Miscellanea

Abstract

Let $M_d$ be the Euclidean space of symmetric $d\times d$ matrices with the scalar product $\langle A_1,A_2\rangle :=\mathrm tr (A_1 A_2)$, $A_1, A_2\in M_d$, $M_d^+\subset M_d$ be the cone of nonnegative definite matrices and $\fancyscript{P}(M^+_d)$ be a class of probability measures on $M^+_d$. Here $\mathrm tr A$ denotes the trace of a matrix $A$.

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7.1 Mixed Moments of Student’s $t$-Distributions

Let $M_d$ be the Euclidean space of symmetric $d\times d$ matrices with the scalar product $\langle A_1,A_2\rangle :=\mathrm tr (A_1 A_2)$, $A_1, A_2\in M_d$, $M_d^+\subset M_d$ be the cone of nonnegative definite matrices and $\fancyscript{P}(M^+_d)$ be a class of probability measures on $M^+_d$. Here $\mathrm tr A$ denotes the trace of a matrix $A$.

The probability distribution of a d-dimensional random vector $X$ is said to be the mixture of centered Gaussian distributions with the mixing distribution $U\in \fancyscript{P}(M^+_d)$ ($U$-mixture for short) if, for all $z\in R^d$,

$$\begin{aligned} \mathrm E e^{i\langle z,X\rangle }=\int \limits _{M^+_d}e^{-\frac{1}{2}\langle zA,z\rangle }U(\text{d}A). \end{aligned}$$

(7.1)

The distributional properties of such mixtures are well studied (see, e.g., [1, 2] and references therein).

Let $c_j=(c_{j_1},\ldots ,c_{j_d})\in R^d$, $j=1,2,\ldots ,2n$. We shall derive formulas evaluating $\mathrm E \left(\prod\nolimits ^{2n}_{j=1}\langle c_j,X\rangle \right)$ for $U$-mixtures of Gaussian distributions, including Student’s $t$-distribution.

Let $\Pi _{2n}$ be the class of pairings $\sigma$ on the set $I_{2n}=\left\{ 1,2,\ldots ,2n\right\}$, i.e. the partitions of $I_{2n}$ into $n$ disjoint pairs, implying that

$$\begin{aligned} \mathrm card \Pi _{2n}=\frac{(2n)!}{2^nn!}. \end{aligned}$$

For each $\sigma \in \Pi _{2n}$, we define uniquely the subsets $I_{2n\backslash \sigma }$ and integers $\sigma (j)$, $j\in I_{2n\backslash \sigma }$, by the equality

$$\begin{aligned} \sigma =\left\{ \left(j,\sigma (j)\right),j\in I_{2n\backslash \sigma }\right\} . \end{aligned}$$

If $U=\varepsilon _\Sigma$ is a Dirac measure with fixed $\Sigma \in M^+_d$, i.e. the Gaussian case, Isserlis theorem (in mathematical physics known as Wick theorem) says (see, e.g., [3–5]) that

$$\begin{aligned} \mathrm E \left[\prod ^{2n}_{j=1}\langle c_j,X\rangle \right]=\sum _{\sigma \in \Pi _{2n}}\prod _{j\in I_{2n\backslash \sigma }}\langle c_j \Sigma , c_{\sigma (j)}\rangle :=m_{2n}(c,\Sigma ). \end{aligned}$$

(7.2)

Write

$$\begin{aligned} \phi _U(\Theta ):=\int \limits _{M^+_d}e^{-\mathrm tr (A\Theta )}U(\text{d}A), \quad \Theta \in M^+_d. \end{aligned}$$

(7.3)

Theorem 7.1

[6] The following statements hold:

• (i) The probability distribution of a d-dimensional random vector$X$is the$U$-mixture of centered Gaussian distributions if and only if

  $$\begin{aligned} \mathrm E e^{i\langle z, X\rangle }=\phi _U\left(\frac{1}{2}z^Tz\right), \end{aligned}$$

  (7.4)

  where$z^{T}$is the transposed vector$z$.

• (ii) If the probability distribution of$X$is the$U$-mixture of centered Gaussian distributions and, for$j=1,2, \ldots ,2n$,

  $$\begin{aligned} \int \limits _{M^+_d}\langle c_jA,c_j\rangle ^nU(\text{d}A)<\infty , \end{aligned}$$

  (7.5)

  then

  $$\begin{aligned} \mathrm E \left[\prod ^{2n}_{j=1}\langle c_j,X\rangle \right]=\sum _{\sigma \in \Pi _{2n}}\int \limits _{M^+_d}m^{\sigma }_{2n}(c,A)U(\text{d}A), \end{aligned}$$

  (7.6)

  where

  $$\begin{aligned} m^{\sigma }_{2n}(c,A)=\prod _{j\in I_{2n\backslash \sigma }}\langle c_jA,c_{\sigma (j)}\rangle . \end{aligned}$$

Proof

• (i) The statement follows from (7.1) and (7.3), because, obviously,

  $$\begin{aligned} \mathrm tr \left((z^Tz)A\right)=\langle zA,z\rangle . \end{aligned}$$

