## Abstract

Let M be a continuous martingale,h:R_{+}→R_{+} continuous and increasing such that M(t)/h(F<M>_{t} → 0 (a.s.) as t → ∞. It is shown that w.p.l, large deviations type limits exist for a class of conditional probabilities which are induced on (C([0, ∞),{norm of matrix}·|_{∞}) by the tail processes y^{t}(·) = M(t + ·)/h(<M>_{t+.}). This is obtained via a simple use of the Borell inequality for Gaussian processes, combined with a random time change argument. Results are applied to obtain convergence rates for the (conditional) tail probabilities of consistent parameter estimators in diffusion processes. This is followed by the derivation of efficient stopping rules. Finally, unconditional large deviations lower bounds for the tails of consistent estimators in diffusions are investigated via an extension of a well known direct method.

Original language | English |
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Pages (from-to) | 117-134 |

Number of pages | 18 |

Journal | Stochastic Processes and their Applications |

Volume | 51 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 1994 |

Externally published | Yes |

## Keywords

- Borell inequality
- Diffusions
- Large deviations
- Martingale LLN
- Parameter estimation
- Tail probabilities

## ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics