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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
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##### References:
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