• (ii) Observe that card $I_{2n\backslash \sigma }=n$ and, for all $\sigma \in \Pi _{2n}$ and $A\in M^+_d$,

  $$\begin{aligned} \prod _{j\in I_{2n\backslash \sigma }}\left|\langle c_j A,c_{\sigma (j)}\rangle \right|^n&\le n^{-n}\left(\sum _{j\in I_{2n\backslash \sigma }}\left|\langle c_jA,c_{\sigma (j)}\rangle \right|\right)^n \nonumber \\&\le n^{-1}\sum _{j\in I_{2n\backslash \sigma }}\left|\langle c_jA, c_{\sigma (j)}\rangle \right|^n \nonumber \\&\le \frac{2^{n-1}}{n}\sum _{j\in I_{2n\backslash \sigma }}\left[\langle c_jA,c_j\rangle ^n+\langle c_{\sigma (j)}A,c_{\sigma (j)}\rangle ^n\right] \nonumber \\&= \frac{2^{n-1}}{n}\sum ^{2n}_{j=1}\langle c_jA,c_j\rangle ^n. \end{aligned}$$

  (7.7)

  Using (7.5) and (7.7), we find that

  $$\begin{aligned} \mathrm E \left[\prod ^{2n}_{j=1}\langle c_j, X\rangle \right]&= \int \limits _{M^+_d}m_{2n}(c,A)U(\text{d}A)\\&= \sum _{\sigma \in \Pi _{2n}}\int \limits _{M^+_d}m^{\sigma }_{2n}(c,A)U(\text{d}A). \end{aligned}$$

$\square$

Taking (see also [7])

$$\begin{aligned} U=\fancyscript{L}(Y\Sigma ), \end{aligned}$$

where $\Sigma \in M^+_d$ is fixed and

$$\begin{aligned} \fancyscript{L}(Y)=GIG\left(-\frac{\nu }{2}, \nu , 0\right) \end{aligned}$$

we have that

$$\begin{aligned} \phi _U(\Theta )=\frac{2\left(\frac{\nu }{2}\right)^{\frac{\nu }{4}}\left(\mathrm tr (\Sigma \Theta )\right)^{\frac{\nu }{4}}}{\Gamma \left(\frac{\nu }{2}\right)}K_{\frac{\nu }{2}}\left(\sqrt{2\mathrm tr (\Sigma \Theta )}\right), \end{aligned}$$

(7.8)

$$\begin{aligned} \fancyscript{L}(X)=T_d(\nu ,\Sigma ,0) \end{aligned}$$

(7.9)

and, for $j=1,2,\ldots ,2n$

$$\begin{aligned} \int \limits _{M^+_d}\langle c_jA,c_j\rangle ^nU(\text{d}A)=\left\{ \begin{array}{l} \dfrac{\Gamma \left(\frac{\nu }{2}-n\right)}{\left(\dfrac{\nu }{2}\right)^{\dfrac{\nu }{2}-n}}\langle c_j\Sigma ,c_j\rangle ^n, \quad \text {if}\, 2n<\nu , \\ \infty , \quad \text {if}\, 2n\ge \nu . \end{array} \right. \end{aligned}$$

Thus, for $2n<\nu$,

$$\begin{aligned} \int \limits _{R^d}\prod ^{2n}_{j=1}\langle c_j,x\rangle T_d\left(\nu ,\Sigma ,0\right)(\text{d}x)=\frac{\Gamma \left(\frac{\nu }{2}-n\right)}{\left(\frac{\nu }{2}\right)^{\frac{\nu }{2}-n}}m_{2n}(c,\Sigma ), \end{aligned}$$

(7.10)

$$\begin{aligned} \int \limits _{R^d}\prod ^{2n}_{j=1}\langle c_j,x\rangle T_d(\nu ,\Sigma ,\alpha )(\text{d}x)=\int \limits _{R^d}\prod ^{2n}_{j=1}\left[\langle c_j,y\rangle +\langle c_j,\alpha \rangle \right]T_d(\nu ,\Sigma ,0)(\text{d}y) \end{aligned}$$

and because of anti-symmetry, for $2k+1<\nu$,

$$\begin{aligned} \int \limits _{R^d}\prod ^{2k+1}_{j=1}\langle c_j,x\rangle T_d(\nu ,\Sigma ,0)(\text{d}x)=0. \end{aligned}$$

