|u₂|) is a polynomial of |u₂| with degree − 2ν − d.

For (LS.5), we focus on the first-order partial derivatives in detail and omit details for the second-order partial derivatives, as similar arguments can be applied. Straightforward calculation shows that the (absolute value of) partial derivatives of h(u₁,u₂) can be bounded by products of two positive functions, $\widetilde {\gamma }_{2}(\Vert \boldsymbol {u}_{1}\Vert )$ and $\widetilde {\gamma }_{3}(|u_{2}|)$. Moreover, $\widetilde {\gamma }_{2}(\Vert \boldsymbol {u}_{1}\Vert )$ is a linear combination of a polynomial of ∥u₁∥ with degree at least ν and a modified Bessel function of the second kind, and $\widetilde {\gamma }_{3}(|u_{2}|)$ is a polynomial of |u₂| with degree at most − 2ν − d.

Since the partial derivatives of h(u₁,u₂) with respect to 𝜃 is continuous in ∥u₁∥ and |u₂|, it is bounded if ∥u₁∥ and |u₂| are bounded. To show (LS.3) and (LS.5), it suffices to show that, for k, l > 0, there exists M > 0 such that

• (i) ${\int \limits }_{M}^{\infty } u^{k}K_{l}(u)du<\infty$,

• (ii) u^kK_l(u) is bounded by a nonincreasing function on $(M,\infty )$.

Since d ≥ 1 and k > 0, u^− 2k−d is bounded on $(M,\infty )$ and ${\int \limits }_{M}^{\infty } u^{-2k-d}du=M^{-2k-d+1}/(2k+d-1) <\infty$. The last two conditions hold. By the property of Bessel function, $K_{l}(u)\propto e^{-u}u^{-1/2}\{1+\mathcal {O}(1/u)\}$, as $|u|\to \infty$. We can find M₁,M₂ such that K_l(u) ≤ M₁e^−uu^− 1/2(1 + M₂/u), when u ≥ M₂. So (i) holds since

$$ \begin{array}{@{}rcl@{}} {\int}_{M_{2}}^{\infty} u^{k}K_{l}(u)du \leq &&{\int}_{M_{2}}^{\infty} M_{1}u^{k-1/2}e^{-u}(1+M_{2}/u)du\\ \leq && 2M_{1}{\int}_{M_{2}}^{\infty} u^{k-1/2}e^{-u}du < 2M_{1}{\Gamma}(k+1/2) < \infty. \end{array} $$

For (ii), we have u^kK_l(u) ≤ M₁e^−uu^k− 1/2(1 + M₂/u) ≤ 2M₁e^−uu^k− 1/2, when u ≥ M₂. Since e^−uu^k− 1/2 is decreasing on $(k-1/2,\infty )$, (ii) is satisfied. □

Appendix A.2: Generalized Exponential Spatio-temporal Covariance Function

In this section, we show that the following exponential spatio-temporal covariance function used in a simulation study satisfies conditions (LS.1)–(LS.5). The covariance function can be written as

$$ \begin{array}{@{}rcl@{}} \begin{array}{ll} &\gamma_{n}((\boldsymbol{s},t), (\boldsymbol{s}^{\prime},t^{\prime});\boldsymbol{\theta}) \\ =&\left\{ \begin{array}{ll} D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})\sigma^{2}\exp\{-\Vert\boldsymbol{u}_{1}\Vert/c_{s} - |u_{2}|/c_{t}\}, & \text{if}~\Vert \boldsymbol{u}_{1}\Vert>0~\text{or}~|u_{2}|> 0; \\ D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})\sigma^{2} + \tau^{2}, & \text{otherwise}, \end{array} \right. \end{array} \end{array} $$

where 𝜃 = (c_s,c_t,σ²,τ²)^⊤ is the vector of spatio-temporal parameters with the scaling parameter in space, c_s ≥ 0, and the scaling parameter in time, c_t ≥ 0. In addition, $\boldsymbol {u}_{1} = \varrho _{1,n}(\boldsymbol {s}-\boldsymbol {s}^{\prime })$ is the spatial lag scaled to the spatially expanding domain, and $u_{2} = \varrho _{2,n}(t-t^{\prime })$ is the temporal lag scaled to the temporally expanding domain, where ϱ_1,n and ϱ_2,n are positive constants. Further, D(s, t) is some fixed positive spatio-temporal function with D(0,0) = 1. Note that D(s, t)²σ² + τ² is the variance of Y (s, t).

