×

zbMATH — the first resource for mathematics

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.

MSC:
60J99 Markov processes
60G99 Stochastic processes
60G52 Stable stochastic processes
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] V. Yu. Korolev, V. E. Bening, and S. Ya. Shorgin, Mathematical Foundations of the Risk Theory [in Russian], Moscow (2007). · Zbl 1234.60004
[2] O. E. Barndorff-Nielsen, ”Superposition of Ornstein-Uhlenbeck type processes,” Teor. Veroyatn. Primen., 45 (2), 289–311 (2000). · Zbl 1003.60039
[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
[4] Z. J. Jurec and J. D. Mason, Operator–Limit Distributions in Probability Theory, Wiley (1993).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.