Remark 7.2

Let $\nu \ge d$ be an integer, $Y_1,\ldots ,Y_{\nu }$ be i.i.d. d-dimensional centered Gaussian vectors with a covariance matrix $\Sigma$, $|\Sigma |>0$, and $U=\fancyscript{L}\left(\nu \Sigma ^{-1}_{\nu }\right)$, where the matrix

$$\begin{aligned} W_{\nu }=\sum ^{\nu }_{j=1}Y^T_jY_j. \end{aligned}$$

If $\nu \ge d$, the matrix $W_{\nu }$ is invertible with probability 1, because it is well known that the Wishart distribution

$$\begin{aligned} \fancyscript{L}(W_{\nu }):=W_d(\Sigma ,\nu ) \end{aligned}$$

has a density

$$\begin{aligned} W_d(\Sigma ,\nu ,A)=\left\{ \begin{array}{l} \dfrac{|A|^{\dfrac{\nu -d-1}{2}}\exp \left\{ -\dfrac{1}{2}\mathrm tr \left(\Sigma ^{-1}A\right)\right\} }{\left(2^d|\Sigma |\right)^{\dfrac{\nu }{2}} \pi ^{\dfrac{d(d-1)}{4}}\prod \limits ^{d}_{j=1}\Gamma \left(\dfrac{k-j+1}{2}\right)}, \quad \text {if}\, |A|>0, \\ 0, \quad \text {otherwise} . \end{array} \right. \end{aligned}$$

Because (see, e.g., [2, 8, 9])

$$\begin{aligned} \int \limits _{M^+_d}e^{-\frac{1}{2}\langle zA,z\rangle }U(\text{d}A)&=\int \limits _{R^d}e^{i\langle z,x\rangle }T_d(\nu ,\Sigma ,0)(\text{d}x) \nonumber \\&=\mathrm E [e^{-\frac{1}{2}\langle z\Sigma ,z\rangle Y}], \quad z\in R^d, \end{aligned}$$

(7.11)

taking $z=tc$, $t\in R^1$, $c\in R^d$, we find that

$$\begin{aligned} \int \limits _{M^+_d}e^{-\frac{t^2}{2}\langle cA,c\rangle }U(\text{d}A)=\mathrm E \left[e^{-\frac{t^2}{2}\langle c\Sigma ,c\rangle Y}\right]. \end{aligned}$$

Thus, for all $c\in R^d$,

$$\begin{aligned} \fancyscript{L}\left(\nu \langle cW^{-1},c\rangle \right)=\fancyscript{L}\left(\langle c\Sigma ,c\rangle Y\right), \end{aligned}$$

contradicting to the formula

$$\begin{aligned} \fancyscript{L}\left(\langle cW^{-1}_{\nu },c\rangle \right)=\fancyscript{L}\left(\langle c\Sigma ^{-1}, c\rangle \frac{1}{\chi ^2_{\nu -d+1}}\right) \end{aligned}$$

in [9].

Unfortunately, the last formula was used in [6, Example 3.

From (7.11) we easily find that

$$\begin{aligned} \int \limits _{R^d}e^{i\langle z,x\rangle }T_d(\nu ,\Sigma ,\alpha )(\text{d}x)&= \frac{e^{i\langle z,\alpha \rangle }}{2^{\frac{\nu }{2}-1}\Gamma \left(\frac{\nu }{2}\right)}\left(\nu \langle z\Sigma ,z\rangle \right)^{\frac{\nu }{4}}\\\quad&\times K_{\frac{\nu }{2}}\left(\sqrt{\nu \langle z\Sigma ,z\rangle }\right), \quad z\in R^d, \end{aligned}$$

(see [10, 11]).

7.2 Long-Range Dependent Stationary Student Processes

It is well known (see, e.g., [12]) that a real square integrable and continuous in quadratic mean stochastic process $X=\left\{ X_t,t\in R^1\right\}$ is second order stationary if and only if it has the following spectral decomposition:

$$\begin{aligned} X_t=\alpha +\int \limits ^{\infty }_{-\infty }\cos {(\lambda t)}v(\text{d}\lambda )+\int \limits ^{\infty }_{-\infty }\sin {(\lambda t)}w(\text{d}\lambda ), \quad t\in R^1, \end{aligned}$$

where $\alpha =\mathrm E X_0$, $v(\text{d}\lambda )$ and $w(\text{d}\lambda )$ are mean 0 and square integrable real random measures such that, for each $A, A_1, A_2\in \fancyscript{B}(R^1)$,

$$\begin{aligned} \mathrm E \left[v(A_1)v(A_2)\right]=\mathrm E v^2(A_1\cap A_2), \end{aligned}$$

(7.12)

$$\begin{aligned} \mathrm E \left[w(A_1)w(A_2)\right]=\mathrm E w^2(A_1\cap A_2), \end{aligned}$$

(7.13)

$$\begin{aligned} \mathrm E \left[v(A_1)w(A_2)\right]=0, \end{aligned}$$

(7.14)

$$\begin{aligned} \tilde{F}(A):=\mathrm E v^2(A)=\mathrm E w^2(A). \end{aligned}$$

(7.15)

The correlation function $r$ satisfies

$$\begin{aligned} r(t)=\int \limits ^{\infty }_{-\infty }\cos {(\lambda t)}F(\text{d}\lambda ), \end{aligned}$$

where

$$\begin{aligned} F(A)=\frac{\tilde{F}(A)}{\tilde{F}(R^1)}, \quad A\in \fancyscript{B}(R^1). \end{aligned}$$

Following [13], we shall construct a class of strictly stationary stochastic processes $X=\left\{ X_t, t\in R^1\right\}$ such that

$$\begin{aligned} \fancyscript{L}(X_t)\equiv T_1\left(\nu ,\sigma ^2,\alpha \right), \quad \nu >2, \end{aligned}$$

called the Student’s stationary processes.

