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Zapiski Nauchnykh Seminarov LOMI, 1976, Volume 55, Pages 128–164 (Mi znsl2846)  

On connection between random curves, changes of time and regenerative times of random processes

B. P. Harlamov
Abstract: A product space $\Phi\times D$ is considered where $\Phi$ is a set of all continuous non-decreasing functions $\varphi\colon[0,\infty)\to(0,\infty)$, $\varphi(0)=0$, $\varphi(t)\to+\infty$ ($t\to\infty$); $D$ is a set of all right-continuous functions $\xi\colon(0,\infty)\to X$, $X$ is some metric space. Two maps $\Phi\times D\to D$: are defined. The first is the projection $q(\varphi,\xi)=\xi$, and the second is change of time $u(\varphi,\xi)=\xi\circ\varphi$. The following equivalence relation in $D$ is defined:
$$ \zeta_1\sim\xi_2\Leftrightarrow\exists\varphi_1,\varphi_2\in\Phi: \xi_1\circ\varphi_1=\xi_2\circ\varphi_2. $$
Let $M$ is a set of all equivalence classes. Then $L$ is the map $D\to M$: $L\xi_1=L\xi_2\Leftrightarrow\xi_1\sim\xi_2$. $L\xi$ is called the curve corresponding to $\xi$. The following theorem is proved: two random processes with probability measures $P^1$ and $P^2$ on $D$ possess of identical random curves (i.e. $P^1\circ L^{-1}=P^2\circ L^{-1}$) if and only if two random changes of time exist (i.e. two probability measures $Q^1$ and $Q^2$ on $\Phi\times D$) for which $P^1=Q^1\circ q^{-1}$, $P^2=Q^2\circ q^{-1}$) which transform these two processes in a process with a measure $\widetilde{P}$ (i.e. $Q^1\circ u^{-1}=Q^2\circ u^{-1}=\widetilde{P}$). If $(P_x^1)_{x\in X}$ and $(P_x^2)_{x\in X}$ are two families of probability measures for which $P_x^1\circ L^{-1}=P_x^2\circ L^{-1}$ $\forall x\in X$ then for each $x\in X$ corresponding measures $Q^1_x$ and $Q^2_x$ may be found as follows. The set of regenerative times of the family $(\widetilde{P}_x)_{x\in X}$ contains all stopping times which are simultaneously regenerative times of the families $(P^1_x)_{x\in X}$ and $(P^2_x)_{x\in X}$ and have a special first passage time property.
English version:
Journal of Soviet Mathematics, 1981, Volume 16, Issue 2, Pages 1005–1027
DOI: https://doi.org/10.1007/BF01676144
Bibliographic databases:
UDC: 519.21
Language: Russian
Citation: B. P. Harlamov, “On connection between random curves, changes of time and regenerative times of random processes”, Problems of the theory of probability distributions. Part 3, Zap. Nauchn. Sem. LOMI, 55, "Nauka", Leningrad. Otdel., Leningrad, 1976, 128–164; J. Soviet Math., 16:2 (1981), 1005–1027
Citation in format AMSBIB
\Bibitem{Har76}
\by B.~P.~Harlamov
\paper On connection between random curves, changes of time and regenerative times of random processes
\inbook Problems of the theory of probability distributions. Part~3
\serial Zap. Nauchn. Sem. LOMI
\yr 1976
\vol 55
\pages 128--164
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl2846}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=433592}
\zmath{https://zbmath.org/?q=an:0345.60021|0462.60038}
\transl
\jour J. Soviet Math.
\yr 1981
\vol 16
\issue 2
\pages 1005--1027
\crossref{https://doi.org/10.1007/BF01676144}
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