By arguments similar to Section Appendix, we show (LS.1), (LS.2) and (LS.4). For (LS.3), we can see that, for all $(\boldsymbol {s},t), (\boldsymbol {s}^{\prime },t^{\prime }) \in \mathcal {R}\times \mathcal {T}$,

$$ \begin{array}{@{}rcl@{}} \gamma_{n}((\boldsymbol{s},t),(\boldsymbol{s}^{\prime},t^{\prime})) &\leq&\{\max D(\boldsymbol{s},t)\}^{2}(\sigma^{2} +\tau^{2})\exp\{-\Vert\boldsymbol{u}_{1}\Vert/c_{s}\} \exp\{- |u_{2}|/c_{t}\} \\&=& {\gamma}_{0}(\|\boldsymbol{u}_{1}\|){\gamma}_{1}(|u_{2}|). \end{array} $$

Here, both γ₀(∥u₁∥) and γ₁(|u₂|) are nonincreasing positive functions. Moreover, we have ${\int \limits }_{0}^{\infty } e^{-u/c_{s}} du = 1/c_{s} < \infty$ and ${\int \limits }_{0}^{\infty } e^{-u/c_{t}} du = 1/c_{t} < \infty$. Thus, (LS.3) is satisfied.

Further, we have

$$ \begin{array}{ll} &\partial\gamma_{n}/\partial\tau^{2} = 1_{\{\Vert\boldsymbol{u}_{1}\Vert=0, |u_{2}|=0\}},\\ &\partial\gamma_{n}/\partial\sigma^{2} = D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})\exp\{-\Vert\boldsymbol{u}_{1}\Vert/c_{s} - |u_{2}|/c_{t}\},\\ &\partial \gamma_{n}/\partial c_{s} = D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})\sigma^{2}\Vert\boldsymbol{u}_{1}\Vert\exp\{-\Vert\boldsymbol{u}_{1}\Vert/c_{s} - |u_{2}|/c_{t}\}/{c_{s}^{2}},\\ &\partial \gamma_{n}/\partial c_{t} = D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})\sigma^{2}|u_{2}|\exp\{-\Vert\boldsymbol{u}_{1}\Vert/c_{s} - |u_{2}|/c_{t}\}/{c_{t}^{2}},\\ &\partial^{2}\gamma_{n}/\partial\sigma^{2}\partial c_{s} = D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})\Vert\boldsymbol{u}_{1}\Vert\exp\{-\Vert\boldsymbol{u}_{1}\Vert/c_{s} - |u_{2}|/c_{t}\}/{c_{s}^{2}},\\ &\partial^{2}\gamma_{n}/\partial\sigma^{2}\partial c_{t} = D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})|u_{2}|\exp\{-\Vert\boldsymbol{u}_{1}\Vert/c_{s} - |u_{2}|/c_{t}\}/{c_{t}^{2}},\\ &\partial^{2} \gamma_{n}/\partial c_{s}\partial c_{t} = D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})\sigma^{2}\Vert\boldsymbol{u}_{1}\Vert|u_{2}|\exp\{-\Vert\boldsymbol{u}_{1}\Vert/c_{s} - |u_{2}|/c_{t}\}/({c_{s}^{2}}{c_{t}^{2}}),\\ &\partial^{2} \gamma_{n}/\partial {c_{s}^{2}} = D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})\sigma^{2}\Vert\boldsymbol{u}_{1}\Vert\exp\{-\Vert\boldsymbol{u}_{1}\Vert/c_{s} - |u_{2}|/c_{t}\}(\Vert\boldsymbol{u}_{1}\Vert/{c_{s}^{4}}-2/{c_{s}^{3}}),\\ &\partial^{2} \gamma_{n}/\partial {c_{t}^{2}} = D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})\sigma^{2}|u_{2}|\exp\{-\Vert\boldsymbol{u}_{1}\Vert/c_{s} - |u_{2}|/c_{t}\}(|u_{2}|/{c_{t}^{4}}-2/{c_{t}^{3}}). \end{array} $$