Recall the notion and some properties of the independently scattered random measures (i.s.r.m.) (see [13–15]).

Let $T\in \fancyscript{B}(R^d)$, $\fancyscript{S}$ be a $\sigma$-ring of subsets of T (i.e. countable unions of sets in $\fancyscript{S}$ belong to $\fancyscript{S}$ and, if $A,B\in \fancyscript{S}$, $A\subset B$, then $B\backslash A\in \fancyscript{S}$). The $\sigma$ algebra generated by $\fancyscript{S}$ is denoted $\sigma (\fancyscript{S})$.

A collection of random variables $v=\left\{ v(A),A\in \fancyscript{S}\right\}$ defined on a probability space $\left(\Omega ,\fancyscript{F},\mathrm P \right)$ is said to be an i.s.r.m. if, for every sequence $\left\{ A_n,n\ge 1\right\}$ of disjoint sets in $\fancyscript{S}$, the random variables $v(A_n)$, $n=1,2,\ldots$, are independent and

$$\begin{aligned} v\left(\bigcup ^{\infty }_{n=1}A_n\right)=\sum ^{\infty }_{n=1}v(A_n) \quad a.s., \end{aligned}$$

whenever $\bigcup\nolimits ^{\infty }_{n=1}\,A_n\in \fancyscript{S}$.

Let $v(A)$, $A\in \fancyscript{S}$, be infinitely divisible,

$$\begin{aligned} \log \mathrm{E e^{izv(A)}}=izm_0(A)-\frac{1}{2}z^2m_1(A)+\int \limits _{R^+_0}\left(e^{izu}-1-iz\tau (u)\right)\Pi (A,\text{d}u), \end{aligned}$$

where $m_0$ is a signed measure, $\Pi (A,\text{d}u)$ for fixed $A$ is a measure on $\fancyscript{B}(R^1_0)$ such that

$$\begin{aligned} \int \limits _{R^1_0}\left(1\wedge u^2\right)\Pi (A,\text{d}u)<\infty ; \end{aligned}$$

$$\begin{aligned} \tau (u)=\left\{ \begin{array}{l} u, \quad \text {if}\, |u|\le 1, \\ \dfrac{u}{|u|}, \quad \text {if}\, |u|>1. \end{array} \right. \end{aligned}$$

Assume now that $m_0=m_1=0$ and

$$\begin{aligned} \Pi (A,\text{d}u)=M(A)\Pi (\text{d}u), \end{aligned}$$

where $M(A)$ is some measure on $T$ and $\Pi (\text{d}u)$ is some Lévy measure on $R^1_0$.

Integration of functions on $T$ with respect to $v$ is defined first for real simple functions $f=\sum\nolimits ^{n}_{j=1}x_j1_{A_j}$, $A_j\in \fancyscript{S}$, $j=1,\ldots ,n$, by

$$\begin{aligned} \int \limits _{A}f(x)v(\text{d}x)=\sum ^{n}_{j=1}x_jv(A\cap A_j), \end{aligned}$$

where A is any subset of $T$, for which $A\in \sigma (\fancyscript{S})$ and $A\cap A_j\in \fancyscript{S}$, $j=1,\ldots ,n$.

In general, a function $f{:}\left(T,\sigma (\fancyscript{S})\right)\rightarrow \left(R^1,\fancyscript{B}(R^1)\right)$ is said to be $v$-integrable if there exists a sequence $\left\{ f_n, n=1,2,\ldots \right\}$ of simple functions as above such that $f_n\rightarrow f$ $M$-a.e. and, for every $A\in \sigma (\fancyscript{S})$, the sequence $\left\{ \int _{A}f_n(x)v(\text{d}x),\right.$ $\left.n=1,2,\ldots \right\}$ converges in probability, as $n\rightarrow \infty$. If $f$ is $v$-integrable, we write

$$\begin{aligned} \int \limits _{A}f(x)v(\text{d}x)=p-\lim _{n\rightarrow \infty }\int \limits _{A}f_n(x)v(\text{d}x). \end{aligned}$$

The integrand $\int _{A}f(x)v(\text{d}x)$ does not depend on the approximating sequence.

