We consider the one-sided exit problem for (fractionally) integrated random walks and L 'evy processes. We prove that the rate of decrease of the non-exit probability -- the so-called survival exponent -- is universal in this class of processes. In particular, the survival exponent can be inferred from the (fractionally) integrated Brownian motion. This, in particular, extends Sinai's result on the survival exponent for the integrated simple random walk to general random walks with some finite exponential moment. Further, we prove existence and monotonicity of the survival exponent of fractionally integrated processes. We show that this exponent is related to a constant appearing in the study of random polynomials.
Li and Shao @cite_10 @cite_20 are the first who aim at building a theory for a whole class of processes. In the mentioned works, the lower tail probability problem ) is studied for Gaussian processes. It is shown that the decrease in ) is indeed on the polynomial scale for many one-dimensional Gaussian processes. However, the technique does not yield values for the survival exponent. An important tool in the study of the above problems for Gaussian processes is the Slepian lemma @cite_2 and a comparable opposite inequality from @cite_17 .
@cite_10 @cite_17 @cite_20 @cite_2
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Let X=(Xt)t∈S be a real-valued Gaussian random process indexed by S with mean zero. General upper and lower estimates are given for the lower tail probability P(supt∈S(Xt−Xt0)≤x) as x→0, with t0∈S fixed. In particular, sharp rates are given for fractional Brownian sheet. Furthermore, connections between lower tail probabilities for Gaussian processes with stationary increments and level crossing probabilities for stationary Gaussian processes are studied. Our methods also provide useful information on a random pursuit problem for fractional Brownian particles. Let @math and @math be standard normal random variables with covariance matrices @math and @math , respectively. Slepian's lemma says that if @math for @math , the lower bound @math is at least @math . In this paper an upper bound is given. The usefulness of the upper bound is justified with three concrete applications: (i) the new law of the iterated logarithm of Erdős and Revesz, (ii) the probability that a random polynomial does not have a real zero and (iii) the random pursuit problem for fractional Brownian particles. In particular, a conjecture of Kesten (1992) on the random pursuit problem for Brownian particles is confirmed, which leads to estimates of principal eigenvalues. This paper surveys briefly some recent developments on lower tail probabilities for real valued Gaussian processes. Connections and applications to various problems are discussed. A new and simplified argument is given and it is of independent interest. This paper is concerned with the probability, P[T, r(τ)], that a stationary Gaussian process with mean zero and covariance function r(τ) be nonnegative throughout a given interval of duration T. Several strict upper and lower bounds for P are given, along with some comparison theorems that relate P's for different covariance functions. Similar results are given for F[T, r(τ)], the probability distribution for the interval between two successive zeros of the process.