Here, all the first- and second-order partial derivatives are continuous in ∥u₁∥ and |u₂| and hence bounded when ∥u₁∥ and |u₂| are bounded. In addition, they are bounded by a product of two functions $\widetilde {\gamma }_{2}(\Vert \boldsymbol {u}_{1}\Vert )$ and $\widetilde {\gamma }_{3}(|u_{2}|)$, where $\widetilde {\gamma }_{2}(u)$ and $\widetilde {\gamma }_{3}(u)$ are products of a polynomial of u and an exponential function of u. (LS.5) is satisfied, since for k > 0, u^ke^−u is nonincreasing on $[k,\infty )$, and ${\int \limits }_{k}^{\infty } u^{k}e^{-u}du<\infty$.

Appendix A.3: A Remark on Assumption (C.1)

In this section, we provide two sufficient conditions for (C.1). Further, we will demonstrate that, if 𝜃₃ > 1, the generalized spatio-temporal Matérn covariance function (1) satisfies Assumption (C.1). The two sufficient conditions are stated as follows:

• (E.1) For 1 ≤ k ≤ q, $|\gamma _{n,k}((\boldsymbol {s},t), (\boldsymbol {s}^{\prime },t^{\prime }))|$ satisfies one of the following two conditions: (i) |γ_{n, k}((s, t),(s, t))| > 0; (ii) For $\|\boldsymbol {u}_{1}\|, |u_{2}| \in [M,\infty )$ for some constant M > 0 such that $(\boldsymbol {s},t), (\boldsymbol {s}+{\boldsymbol {u}_{1}}/{A_{n}},t+{u_{2}}/{B_{n}}) \in \mathcal {R}\times \mathcal {T}$, we have $|\gamma _{n,k}((\boldsymbol {s},t), (\boldsymbol {s}+{\boldsymbol {u}_{1}}/{A_{n}},t+{u_{2}}/{B_{n}}))| \geq C_{3}\exp (-C_{4}\Vert \boldsymbol {u}_{1}\Vert - C_{5}|u_{2}|)$ for all n, where C₃,C₄,C₅ > 0 are constants.

• (E.2) (i) For any two given positive constants M₁, M₂, there exist $M_{1}^{\prime }$ and $M_{2}^{\prime }$ with $M_{1} < M_{1}^{\prime } < \infty$ and $M_{2} < M_{2}^{\prime } < \infty$ such that ${\sum }_{i}{\sum }_{j} {1}(\Vert \boldsymbol {s}_{i} - \boldsymbol {s}_{j}\Vert \in [M_{1}\delta _{n},M_{1}^{\prime }\delta _{n}]){1}(|t_{i}-t_{j}| \in [M_{2}\zeta _{n},M_{2}^{\prime }\zeta _{n}]) \geq C_{6}N_{n}^{1/2+\iota _{1}}$ for some C₆ > 0 and ι₁ > 0. (ii) $A_{n}\delta _{n} = \mathcal {O}(b_{n})$ and $B_{n}\zeta _{n} = \mathcal {O}(b_{n})$, where $b_{n} = \log \log N_{n}$.