A function $f$ on $T$ is $v$-integrable if and only if

$$\begin{aligned} \int \limits _{T}Z_0\left(f(x)\right)M(\text{d}x)<\infty \end{aligned}$$

and

$$\begin{aligned} \int \limits _{T}\left|Z\left(f(x)\right)\right|M(\text{d}x)<\infty , \end{aligned}$$

where

$$\begin{aligned} Z_0(y)=\int \limits _{R^1_0}\left(1\wedge (uy)^2\right)\Pi (\text{d}u), \end{aligned}$$

and

$$\begin{aligned} Z(y)=\int \limits _{R^1_0}\left(\tau (uy)-y\tau (u)\right)\Pi (\text{d}u). \end{aligned}$$

For such functions $f$

$$\begin{aligned} \log \mathrm E \exp \left\{ i\xi \int \limits _{A}f(x)v(\text{d}x)\right\} =\int \limits _{A}\varkappa \left(\xi f(x)\right)M(\text{d}x), \end{aligned}$$

where

$$\begin{aligned} \varkappa (\xi )=\int \limits _{R^1_0}\left(e^{i\xi u}-1-i\xi \tau (u)\right)\Pi (\text{d}u). \end{aligned}$$

Let now $Y_t=\left(Y^1_t,Y^2_t\right)$, $t\ge 0$, be a bivariate Student-Lévy process such that

$$\begin{aligned} \fancyscript{L}(Y_1)=T_2(\nu ,\sigma ^2I_2,0), \quad I_2=\left(\begin{array}{cc} 1&0 \\ 0&1 \end{array}\right), \end{aligned}$$

and $F$ be an arbitrary probability distribution on $R^1$.

Let $T=R^1$, $\fancyscript{S}$ be the $\sigma$-ring of subsets $A=\bigcup\nolimits ^{\infty }_{j=1}\left(a_j,b_j\right]$, where the intervals $\left(a_j,b_j\right]$, $j=1,2,\ldots$, are disjoint. Define i.m.r.m. $v$ and $w$ by the equalities:

$$\begin{aligned} v(A)=\sum ^{\infty }_{j=1}\left(Y^1_{F(b_j)}-Y^1_{F(a_j)}\right) \end{aligned}$$

and

$$\begin{aligned} w(A)=\sum ^{\infty }_{j=1}\left(Y^2_{F(b_j)}-Y^2_{F(a_j)}\right), \quad A=\bigcup ^{\infty }_{j=1}\left(a_j,b_j\right]\in \fancyscript{S}. \end{aligned}$$

Because, for $i=1,2,$ $j=1,2,\ldots$, $\nu >2$,

$$\begin{aligned} \mathrm E (Y^{i}_{F(b_j)}-Y^i_{F(a_j)})=0, \end{aligned}$$

$$\begin{aligned} \mathrm E (Y^{i}_{F(b_j)}-Y^{i}_{F(a_j)})^2=\frac{\sigma ^2\nu}{\nu -2}\left(F(b_j)-F(a_j)\right) \end{aligned}$$

and

$$\begin{aligned} \sum ^{\infty }_{j=1}\mathrm E (Y^i_{F(b_j)}-Y^i_{F(a_j)})^2\le \frac{\sigma ^2\nu}{\nu -2}<\infty , \end{aligned}$$

the definition of $v$ and $w$ is correct.

From (7.10) it follows that $v$ and $w$ satisfies (7.12)–(7.15) with

$$\begin{aligned} \tilde{F}(A)=\frac{\sigma ^2\nu}{\nu -2}F(A), \quad A\in \fancyscript{S}. \end{aligned}$$

Thus, the process

$$\begin{aligned} X_t=\alpha +\int \limits ^{\infty }_{-\infty }\cos (ut)v(\text{d}u)+\int \limits ^{\infty }_{-\infty }\sin {(ut)}w(\text{d}u), \quad t\in R^1, \end{aligned}$$

is well defined, strictly stationary,

$$\begin{aligned} \fancyscript{L}(X_t)\equiv T_1(\nu ,\sigma ^2,\alpha ) \end{aligned}$$

and the correlation function $r$ satisfies

$$\begin{aligned} r(t)=\int \limits ^{\infty }_{-\infty }\cos {(ut)}F(\text{d}u), \quad t\in R^1. \end{aligned}$$

Strict stationarity of $X$ follows from the formula (see [13]):

$$\begin{aligned} \mathrm E e^{i \sum \limits ^{n}_{j=1}\eta _jX_{t_j}}&= e^{i\alpha \sum \limits ^{n}_{j=1}\eta _j}\\&\quad\times \exp \left\{ \int \limits ^{\infty }_{-\infty }\log {\hat{h}_{\nu ,\sigma }}\left(\frac{1}{2}\sum ^{n}_{j,k=1}\eta _j\eta _k\cos \left(u(t_j-t_k)\right)\right)F(\text{d}u)\right\} , \\&\quad\eta _j, t_j\in R^1, \quad j=1,\ldots ,n, \end{aligned}$$

where

$$\begin{aligned} \hat{h}_{\nu ,\sigma }(\theta )&:= \int \limits ^{\infty }_{0}e^{-\theta u}\frac{1}{\sigma ^2}gig\left(\frac{u}{\sigma ^2};-\frac{\nu }{2},\nu ,0\right)\text{d}u\\&= \frac{2}{\Gamma\left(\frac{\nu}{2}\right)}\left(\frac{\theta \sigma ^2 \nu }{2}\right)^{\frac{\nu }{4}}K_{\frac{\nu }{2}}\left(\sqrt{2\sigma ^2 \theta \nu }\right), \quad \theta >0. \end{aligned}$$

