Amarts: A class of asymptotic martingales B. Continuous parameter
| Autor: | Louis Sucheston, G. A. Edgar |
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| Rok vydání: | 1976 |
| Předmět: |
Discrete mathematics
Statistics and Probability Pure mathematics Numerical Analysis Stochastic process Monotonic function Quasimartingale Continuity of trajectories Doob decomposition Martingale Hyperamart Riesz decomposition Bounded function Right and left limits Continuous parameter Amart Statistics Probability and Uncertainty Martingale (probability theory) Mathematics Decomposition theorem |
| Zdroj: | Journal of Multivariate Analysis. 6(4):572-591 |
| ISSN: | 0047-259X |
| DOI: | 10.1016/0047-259x(76)90005-1 |
| Popis: | A continuous-parameter ascending amart is a stochastic process (Xt)t∈ R + such that E[Xτn] converges for every ascending sequence (τn) of optional times taking finitely many values. A descending amart is a process (Xt)t∈ R + such that E[Xτn] converges for every descending sequence (τn), and an amart is a process which is both an ascending amart and a descending amart. Amarts include martingales and quasimartingales. The theory of continuous-parameter amarts parallels the theory of continuous-parameter martingales. For example, an amart has a modification every trajectory of which has right and left limits (in the ascending case, if it satisfies a mild boundedness condition). If an amart is right continuous in probability, then it has a modification every trajectory of which is right continuous. The Riesz and Doob-Meyer decomposition theorems are proved by applying the corresponding discrete-parameter decompositions. The Doob-Meyer decomposition theorem applies to general processes and generalizes the known Doob decompositions for continuous-parameter quasimartingales, submartingales, and supermartingales. A hyperamart is a process (Xt) such that E[Xτn] converges for any monotone sequence (τn) of bounded optional times, possibly not having finitely many values. Stronger limit theorems are available for hyperamarts. For example: A hyperamart (which satisfies mild regularity and boundedness conditions) is indistinguishable from a process all of whose trajectories have right and left limits. |
| Databáze: | OpenAIRE |
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