To see the sufficiency of (E.1) and (E.2), we first note that if $A_{n}\delta _{n} = \mathcal {O}(1)$ and $B_{n}\zeta _{n} = \mathcal {O}(1)$, then $\|{{~}^{n}\boldsymbol {\Gamma }}_{k}\|_{F}^{2} \geq CN_{n}^{1/2+\iota _{1}}$ for some C > 0. Thus, (C.1) is satisfied with ι = ι₁. If $A_{n}\delta _{n} \to \infty$ or $B_{n}\zeta _{n} \to \infty$, by (E.1)–(E.2), we have $\|{{~}^{n}\boldsymbol {\Gamma }}_{k}\|_{F}^{2} \geq CN_{n}^{1/2+\iota _{1}^{\prime }}$ for some C > 0 and any $\iota _{1}^{\prime }$ such that $0<\iota _{1}^{\prime }<\iota _{1}$, so (C.1) is satisfied with $\iota = \iota _{1}^{\prime }$.

Next, we will show that the generalized spatio-temporal Matérn covariance function (1) satisfies (E.1), when 𝜃₃ > 1. Since $\left |\frac {\partial \gamma _{n}((\boldsymbol {s},t),(\boldsymbol {s},t))}{\partial \sigma ^{2}}\right | = D(\boldsymbol {s},t)^{2} > 0$ and $\left |\frac {\partial \gamma _{n}((\boldsymbol {s},t),(\boldsymbol {s},t))}{\partial \tau ^{2}}\right | = 1$, ∂γ_n/∂σ² and ∂γ_n/∂τ² satisfy (E.1)(i).

Further, we show that ∂γ_n/∂𝜃_i satisfies (E.1)(ii) for i = 1,2,3. Recall that for all $(\boldsymbol {s},t), (\boldsymbol {s}+{\boldsymbol {u}_{1}}/{\varrho _{1,n}},t+{u_{2}}/{\varrho _{2,n}}) \in \mathcal {R}\times \mathcal {T}$ with ∥u₁∥ > 0 or |u₂| > 0, we have

$$ \begin{array}{@{}rcl@{}} \left|\frac{\partial \gamma_{n}}{\partial \theta_{1}}\right| &=& \frac{D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})\sigma^{2}2^{1-\nu}\theta_{1}\theta_{3}^{d/2}{u_{2}^{2}} }{\Gamma(\nu)({\theta_{1}^{2}}{u_{2}^{2}}+1)^{\nu+1}({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})^{d/2+1}}\left\{(\theta_{3}-1)m^{\nu+1}K_{\nu-1}(m)\right.\\ &&\left.+\left( 2\nu({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})+d({\theta_{1}^{2}}{u_{2}^{2}}+1)\right)m^{\nu}K_{\nu}(m) \right\} ,\\ \left|\frac{\partial \gamma_{n}}{\partial \theta_{2}}\right| &=& \frac{D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})\sigma^{2}2^{1-\nu}\theta_{3}^{d/2}\{m^{\nu+1}K_{\nu-1}(m)\}}{\Gamma(\nu)\theta_{2}({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})^{d/2}({\theta_{1}^{2}}{u_{2}^{2}}+1)^{\nu}} ,\\ \left|\frac{\partial \gamma_{n}}{\partial \theta_{3}}\right| &=\!& \frac{\!D(\boldsymbol{s},t)D(\boldsymbol{s}^{\prime},t^{\prime})\sigma^{2}2^{-\nu}\theta_{3}^{d/2-1}\{d\theta_{1}{u_{2}^{2}}m^{\nu}K_{\nu}(m) + \theta_{3}m^{\nu+1}K_{\nu-\!1}(m)\}}{\Gamma(\nu)({\theta_{1}^{2}}{u_{2}^{2}}+\theta_{3})^{d/2+1}({\theta_{1}^{2}}{u_{2}^{2}}+1)^{\nu}}. \end{array} $$

