Modern Stochastics: Theory and Applications

Let $\mathcal{M}$ be a set. A symmetric funtion $\phi \hspace{2.5pt}:\hspace{2.5pt}\mathcal{M}\times \mathcal{M}\to \mathbb{R}$ is said to be of positive type if for every ${M_{1}},\dots ,{M_{n}}\in \mathcal{M}$ and every ${\lambda _{1}},\dots ,{\lambda _{n}}\in \mathbb{R}$,

Let $\mathcal{M}$ be a set. A symmetric funtion $\psi \hspace{2.5pt}:\hspace{2.5pt}\mathcal{M}\times \mathcal{M}\to \mathbb{R}$ is said to be (conditionally) of negative type if $\psi \left(M,M\right)=0$ for all $M\in \mathcal{M}$, and for every ${M_{1}},\dots ,{M_{n}}\in \mathcal{M}$ and every ${\lambda _{1}},\dots ,{\lambda _{n}}\in \mathbb{R}$ such that ${\textstyle\sum _{j=1}^{n}}{\lambda _{j}}=0$ we have

The spherical bifractional Browninan motion ${B_{H,K}}$ exists, if and only if, $0<H\le 1/2$ and $K\in (0,1]$.

The case $K=1$ reduces ${B_{H,K}}$ to the spherical fBm, which is treated in [11]. We shall treat the case $H\le 1/2$, $K<1$. Let $M,N\in \mathbb{S}$. We need to show that $R(\cdot ,\cdot )$ given in (1) is of positive type. Using the fact that for every $K\in (0,1)$ and $\lambda \ge 0$, we obtain where ${\rho _{K}}=\frac{K}{{2^{K}}\Gamma (1-K)}$. Let ${M_{1}},\dots ,{M_{n}}\in \mathbb{S}$ and ${\lambda _{1}},\dots ,{\lambda _{n}}\in \mathbb{R}$. From (3) it follows that Let $\xi (u)={\textstyle\sum _{j=1}^{n}}{\lambda _{j}}\left(1-{e^{-i\sqrt{2u}{B_{H}}({M_{j}})}}\right)$, where $i=\sqrt{-1}$ and ${B_{H}}$ is the spherical fBm vanishing at O (which exists because $H\le 1/2$ (see [11, Theorem 3.1])). We have On the other hand, Combining (5)–(6) with (4) yields and we conclude that ${B_{H,K}}$ exists.

Now let $H>1/2$. We shall prove that $R(\cdot ,\cdot )$ is not of positive type. Set $2H=1+2{H^{\prime }}$ with ${H^{\prime }}<1/2$. After a change of variables $v=ud(M,N)$, formula (3) can be rewritten as where It is easy to see that ${\Phi _{j}}$, $j=1,2$, are symmetric and of negative type by Lemma 1 and the condition ${H^{\prime }}<1/2$. Note ${d^{K}}$ is of negative type since ${\beta _{\mathbb{S}}}=1$ and $K\le 1$ (see [12, Proposition 2.6]). The function ${\Phi _{3}}$ is not of negative type because it does not vanish on the diagonal. However, we have ${\textstyle\sum _{j,l}}{\lambda _{j}}{\lambda _{l}}{\Phi _{3}}({M_{j}},{M_{l}})\le 0$, for all ${M_{1}},\dots ,{M_{n}}\in \mathbb{S}$ and ${\lambda _{1}},\dots ,{\lambda _{n}}\in \mathbb{R}$. As a result, ${\textstyle\sum _{j,l=1}^{n}}{\lambda _{j}}{\lambda _{l}}R\left({M_{j}},{M_{l}}\right)\le 0$, for all ${M_{1}},\dots ,{M_{n}}\in \mathbb{S}$ and ${\lambda _{1}},\dots ,{\lambda _{n}}\in \mathbb{R}$ such that ${\textstyle\sum _{j=1}^{n}}{\lambda _{j}}=0$. Suppose that $R(\cdot ,\cdot )$ is of positive type, then the last inequality becomes equality. By choosing ${\lambda _{1}}=-{\lambda _{2}}=1$ and ${M_{1}},{M_{2}}\in \mathbb{S}$ so that $d(O,M)=d(O,N)=d(M,N)/2$ we obtain a contradiction.  □

Let $\theta >0$. The symmetric function for all $M,N\in \mathbb{S}$, defines a covariance function if $K\in (-\infty ,1]\setminus \{0\}$ and $H\in (0,1/2]$.