As it was checked in [16], if

$$\begin{aligned} F(\text{d}u)=f_{\beta ,\gamma }(u)\text{d}u, \quad 0<\beta \le 1, \quad \gamma \in R^1, \end{aligned}$$

where

$$\begin{aligned} f_{\beta ,\gamma }(u)=\frac{1}{2}\left[f_{\beta ,0}(u+\gamma )+f_{\beta ,0}(u-\gamma )\right], \quad u\in R^1, \end{aligned}$$

with

$$\begin{aligned} f_{\beta ,0}(u)=\frac{2^{\frac{1-\beta }{2}}}{\sqrt{\pi }\Gamma \left(\frac{\beta }{2}\right)}K_{1-\beta }\left(|u|\right)|u|^{\frac{(1-\beta )}{2}}, \end{aligned}$$

then

$$\begin{aligned} r(t)=\frac{\cos {\gamma t}}{(1+t^2)^{\frac{\beta }{2}}}, \quad t\in R^1, \end{aligned}$$

and

$$\begin{aligned} \int \limits ^{\infty }_{-\infty }\left|r(t)\right|\text{d}t=\infty , \end{aligned}$$

implying long-range dependence of $X$ (see also [17–20]).

Remark 7.3

Defining Student-Lamperti process $X^{\star }$ as (see [21])

$$\begin{aligned} X^{\star }_t=t^HX_{\log t}, \quad t>0, \quad X^{\star }_0=0, \quad H>0. \end{aligned}$$

we have that $X^{\star }$ is $H$-self-similar, i.e., for each $c>0$, processes $\left\{ X^{\star }_{ct}, t\ge 0\right\}$ and $\left\{ c^{H}X^{\star }_t,t\ge 0\right\}$ have the same finite dimensional distributions, and (see [13])

$$\begin{aligned} \mathrm E e^{i\sum \limits ^{n}_{j=1}\eta _jX^{\star }_{t_j}}&= e^{i\alpha \sum \limits ^{n}_{j=1}t^H_j\eta _j}\\&\quad\times \exp \!\left\{ \!\int \limits ^{\infty }_{-\infty }\!\left[\log {\hat{h}_{\nu ,\sigma }}\!\left(\!\frac{1}{2}\sum ^{n}_{j,k=1}\eta _j\eta _k t^H_jt^H_k\cos \!\left(\!u\log \frac{t_j}{t_k}\!\right)\!\right)\!\right]\!F(\text{d}u)\!\right\} \!,\\&\quad t_j>0, \quad \eta _j\in R^1, \quad j=1,\ldots ,n. \end{aligned}$$

In particular,

$$\begin{aligned} \mathrm E e^{i\eta X^{\star }_t}=e^{i\alpha t^{H}\eta }\hat{h}_{\nu ,\sigma }\left(t^{2H}\frac{\eta ^2}{2}\right), \quad t>0, \quad \eta \in R^1, \end{aligned}$$

and

$$\begin{aligned} \mathrm{E}e^{i\eta\left(X^{\star}_t-X^{\star}_s\right)}&=&e^{i\alpha\left(t^H-s^H\right)\eta}\exp\Bigg\{\int\limits^{\infty}_{-\infty}\Bigg[\!\log\hat{h}_{\nu,\sigma}\!\Bigg(\frac{1}{2}\eta^2\Bigg(s^{2H}+\,t^{2H}\Bigg.\Bigg.\\ \Bigg.&&\quad-2s^Ht^H\cos\!\left(u \log \frac{t}{s}\right)\Bigg)\!\Bigg]F(\text{d}u)\Bigg\},\quad s,t>0, \quad \eta\in R^1.\end{aligned}$$

7.3 Lévy Copulas

Considering the probability distributions $F$ on $R^d$ with the 1-dimensional Student’s $t$ marginals $F_{j,j}=1,\ldots ,d$, and having in mind their relationship with stochastic processes, we restricted ourselves to the cases when $F$ is a mixture of the d-dimensional Gaussian distributions .

Denoting

$$\begin{aligned} C(u_1,\ldots ,u_d):=F\left(F^{-1}_1(u_1),\ldots ,F^{-1}_d(u_d)\right), \quad u_j\in [0, 1], \quad j=1,\ldots ,d, \end{aligned}$$

it is obvious that this function is the probability distribution function on the d-cube [0,1]$^{d}$ with uniform one-dimensional marginals, called the d-copula (see, e.g., [22]). Trivially,

$$\begin{aligned} F(x_1,\ldots ,x_d)=C\left(F_1 (x_1),\ldots ,F_d(x_d)\right), \quad (x_1,\dots ,x_d)\in R^d. \end{aligned}$$

(7.16)

Formula (7.16) with the arbitrary d-copula defines uniquely the probability distributions on $R^d$ with the given Student’s 1-dimensional marginals. These statements are very special cases of well known Sklar’s theorem (see [23, 24]).

Thus, taking concrete d-copulas we shall obtain a wide class of multivariate generalizations of Student’s $t$-distributions.