Up to some constant scale, we have

$$ \begin{array}{@{}rcl@{}} \left|\frac{\partial \gamma_{n}}{\partial \theta_{1}}\right| &\geq&|u_{2}|^{-2\nu-d-2}m^{\nu+1}K_{\nu-1}(m) + |u_{2}|^{-2\nu-d}m^{\nu}K_{\nu}(m)\\ &\geq&|u_{2}|^{-2\nu-d-2}\left( \theta_{2}\theta_{3}^{-1/2}\|\boldsymbol{u}_{1}\|\right)^{\nu+1}K_{\nu-1}(\theta_{2}\|\boldsymbol{u}_{1}\|) \\ &&+ |u_{2}|^{-2\nu-d}\left( \theta_{2}\theta_{3}^{-1/2}\|\boldsymbol{u}_{1}\|\right)^{\nu}K_{\nu}(\theta_{2}\|\boldsymbol{u}_{1}\|),\\ \left|\frac{\partial \gamma_{n}}{\partial \theta_{2}}\right| &\geq&|u_{2}|^{-2\nu-d}m^{\nu+1}K_{\nu-1}(m)\\ &\geq&|u_{2}|^{-2\nu-d}\left( \theta_{2}\theta_{3}^{-1/2}\|\boldsymbol{u}_{1}\|\right)^{\nu+1}K_{\nu-1}(\theta_{2}\|\boldsymbol{u}_{1}\|),\\ \left|\frac{\partial \gamma_{n}}{\partial \theta_{3}}\right| &\geq&|u_{2}|^{-2\nu-d-2}m^{\nu+1}K_{\nu-1}(m) + |u_{2}|^{-2\nu-d}m^{\nu}K_{\nu}(m)\\ &\geq&|u_{2}|^{-2\nu-d-2}\left( \theta_{2}\theta_{3}^{-1/2}\|\boldsymbol{u}_{1}\|\right)^{\nu+1}K_{\nu-1}(\theta_{2}\|\boldsymbol{u}_{1}\|)\\ && + |u_{2}|^{-2\nu-d}\left( \theta_{2}\theta_{3}^{-1/2}\|\boldsymbol{u}_{1}\|\right)^{\nu}K_{\nu}(\theta_{2}\|\boldsymbol{u}_{1}\|),\\ \end{array} $$

since $\theta _{2}\theta _{3}^{-1/2}\|\boldsymbol {u}_{1}\|\leq m(\boldsymbol {u}_{1},u_{2};\boldsymbol {\theta })\leq \theta _{2}\|\boldsymbol {u}_{1}\|$. In addition, by property of Bessel function, $K_{l}(u)\propto e^{-u}u^{-1/2}\{1+\mathcal {O}(1/u)\}$, as $|u|\to \infty$. We can find M₁,M₂ such that K_l(u) ≥ M₁e^−uu^− 1/2, when u ≥ M₂; thus, (E.1)(ii) follows.

Remark 1.

It can be seen that the generalized exponential covariance function also satisfies (E.1). Note that $\left |\frac {\partial \gamma _{n}((\boldsymbol {s},t),(\boldsymbol {s},t))}{\partial \sigma ^{2}}\right | = D(\boldsymbol {s},t)^{2} > 0$ and $\left |\frac {\partial \gamma _{n}((\boldsymbol {s},t),(\boldsymbol {s},t))}{\partial \tau ^{2}}\right | = 1$, and (E.1)(i) holds. Next, ∂γ_n/∂σ², ∂γ_n/∂c_s and ∂γ_n/∂c_t are positive when ∥u₁∥ > 2c_s and |u₂| > 2c_t and can be written as linear combinations of products of $|u_{2}|^{j}\exp (-a_{1}|u_{2}|)$ and $\Vert \boldsymbol {u}_{1}\Vert ^{k} \exp (-a_{2}\Vert \boldsymbol {u}_{1}\Vert )$ for j, k ≥ 0 and some constants a₁,a₂ > 0. Hence, (E.1)(ii) follows.

Appendix A.4: Proof of Theorem 1

Proof.

To prove Theorem 1, it suffices to show that $\|\boldsymbol {\Gamma }\|_{2}=\mathcal {O}(1)$, $\|\boldsymbol {\Gamma }_{k}\|_{2} = \mathcal {O}(1)$ and $\|\boldsymbol {\Gamma }_{kk^{\prime }}\|_{2} = \mathcal {O}(1)$, for all $k,k^{\prime } = 1,\ldots , q$ (Mardia and Marshall, 1984). Note that $\|\boldsymbol {A}\|_{2}\leq \|\boldsymbol {A}\|_{\infty }$ for any positive definite matrix A. We only need to show that $\|\boldsymbol {\Gamma }\|_{\infty } = \mathcal {O}(1)$, $\|\boldsymbol {\Gamma }_{k}\|_{\infty } = \mathcal {O}(1)$ and $\|\boldsymbol {\Gamma }_{kk^{\prime }}\|_{\infty } = \mathcal {O}(1)$, for all $k,k^{\prime } = 1,\ldots , q$.

