## Limit Theorems for Stochastic ProcessesInitially the theory of convergence in law of stochastic processes was developed quite independently from the theory of martingales, semimartingales and stochastic integrals. Apart from a few exceptions essentially concerning diffusion processes, it is only recently that the relation between the two theories has been thoroughly studied. The authors of this Grundlehren volume, two of the international leaders in the field, propose a systematic exposition of convergence in law for stochastic processes, from the point of view of semimartingale theory, with emphasis on results that are useful for mathematical theory and mathematical statistics. This leads them to develop in detail some particularly useful parts of the general theory of stochastic processes, such as martingale problems, and absolute continuity or contiguity results. The book contains an introduction to the theory of martingales and semimartingales, random measures stochastic integrales, Skorokhod topology, etc., as well as a large number of results which have never appeared in book form, and some entirely new results. The second edition contains some additions to the text and references. Some parts are completely rewritten. |

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### Contents

1 | |

5 | |

16 | |

Increasing Processes | 27 |

Semimartingales and Stochastic Integrals | 38 |

Characteristics of Semimartingales and Processes | 64 |

Characteristics of Semimartingales | 75 |

Some Examples | 91 |

Weak Convergence | 347 |

The QuasiLeft Continuous Case | 355 |

The General Case | 362 |

Convergence Quadratic Variation Stochastic Integrals | 376 |

Convergence of Processes with Independent Increments | 389 |

Functional Convergence and Characteristics | 413 |

More on the General Case | 428 |

The Central Limit Theorem | 444 |

Semimartingales with Independent Increments | 101 |

Processes with Independent Increments | 112 |

Processes with Conditionally Independent Increments | 124 |

Semimartingales Stochastic Exponential and Stochastic Logarithm | 134 |

Martingale Problems and Changes of Measures | 142 |

Martingale Problems and Semimartingales | 151 |

Absolutely Continuous Changes of Measures | 165 |

Representation Theorem for Martingales | 179 |

Integrals of VectorValued Processes and omartingales | 203 |

Laplace Cumulant Processes and Esschers Change of Measures | 219 |

Hellinger Processes Absolute Continuity | 227 |

Predictable Criteria for Absolute Continuity and Singularity | 245 |

Hellinger Processes for Solutions of Martingale Problems | 254 |

3c The Case Where Local Uniqueness Holds | 266 |

Examples | 272 |

Contiguity Entire Separation Convergence in Variation | 284 |

Predictable Criteria for Contiguity and Entire Separation | 291 |

Examples | 304 |

Skorokhod Topology and Convergence of Processes | 324 |

Continuity for the Skorokhod Topology | 337 |

5d Functional Convergence of PIIs to a Gaussian Martingale | 452 |

Convergence to a PII Without Fixed Time of Discontinuity | 460 |

Applications | 469 |

Convergence to a General Process with Independent Increments | 499 |

Convergence to a Semimartingale | 521 |

Identification of the Limit | 527 |

Limit Theorems for Semimartingales | 540 |

Applications | 554 |

Convergence of Stochastic Integrals | 564 |

Stability for Stochastic Differential Equation | 575 |

Stable Convergence to a Progressive Conditional Continuous PII | 583 |

Limit Theorems Density Processes and Contiguity | 592 |

Convergence of the LogLikelihood to a Process | 612 |

The Statistical Invariance Principle | 620 |

Bibliographical Comments | 629 |

641 | |

652 | |

659 | |

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3-measurable a-field absolute continuity adapted process apply assume assumption belongs C-tight câdläg canonical canonical decomposition characteristic function compensator consider converges in law Corollary d-dimensional semimartingale decomposition deduce defined definition denote density process deterministic equivalent evanescent set exists finite variation formula function h Gaussian gives Hellinger process hence holds hypothesis implies increasing process independent increments inf(t integrable martingale jumps Lemma lim sup liminf limit theorems locally bounded locally square-integrable martingale problem Moreover nonnegative notation obtain obviously P-local martingale P-UT point process Poisson process predictable process probability measure process H Proof of Theorem Proposition purely discontinuous random measure random set random variables recall relative remains to prove resp result satisfies Section similarly Skorokhod topology space special semimartingale square-integrable stochastic basis stochastic integral strict stopping sufficient condition suppose tight trivial truncation function uniformly integrable uniqueness Var(A Wiener process yields ys-D