For instance, the Archimedean copulas have the from

$$\begin{aligned} C(u_1,\ldots ,u_d)=\psi \left(\psi ^{-1}(u_1)+\cdot +\psi ^{-1}(u_d)\right), \quad u_j\in [0,1], \quad j=1,\ldots ,d, \end{aligned}$$

where $\psi$ is a d-monotone function on $[0,\infty )$, i.e., for each $x\ge 0$ and $k=0,1,\ldots ,d-2$,

$$\begin{aligned} (-1)^k\frac{\text{d}^k}{\text{d}x^k}\psi (x)\ge 0, \end{aligned}$$

$(-1)^{d-2}\psi ^{(d-2)}(x)$, $x\ge 0$, is nonincreasing and convex function.

In particular, if

$$\begin{aligned} \psi (x)=(1+x)^{-\frac{1}{\theta }}, \quad \theta \in (0,\infty ), \quad x\ge 0, \end{aligned}$$

we have the Clayton’s copula

$$\begin{aligned} C(u_1,\ldots ,u_d)=\left(\sum ^{d}_{j=1}u^{-\theta }_j-d+1\right)^{-\frac{1}{\theta }}, \quad u_j\in [0,1], \quad j=1,\ldots ,d. \end{aligned}$$

If $\phi (x)=\exp \left\{ -x^{\frac{1}{\theta }}\right\}$, $\theta \ge 1$, $x\ge 0$, we obtain the Gumbel copula

$$\begin{aligned} C(u_1,\ldots ,u_d)=\exp \left\{ -\left(\sum ^d_{j=1}\left(-\log u_j\right)^{\theta }\right)^{\frac{1}{\theta }}\right\} , \quad u_j\in [0, 1], \quad j=1,\ldots ,d. \end{aligned}$$

Unfortunately, it is difficult to describe if the copulation preserves such important for us properties of marginal distributions as infinite divisibility or self-decomposability.

A promising direction for future work is a notion of Lévy copulas and, analogously to the classical copulas, construction of new Lévy measures on $R^d$ using marginal ones (see [25–28]). Following [28], we briefly describe an analogue of Sklar’s theorem in this context.

Let $\bar{R}:=\left(-\infty ,\infty \right]$. For $a,b\in \bar{R}^d$ we write $a\le b$, if $a_k\le b_k$, $k=1,\ldots ,d$ and, in this case, denote

$$\begin{aligned} \left(a,b\right]:=\left(a_1,b_1\right]\times \ldots \times \left(a_d,b_d\right]. \end{aligned}$$

Let $F:S\rightarrow \bar{R}$ for some subset $S\subset \bar{R}^d$. For $a,b\in S$ with $a\le b$ and $\overline{(a,b]}\subset S$, the $F$-volume of $(a,b]$ is defined by

$$\begin{aligned} V_F\left((a,b]\right):=\sum _{u\in \{a_1,b_1\}\times \cdots \times \{a_d,b_d\}} (-1)^{N(u)}F(u), \end{aligned}$$

where $N(u):=\sharp \{k:u_k=a_k\}$.

A function $F:S\rightarrow \bar{R}$ is called d-increasing if $V_F\left((a,b]\right)\ge 0$ for all $a,b\in S$ with $a\le b$ and $\overline{(a,b]}\subset S$.

Definition 7.4

Let $F:\bar{R}^d\rightarrow \bar{R}$ be a d-increasing function such that $F(u_1,\ldots , u_d)=0$ if $u_i=0$ for at least one $i\in \{1,\ldots ,d\}$. For any non-empty index set $I\subset \{1,\ldots ,d\}$ the $I$-marginal of $F$ is the function $F_I:\bar{R}^{|I|}\rightarrow \bar{R}$, defined by

$$\begin{aligned} F^I\left((u)i)_{i\in I}\right):=\lim _{a\rightarrow \infty }\sum _{(u_i)_{i\in I^c}\in \{-a,\infty \}^{|I^c|}}F(u_1,\ldots ,u_d)\prod _{i\in I^c}\mathrm sgn u_i, \end{aligned}$$

where $I^c=\{1,\ldots ,d\}\backslash I$, $|I|:=\mathrm card I$, and

$$\begin{aligned} \mathrm sgn x=\left\{ \begin{array}{l} 1, \quad \text {if}\, x\ge 0, \\ -1, \quad \text {if}\, x<0. \end{array} \right. \end{aligned}$$

Definition 7.5

A function $F:\bar{R}^d\rightarrow \bar{R}$ is called a Lévy copula if

• 1. $F(u_1,\ldots ,u_d)\ne \infty$ for $(u_1,\ldots ,u_d)\ne (\infty ,\ldots ,\infty )$,

• 2. $F(u_1,\ldots ,u_d)=0$ if $u_i=0$ for at least one $i\in \{1,\ldots ,d\}$,

• 3. $F$ is d-increasing,

• 4. $F^{\{i\}}(u)=u$ for any $i\in \{1,\ldots ,d\}$, $u\in R^1$.