For each i, let $\mathcal {A}_{1,i}=\{j: \Vert \boldsymbol {s}_{i}-\boldsymbol {s}_{j}\Vert \leq C_{s,n}\}$ and $\mathcal {A}_{2,i}=\{j: |t_{i}-t_{j}|\leq C_{t,n}\}$. Let a₁ = C_{s, n}/δ_n, a₂ = δ_nA_n, b₁ = C_{t, n}/ζ_n and b₂ = ζ_nB_n. Then,

$$ \begin{array}{@{}rcl@{}} \|\boldsymbol{\Gamma}\|_{\infty} &=& \max_{1\leq i\leq N_{n}}{\sum}_{j\in \mathcal{A}_{1,i}\cap\mathcal{A}_{2,i}}{\Gamma}_{ij} + \max_{1\leq i\leq N_{n}}{\sum}_{j\in \mathcal{A}_{1,i}^{c}\cap\mathcal{A}_{2,i}}{\Gamma}_{ij} \\ &&+ \max_{1\leq i\leq N_{n}}{\sum}_{j\in \mathcal{A}_{1,i}\cap\mathcal{A}_{2,i}^{c}}{\Gamma}_{ij} +\max_{1\leq i\leq N_{n}}{\sum}_{j\in \mathcal{A}_{1,i}^{c}\cap\mathcal{A}_{2,i}^{c}}{\Gamma}_{ij} \\ &=& (I_{1}) + (I_{2}) + (I_{3}) + (I_{4}), \end{array} $$

where Γ_ij is the (i, j)th entry of Γ.

Denote $\text {Card}(\mathcal {A})$ as the cardinality of the set $\mathcal {A}$, then

$$ \begin{array}{@{}rcl@{}} (I_{1}) &\leq&\|\boldsymbol{\Gamma}\|_{\max} \cdot \text{Card}(\mathcal{A}_{1,i}\cap\mathcal{A}_{2,i}) \leq \mathcal{O}\left( \frac{C_{s,n}^{d}C_{t,n}}{{\delta_{n}^{d}}\zeta_{n}}\right) = \mathcal{O}({a_{1}^{d}}b_{1}),\\ (I_{2}) &\leq& \text{Card}(\mathcal{A}_{2,i}) {\sum}_{m=\lfloor\left( \frac{C_{s,n}A_{n}}{b}\right)\rfloor} \mathcal{O}\left( \frac{m^{d-1}b^{d}}{{\delta_{n}^{d}}{A_{n}^{d}}}\right) \max_{mb\leq\|\boldsymbol{u}_{1}\|\leq(m+1)b} \gamma_{0}(\|\boldsymbol{u}_{1}\|)\\ &\leq& \mathcal{O}\left( \frac{C_{t,n}}{\zeta_{n}{\delta_{n}^{d}}{A_{n}^{d}}}\right){\int}_{C_{s,n}A_{n}}^{\infty}u^{d-1}\gamma_{0}(u)du = \mathcal{O}(b_{1}/{a_{2}^{d}}){\int}_{a_{1}a_{2}}^{\infty} u^{d-1}\gamma_{0}(u)du,\\ (I_{3}) &\leq& \text{Card}(\mathcal{A}_{1,i}) {\sum}_{m=\lfloor\left( \frac{C_{t,n}B_{n}}{b}\right)\rfloor} \mathcal{O}\left( \frac{b}{\zeta_{n}B_{n}}\right) \max_{mb\leq|u_{2}|\leq(m+1)b} \gamma_{1}(|u_{2}|)\\ &\leq& \mathcal{O}\left( \frac{C_{s,n}^{d}}{{\delta_{n}^{d}}\zeta_{n}B_{n}}\right){\int}_{C_{t,n}B_{n}}^{\infty}\gamma_{1}(u)du=\mathcal{O}({a_{1}^{d}}/b_{2}){\int}_{b_{1}b_{2}}^{\infty} \gamma_{1}(u)du, \end{array} $$