Our motivation for introducing $\theta >0$ in the covariance structures in Theorem 2 and propositions below is threefold. First, we try to give covariance functions more general as in [19, Theorem 1], but we consider negative exponents (as parameters $K<0$ in Theorem 2 or functions ${\nu _{j}}(M)<0$ in propositions below). Second, if negative exponents are considered, one has to face the problem that the covariance may explode. This is what makes the condition $\theta >0$ very crucial and why it cannot be dropped. Third, the necessary condition for the existence of bifBm on ${\mathbb{R}_{+}}$ (see [17]) to exist is trivial in our case $K<0$.

A process indexed by a sphere $\mathbb{S}$ is said to be spherical θ-bifBm with parameters $(\theta ,H,K,)$ if it is centered Gaussian process with covariance function given by (7).

First, note that the necessary condition for the bifractional Brownian motion on ${\mathbb{R}_{+}}$ to exist is that $K\le 2$ and $HK\le 1$ (see [17]). This condition is trivial if $K<0$ and $H\ge 0$. Observe also that for all $\beta ,\lambda >0$ The case $\theta =0$, $K\in (0,1]$ is already treated in Theorem 1, and incorporating the parameter $\theta >0$ does not affect the proof. Let $K<0<\theta $, $H\in (0,1/2]$ and set $\beta =-K>0$. By using (8) we have, for all $M,N\in \mathbb{S}$, Now let ${M_{1}},\dots ,{M_{n}}\in \mathbb{S}$ and ${\lambda _{1}},\dots ,{\lambda _{n}}\in \mathbb{R}$. We have where $\xi (u)={\textstyle\sum _{j=1}^{n}}{\lambda _{j}}\left(1-{e^{-i\sqrt{2u}{B_{H}}({M_{j}})}}\right)$ with $i=\sqrt{-1}$. For the equality (9), see the proof of Theorem 1.  □

Let $\theta >0$. If $\gamma (M,N)$, $M,N\in \mathbb{D}$, is a scalar variogram, and ${\nu _{j}}(M),\dots ,{\nu _{p}}(M)$ are negative functions with ${\nu _{j}}(M)<0$, $j=1,\dots ,p$, for all $M\in \mathbb{D}$, then the symmetric functions define a covariance matrix function.

Let $\theta >0$. If $\gamma (M,N)$, $M,N\in \mathbb{D}$, is a scalar variogram, and ${\nu _{j}}(M),\dots ,{\nu _{p}}(M)$ are negative functions with ${\nu _{j}}(M)<0$, $j=1,\dots ,p$, for all $M\in \mathbb{D}$, then the symmetric functions define a covariance matrix function.

Let $\theta >0$. If ${\psi _{1}}(M)$, …, ${\psi _{p}}(M)$, $M\in \mathbb{D}$, are nonnegative functions, and ${\nu _{j}}(M)$, …, ${\nu _{p}}(M)$ are negative functions with ${\nu _{j}}(M)<0$, $j=1,\dots ,p$, for all $M\in \mathbb{D}$, then the symmetric functions define a covariance matrix function.

If $0<H\le 1/2$ and $K\in (0,1)$, then as $x\to \infty $ we have where ${\mathcal{C}_{n,H,K}}={\pi ^{(2HK-1)n}}{\mathcal{H}_{2HK}^{n}}{\textstyle\int _{{\mathbb{R}^{n}}}}{e^{-{2^{1/(2H)}}HK\left\| u\right\| }}du$ and ${\mathcal{H}_{2HK}^{n}}$ is the Pickands constant (see [1]).

Let $M,N\in {\mathbb{S}^{n}}$ be two points with spherical coordinates $\widetilde{M}\hspace{-0.1667em}=\hspace{-0.1667em}({\theta _{1}},\dots ,{\theta _{n}})$ and $\widetilde{N}=({\varphi _{1}},\dots ,{\varphi _{n}})$, respectively. Let N be fixed. Then as $d(M,N)\to 0$,

We consider the process ${B_{H,K}}$ as Gaussian random field defined on the compact set $\overline{\Theta }\subset {\mathbb{R}^{n}}$. The variance function ${\sigma ^{2}}(\widetilde{M})$ attains its maximum at interior point $\widetilde{{M^{\ast }}}=(\pi /2,\dots ,\pi /2,\pi )$. The proof is based on Lemma 3.3 in [1]. Thus, we shall verify the conditions where C is some nonnegative constant. Since $\sigma (\widetilde{M})=\sigma (M)={\arccos ^{HK}}({x_{n}})$, the condition (13) can be established in a similar fashion as it is done in [1, Lemma 3.2] (just replace β by $HK$). Let $\widetilde{M},\widetilde{N}\in \overline{\Theta }$ and set ${\eta _{K}}=K/(\Gamma (1-K))$. We have where Let $\varepsilon >0$. Using the fact that $\left|{e^{-y}}-{e^{-x}}\right|\le \left|y-x\right|$, for all $x,y\in {\mathbb{R}_{+}}$, we obtain In the last inequality we used the fact that $\left|d(O,M)-d(O,N)\right|\le d(M,N)$, for all M, N. It follows that ${\mathcal{A}_{H,K}}(M,N)=o\left({d^{2HK}}(M,N)\right)$ (because $2HK\le K<1$), and (16) becomes Let $\widetilde{M}=({\theta _{1}},\dots ,{\theta _{n}})$ and $\widetilde{N}=({\varphi _{1}},\dots ,{\varphi _{n}})$. First, observe that $\widetilde{M},\widetilde{N}\to \widetilde{{M^{\ast }}}$ implies $d(M,N)\to 0$. By Lemma 2 we have $d(M,N)\sim \left\| \widetilde{M}-\widetilde{N}\right\| $ as $\widetilde{M},\widetilde{N}\to \widetilde{{M^{\ast }}}$. Thus substituting $d(M,N)$ by $\left\| \widetilde{M}-\widetilde{N}\right\| $ in (17) completes the proof of (14). For the condition (15), we use the concavity of the function $x\mapsto {x^{K}}$ to get and To complete the proof we must show that there is some $\mathcal{C}>0$ such that Set $M=({x_{1}},\dots ,{x_{n+1}})$ and $N=({y_{1}},\dots ,{y_{n+1}})$. Considering the Cartesian coordinates ${x_{j}}$ as functions of (spherical coordinates) ${\theta _{k}}$, $k\le j\wedge n$, and using the mean value theorem one has $\left|{x_{j}}-{y_{j}}\right|\le {c_{j}}\left\| \widetilde{M}-\widetilde{N}\right\| $, for each j, where ${c_{j}}$ are some nonnegative constants depending only on n. We conclude that there exists ${C_{1}}>0$ such that Set $X=d(M,N)/2$ and observe that $X\le \pi /2$. By using the identity (e.g., [18]) and the fact that $0\le x\cos (x)\le \sin (x)$, for all $x\in [0,\pi /2]$, we obtain Hence, $d(M,N)\le \sqrt{1+{\pi ^{2}}/4}\left\| M-N\right\| $. Combining this result with (19) yields $d(M,N)\le \mathcal{C}\left\| \widetilde{M}-\widetilde{N}\right\| $, where $\mathcal{C}={C_{1}}\sqrt{1+{\pi ^{2}}/4}$. Since we have we apply [1, Lemma 3.3] to the process $X={B_{H,K}}/{\pi ^{HK}}$ with $\eta =1$, $\alpha =2HK$, $A=HK{\pi ^{-1}}I$ and $C={2^{-1/(2H)}}{\pi ^{-1}}I$, where I is the identity matrix, and get the desired result.  □

Consider the spherical θ-bifBm ${B_{H,K}^{\theta }}$ with $K<0<\theta $ and $H\in (0,1/2)$. Then as $x\to \infty $ we have where ${\mathcal{B}_{n,H,K}^{\theta }}={\mathcal{H}_{2H}^{n}}{\sigma _{\theta }}{({M^{\ast }})^{\frac{(2H-1)n}{H}}}{\textstyle\int _{{\mathbb{R}^{n}}}}{e^{-{\mathcal{D}_{H,K}^{\theta }}\left\| u\right\| }}du$, $\Psi (x)$ is given in (11) and ${\mathcal{H}_{2HK}^{n}}$ is the Pickands constant (see [1]).

Consider the spherical θ-bifBm ${B_{H,K}^{\theta }}$ with $K<0<\theta $ and $H\in (0,1/2)$. Then there exist ${C^{(l)}}>0$, $l=1,2,3$, such that Here $\widetilde{M}$ denotes the spherical coordinate of the point $M\in {\mathbb{S}^{n}}$ and ${_{\theta }}\left(\cdot ,\cdot \right)$ stands for the correlation function of the process ${B_{H,K}^{\theta }}$.