A general large deviation principle for longest runs

In this note we prove a general large deviation principle (LDP) for the longest success run in a sequence of independent Bernoulli trails. This study not only recovers several recently derived LDPs, but also gives new LDPs for the longest success run. The method is based on the Bryc's inverse Varadhan lemma, which can be intuitively generalized to the longest success run in a two-state (success and failure) Markov chain.