$$ \begin{array}{@{}rcl@{}} (I_{4}) &\leq& {\sum}_{m=\lfloor\left( \frac{C_{s,n}A_{n}}{b}\right)\rfloor} \mathcal{O}\left( \frac{m^{d-1}b^{d}}{{\delta_{n}^{d}}{A_{n}^{d}}}\right) \max_{mb\leq\|\boldsymbol{u}_{1}\|\leq(m+1)b} \gamma_{0}(\|\boldsymbol{u}_{1}\|)\times\\ && {\sum}_{m^{\prime}=\lfloor\left( \frac{C_{t,n}B_{n}}{b}\right)\rfloor} \mathcal{O}\left( \frac{b}{\zeta_{n}B_{n}}\right) \max_{m^{\prime}b\leq|u_{2}|\leq(m^{\prime}+1)b} \gamma_{1}(|u_{2}|)\\ &\leq& \mathcal{O}\left( \frac{1}{\zeta_{n}B_{n}{\delta_{n}^{d}}{A_{n}^{d}}}\right){\int}_{C_{s,n}A_{n}}^{\infty}u\gamma_{0}(u)du{\int}_{C_{t,n}B_{n}}^{\infty}\gamma_{1}(u)du\\ &=& \mathcal{O}(1/{a_{2}^{d}}b_{2}){\int}_{a_{1}a_{2}}^{\infty} u^{d-1}\gamma_{0}(u)du{\int}_{b_{1}b_{2}}^{\infty} \gamma_{1}(u)du. \end{array} $$

To show $\Vert \boldsymbol {\Gamma } \Vert _{\infty }= \mathcal {O}(1)$, it suffices to show that

• (i) ${a_{1}^{d}}b_{1}\in [C_{1},C_{2}]$ for some constants C₁,C₂ > 0,

• (ii) $\mathcal {O}(1/{a_{1}^{d}}{a_{2}^{d}}){\int \limits }_{a_{1}a_{2}}^{\infty } u^{d-1}\gamma _{0}(u)du = \mathcal {O}(1)$,

• (iii) $\mathcal {O}(1/b_{1}b_{2}) {\int \limits }_{b_{1}b_{2}}^{\infty } \gamma _{1}(u)du = \mathcal {O}(1)$.

Let C_{s, n} = 1/A_n and C_{t, n} = 1/B_n. By (A.3), ${a_{1}^{d}}b_{1} = ({\delta _{n}^{d}}{A_{n}^{d}}\zeta _{n}B_{n})^{-1} \leq c_{3}^{-1} = \mathcal {O}(1)$, the above requirements are fulfilled. By (LS.5),

$$ \max\{|\gamma_{n,k}((\boldsymbol{s},t),(\boldsymbol{s}^{\prime},t^{\prime}); \boldsymbol{\theta})|,|\gamma_{n,kk^{\prime}}((\boldsymbol{s},t),(\boldsymbol{s}^{\prime},t^{\prime}); \boldsymbol{\theta})|\} \leq \gamma_{2}(0)\gamma_{3}(0), $$

uniformly for all n and $1< k,k^{\prime } < q$. Thus, we have $\|\boldsymbol {\Gamma }\|_{2} = \mathcal {O}(1)$, and similar arguments can be applied to show that $\|\boldsymbol {\Gamma }_{k}\|_{2} = \mathcal {O}(1)$ and $\|\boldsymbol {\Gamma }_{kk^{\prime }}\|_{2}= \mathcal {O}(1)$.

Together with (C.1)–(C.3) and by Theorem 1 of Sweeting (1980), we have the result of Theorem 1. □