Large deviations for sums and Markov chains:heavy-tailed case

• BY : Dima Korshunov
• ABSTRACT :
  

  In our talk, we'll consider a random walk $\{S_n\}$ with i.i.d. heavy-tailed increments and negative drift. The exact asymptotic behaviour of $\Prob(\sup_n S_n>x)$ is well-known and may be described in terms of integrated tail. We are going to present the exact asymptotics for the tail probability of partial supremum $\Prob(\sup_{kx)$, as $x\to\infty$. The whole range of values of $n$ will be considered. We also present large deviations results for one-dimensional Markov chains with heavy-tailed jumps. Tail behaviour of a stationary and pre-stationary distributions will be discussed.