The spectral method and ergodic theorems for general Markov chains
We study the ergodic properties of Markov chains with an arbitrary state space and prove a geometric ergodic theorem. The method of the proof is new: it may be described as an operator method. Our main result is an ergodic theorem for Harris-Markov chains in the case when the return time to some fix...
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| Veröffentlicht in: | Izvestiya. Mathematics 2015-01, Vol.79 (2), p.311-345 |
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| container_title | Izvestiya. Mathematics |
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| creator | Nagaev, S. V. |
| description | We study the ergodic properties of Markov chains with an arbitrary state space and prove a geometric ergodic theorem. The method of the proof is new: it may be described as an operator method. Our main result is an ergodic theorem for Harris-Markov chains in the case when the return time to some fixed set has finite expectation. Our conditions for the transition function are more general than those used by Athreya-Ney and Nummelin. Unlike them, we impose restrictions not on the original transition function but on the transition function of an embedded Markov chain constructed from the return times to the fixed set mentioned above. The proof uses the spectral theory of linear operators on a Banach space. |
| doi_str_mv | 10.1070/IM2015v079n02ABEH002744 |
| format | Article |
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The proof uses the spectral theory of linear operators on a Banach space.</description><subject>embedded Markov chain</subject><subject>Ergodic processes</subject><subject>Functions (mathematics)</subject><subject>Linear operators</subject><subject>Markov chains</subject><subject>Mathematical analysis</subject><subject>resolvent</subject><subject>spectral method</subject><subject>Spectral methods</subject><subject>stationary distribution</subject><subject>Theorem proving</subject><subject>Theorems</subject><subject>uniform ergodicity</subject><issn>1064-5632</issn><issn>1468-4810</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><recordid>eNqFkD9PwzAUxC0EEqXwGfDAwBJ4_hM7Gduq0EqtWAoDi-XETpOSxMFOK_HtSVVGENO74Xene4fQLYEHAhIel2sKJD6ATFugk-l8AUAl52doRLhIIp4QOB80CB7FgtFLdBXCDgA4J2yEppvS4tDZvPe6xo3tS2ewbg22futMleO-tM7bJuDCeby1rT1ya-0_3AHnpa7acI0uCl0He_Nzx-j1ab6ZLaLVy_NyNllFOROsj0iSJamVIE3OCknjoVdKMzCp1gk1GTUkYTQWQ3kt48JQIpnlItOUCyEBCjZG96fczrvPvQ29aqqQ27rWrXX7oAZ_HEtOhRhQeUJz70LwtlCdrxrtvxQBdVxN_bHa4Lw7OSvXqZ3b-3Z4SS3f35RMFVWMENWZYxf2C_Zf-DcDMHra</recordid><startdate>20150101</startdate><enddate>20150101</enddate><creator>Nagaev, S. 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| fulltext | fulltext |
| identifier | ISSN: 1064-5632 |
| ispartof | Izvestiya. Mathematics, 2015-01, Vol.79 (2), p.311-345 |
| issn | 1064-5632 1468-4810 |
| language | eng |
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| source | Institute of Physics Journals; Alma/SFX Local Collection |
| subjects | embedded Markov chain Ergodic processes Functions (mathematics) Linear operators Markov chains Mathematical analysis resolvent spectral method Spectral methods stationary distribution Theorem proving Theorems uniform ergodicity |
| title | The spectral method and ergodic theorems for general Markov chains |
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