On Vague Convergence of Stochastic Processes
Suppose $Y, Y_n$ are stochastic processes in $C\lbrack 0, 1 \rbrack$ and the finite-dimensional distributions of $Y_n$ converge vaguely to those of $Y$. Then a necessary and sufficient condition for the vague convergence of the distributions of $Y_n$ to that of $Y$ is an approximate equicontinuity o...
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| Veröffentlicht in: | The Annals of probability 1975-12, Vol.3 (6), p.1014-1022 |
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| container_title | The Annals of probability |
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| creator | Erickson, R. V. Fabian, Vaclav |
| description | Suppose $Y, Y_n$ are stochastic processes in $C\lbrack 0, 1 \rbrack$ and the finite-dimensional distributions of $Y_n$ converge vaguely to those of $Y$. Then a necessary and sufficient condition for the vague convergence of the distributions of $Y_n$ to that of $Y$ is an approximate equicontinuity of the sequence $\langle Y_n \rangle$. Dudley (1966) generalized this standard result. We generalize Dudley's result to the case when the values of $X_n$ are in an arbitrary metric space and extend the result also to the case of the Skorohod metric. In our situation vague compactness does not imply tightness and thus a different proof than Dudley's (1966) must be used. The proof we use is simple and of interest even when other proofs are available. |
| doi_str_mv | 10.1214/aop/1176996227 |
| format | Article |
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V. ; Fabian, Vaclav</creator><creatorcontrib>Erickson, R. V. ; Fabian, Vaclav</creatorcontrib><description>Suppose $Y, Y_n$ are stochastic processes in $C\lbrack 0, 1 \rbrack$ and the finite-dimensional distributions of $Y_n$ converge vaguely to those of $Y$. Then a necessary and sufficient condition for the vague convergence of the distributions of $Y_n$ to that of $Y$ is an approximate equicontinuity of the sequence $\langle Y_n \rangle$. Dudley (1966) generalized this standard result. We generalize Dudley's result to the case when the values of $X_n$ are in an arbitrary metric space and extend the result also to the case of the Skorohod metric. In our situation vague compactness does not imply tightness and thus a different proof than Dudley's (1966) must be used. 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V.</creatorcontrib><creatorcontrib>Fabian, Vaclav</creatorcontrib><title>On Vague Convergence of Stochastic Processes</title><title>The Annals of probability</title><description>Suppose $Y, Y_n$ are stochastic processes in $C\lbrack 0, 1 \rbrack$ and the finite-dimensional distributions of $Y_n$ converge vaguely to those of $Y$. Then a necessary and sufficient condition for the vague convergence of the distributions of $Y_n$ to that of $Y$ is an approximate equicontinuity of the sequence $\langle Y_n \rangle$. Dudley (1966) generalized this standard result. We generalize Dudley's result to the case when the values of $X_n$ are in an arbitrary metric space and extend the result also to the case of the Skorohod metric. In our situation vague compactness does not imply tightness and thus a different proof than Dudley's (1966) must be used. 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Then a necessary and sufficient condition for the vague convergence of the distributions of $Y_n$ to that of $Y$ is an approximate equicontinuity of the sequence $\langle Y_n \rangle$. Dudley (1966) generalized this standard result. We generalize Dudley's result to the case when the values of $X_n$ are in an arbitrary metric space and extend the result also to the case of the Skorohod metric. In our situation vague compactness does not imply tightness and thus a different proof than Dudley's (1966) must be used. The proof we use is simple and of interest even when other proofs are available.</abstract><pub>Institute of Mathematical Statistics</pub><doi>10.1214/aop/1176996227</doi><tpages>9</tpages><oa>free_for_read</oa></addata></record> |
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| identifier | ISSN: 0091-1798 |
| ispartof | The Annals of probability, 1975-12, Vol.3 (6), p.1014-1022 |
| issn | 0091-1798 2168-894X |
| language | eng |
| recordid | cdi_projecteuclid_primary_oai_CULeuclid_euclid_aop_1176996227 |
| source | JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; EZB-FREE-00999 freely available EZB journals; Project Euclid Complete |
| subjects | 60B10 60G99 62E20 Mathematical theorems Perceptron convergence procedure Real lines Short Communications Skorohod metric stochastic process Stochastic processes Sufficient conditions tightness Topological theorems Topology uniform metric Vague convergence |
| title | On Vague Convergence of Stochastic Processes |
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