Sums of independent Poisson subordinators and their connection with strictly \(\alpha \)-stable processes of Ornstein-Uhlenbeck type.

*(English. Russian original)*Zbl 1189.60163
J. Math. Sci., New York 159, No. 3, 350-357 (2009); translation from Zap. Nauchn. Semin. POMI 361, 123-137 (2008).

Let \(\{\xi\}=\{\xi_0, \xi_1, \dots\}\) be a sequence of i.i.d., random values. Let \(\Pi =\Pi(s, \lambda ), s\geq 0\) be a Poisson random process with constant intensity \(\lambda >0\). All random processes and variables are independent in this paper. Consider a random change of time in sequence \(\{\xi\} =\{\xi_i, i=0, 1,\dots\}\) determinated by a subordinator \(\Pi\), i.e., consider the process \(\Psi_\Pi(s)=\xi_{\Pi(s)}, s\geq 0\) of Poisson random index. It is the process with piecewise constant and continuous from the right sample paths, defined on \(\mathbb R_+\).

Process \(\Pi\) is called the leading (or guiding) process and the sequence \(\{\xi\}\) is forming for process \[ \Psi_\Pi(s)=\sum^{\infty}_{j=0}\xi_j{\mathbf 1} \{\Pi(s)=j\}. \] It is the Markov process. Its transition probability together with Fourier transformation are obtained. The process \(\Psi_\Pi(s)\) is not a process with independent increments.

The relation of the characteristic function of the random vector \(\eta=(\xi_{\Pi(r)}\); \(\xi_{\Pi(r+s)})\in R^2, r>0, s>0\) and that of random vector \(\xi_0\) is presented. Examples are considered when the i.i.d. values in the forming sequence obey normal, stable distribution or take values \(\pm 1\) with probability \(\frac{1}{2}\).

Sums of i.i.d. processes \(\xi_{{\Pi}_i}(s), 1\leq i\leq N\) are considered in section 2. Let \(\mathbb{E}\xi_0=0\) and \(D\xi_0=1\). Denote

\[ U_N(s)=\frac{1}{\sqrt{N}}\sum^N_{i=j}\xi_{\Pi_i(s)}. \] As \(N\to\infty\), these sums behave like Ornstein-Uhlenbeck processes with viscosity coefficient \(\lambda\). For \(N, s>0, r>0\)

\[ \text{Cov}.\big(U_N(r), U_N(r+s)\big)=\exp (-\lambda s). \]

In case the forming sequence consists of i.i.d. random values with stictly \(\alpha\)-stable distributions with \(\Phi_\xi(\tau)=\exp\{-|\tau|^\alpha\}, 0<\alpha\leq 2\),

\[ L_N(s)=\frac{1}{N^{1/\alpha}}\;\sum^N_{j=1}\,\xi_{\Pi_j(s)} \]

are stationary processes. The characteristic function of limit distribution of random vector \((L_N(s), L_N(t))\) is obtained as \(N\to\infty\). It is proved that in case \(\alpha =2\), this two-dimensional limit distribution coincide with that of Ornstein-Uhlenbeck process. For \(0<\alpha<2\), it is different.

Process \(\Pi\) is called the leading (or guiding) process and the sequence \(\{\xi\}\) is forming for process \[ \Psi_\Pi(s)=\sum^{\infty}_{j=0}\xi_j{\mathbf 1} \{\Pi(s)=j\}. \] It is the Markov process. Its transition probability together with Fourier transformation are obtained. The process \(\Psi_\Pi(s)\) is not a process with independent increments.

The relation of the characteristic function of the random vector \(\eta=(\xi_{\Pi(r)}\); \(\xi_{\Pi(r+s)})\in R^2, r>0, s>0\) and that of random vector \(\xi_0\) is presented. Examples are considered when the i.i.d. values in the forming sequence obey normal, stable distribution or take values \(\pm 1\) with probability \(\frac{1}{2}\).

Sums of i.i.d. processes \(\xi_{{\Pi}_i}(s), 1\leq i\leq N\) are considered in section 2. Let \(\mathbb{E}\xi_0=0\) and \(D\xi_0=1\). Denote

\[ U_N(s)=\frac{1}{\sqrt{N}}\sum^N_{i=j}\xi_{\Pi_i(s)}. \] As \(N\to\infty\), these sums behave like Ornstein-Uhlenbeck processes with viscosity coefficient \(\lambda\). For \(N, s>0, r>0\)

\[ \text{Cov}.\big(U_N(r), U_N(r+s)\big)=\exp (-\lambda s). \]

In case the forming sequence consists of i.i.d. random values with stictly \(\alpha\)-stable distributions with \(\Phi_\xi(\tau)=\exp\{-|\tau|^\alpha\}, 0<\alpha\leq 2\),

\[ L_N(s)=\frac{1}{N^{1/\alpha}}\;\sum^N_{j=1}\,\xi_{\Pi_j(s)} \]

are stationary processes. The characteristic function of limit distribution of random vector \((L_N(s), L_N(t))\) is obtained as \(N\to\infty\). It is proved that in case \(\alpha =2\), this two-dimensional limit distribution coincide with that of Ornstein-Uhlenbeck process. For \(0<\alpha<2\), it is different.

Reviewer: Nijole Kalinauskaitė (Vilnius)

##### Keywords:

processes of random index; sums of random processes; Markov processes; Ornstein-Uhlenbeck type \(\alpha \)-stable processes
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\textit{O. V. Rusakov}, J. Math. Sci., New York 159, No. 3, 350--357 (2009; Zbl 1189.60163); translation from Zap. Nauchn. Semin. POMI 361, 123--137 (2008)

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##### References:

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[3] | O. E. Barndorff-Nielsen and N. Shephard, ”Non-Gaussian Ornstein-Uhlenbeck – based models and some of their uses in financial economics,” J. Royal Statist. Soc., 63 (2), 167–241 (2001). · Zbl 0983.60028 · doi:10.1111/1467-9868.00282 |

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