Write

$$\begin{aligned} \fancyscript{I}(x):=\left\{ \begin{array}{l} (x,\infty ), \quad \text {if}\, x\le 0, \\ (-\infty ,x], \quad \text {if}\, x>0. \end{array} \right. \end{aligned}$$

Definition 7.6

Let $X=(X^1,\ldots ,X^d)$ be an $R^d$-valued Lévy process with the Lévy measure $\Pi$. The tail integral of $X$ is the function $V:\left(R^1\backslash \{0\}\right)^d\rightarrow R^1$ defined by

$$\begin{aligned} V(x_1,\ldots ,x_d):=\prod ^d_{i=1}\mathrm sgn (x_i)\Pi \left(\fancyscript{I}(x_1)\times \cdots \times \fancyscript{I}(x_d)\right) \end{aligned}$$

and, for any non-empty $I\subset \{1,\ldots ,d\}$ the $I$-marginal tail integral $V^I$ of $X$ is the tail integral of the process $X^I:=(X^i)_{i\in I}$.

We denote one-dimensional margins by $V_i:=V^{\{i\}}$.

Observe, that marginal tail integrals {$V^I{:}\!I{\subset }\{1,\ldots ,d\}$ non-empty} are uniquely determined by $\Pi$. Conversely, $\Pi$ is uniquely determined by the set of its marginal tail integral.

Relationship between Lévy copulas and Lévy processes are described by the following analogue of Sklar’s theorem.

Theorem 7.7

[28]

• 1. Let$X=\left(X^1, \ldots ,X^d\right)$be an$R^d$-valued Lévy process. Then there exists a Lévy copula$F$such that the tail integrals of$X$satisfy

  $$\begin{aligned} V\left((x_i)_{i\in I}\right)=F^I\left(\left(V_i (x_i)\right)_{i\in I}\right), \end{aligned}$$

  (7.17)

  for any non-empty$I\subset \{1,\ldots ,d\}$and any$(x_i)_{i\in I}\in \left(R^1\backslash \{0\}\right)^{|I|}$. The Lévy copula$F$is unique on$\mathrm Ran V_1\times \cdots \times \mathrm Ran V_d$.

• 2. Let$F$be a d-dimensional Lévy copula and$V_i$, $i=1,\ldots , d$, be tail integrals of real-valued Lévy processes. Then there exists an$R^d$-valued Lévy process$X$whose components have tail integrals$V_1,\ldots ,V_d$and whose marginal tail integrals satisfy (7.17) for any non-empty$I\subset \{1,\ldots ,d\}$and any$(x_i)_{i\in I}\in \left(R^1\backslash \{0\}\right)^{|I|}$. The Lévy measure$\Pi$of$X$is uniquely determined by$F$and$V_i$, $i=1, \ldots , d$.

In the above formulation $\mathrm Ran V$ means the range of $V$. The reader is referred for proofs to [28].

An analogue of the Archimedean copulas is as follows (see [28]).

Let $\varphi :[-1,1]\rightarrow [-\infty ,\infty ]$ be a strictly increasing continuous function with $\varphi (1)=\infty$, $\varphi (0)=0$, and $\varphi (-1)=-\infty$, having derivatives of orders up to d on $(-1,0)$ and $(0,1)$, and, for any $k=1,\ldots ,d$, satisfying

$$\begin{aligned} \frac{\text{d}^{k}\varphi (u)}{\text{d}u^k}\ge 0, \quad u\in (0,1) \quad \text {and} \quad (-1)^k\frac{\text{d}^k\varphi (u)}{\text{d}u^k}\le 0, \quad u\in (-1,0). \end{aligned}$$

Let

$$\begin{aligned} \tilde{\varphi }(u):=2^{d-2}\left(\varphi (u)-\varphi (-u)\right), \quad u\in [-1,1]. \end{aligned}$$

Then

$$\begin{aligned} F(u_1,\ldots ,u_d):=\varphi \left(\prod ^{d}_{i=1}\tilde{\varphi }^{-1}(u_i)\right) \end{aligned}$$

defines a Lévy copula.

In particular, if

$$\begin{aligned} \varphi (x):=\eta \left(-\log |x|\right)^{-\frac{1}{\vartheta }}1_{\{x>0\}}-(1-\eta )\left(-\log |x|\right)^{-\frac{1}{\vartheta }}1_{\{x<0\}} \end{aligned}$$

with $\vartheta >0$ and $\eta \in (0,1)$, then

$$\begin{aligned} \tilde{\varphi }(x)=2^{d-2}\left(-\log |x|\right)^{-\frac{1}{\vartheta }}\mathrm sgn x, \quad x\in -1,1], \end{aligned}$$

and

$$\begin{aligned} F(u_1,\ldots ,u_d)=2^{2-d}\left(\sum ^d_{i=1}|u_i|^{-\vartheta }\right)^{-\frac{1}{\vartheta }}(\eta 1_{\{u_1\ldots u_d\ge 0\}}-(1-\eta )1_{\{u_1\ldots u_d<0\}}), \end{aligned}$$

resembling the ordinary Clayton copulas.