Markov Processes

在Esscher和Cramer的原著中，通过现在所知的Esscher或Cramer倾斜产生了大偏差，这改变了潜在的分布，使得最初罕见的事件现在变成了非常可能的事件。导致大偏差的准确倾斜度使人们能够计算出大偏差的精确比率。这提供了关于条件分布是什么的线索，假设已经发生了特定的罕见事件。虽然最初的工作是在独立随机变量和的背景下进行的，但随后关于大偏差的工作大大扩展了这一想法。这项建议的目的是在随机环境中随机行走的背景下研究这一点。特别是，如果行走的速度不太可能，它会经历什么？大偏差涉及到估计罕见事件的概率。罕见的事件确实会发生，该理论涉及确定它们到底有多罕见。当一个罕见的事件发生时，它并不是孤立的。其他罕见的事件也会发生。产生罕见事件的同一现象很可能会催生其他罕见事件。该理论涉及到预测其他罕见事件。一种是从模型的世界开始，以及对特定模型的假设。在该模型下发生概率非常小的事件。这改变了一个人对特定模型的信念，并选择了一个与罕见事件一致的新模型。如果有许多这样的模型，则会以某种方式选择最优的一个。这个模型做出了预测，这在旧模型下可能很少见，但现在不再是这样了。

In the original work of Esscher and Cramer Large Deviations arise through what is now known as Esscher or Cramer tilt, that changes the underlying distribution so that what was originally a rare event is now a very likely event. The exact tilt that leads to the large deviation allows one to calculate the precise rate of the large deviation. This provides a clue as to what the conditional distribution is, given that a particular rare event has occurred. While the original work is in the context of sums of independent random variables, subsequent work on large deviations has extended this idea considerably. It is the aim of this proposal to investigate this in the context of a random walk in a random environment. In particular if the walk travels with an unlikely velocity, what would it have experienced?Large deviations deals with estimating probabilities of rare events. Rare events do occur and the theory deals with determining exactly how rare they are. When a rare event occurs it is not isolated. Other rare events happen as well. The same phenomenon that generated the rare event could very well have spawned other rare events. The theory deals with predicting such other rare events.One starts with a world of models as well as an assumption about a specific model. An event with very small probability under this model occurs. This changes one's belief in the particular model and a new model is chosen that is consistent with the rare event.If there are many such models an optimal one is chosen in some way. This model makes predictions, which were perhaps rare under the old model but not any more.