# Persistence probabilities in centered, stationary, Gaussian processes in discrete time

@article{Krishna2016PersistencePI,
title={Persistence probabilities in centered, stationary, Gaussian processes in discrete time},
author={M. Krishna and Manjunath Krishnapur},
journal={Indian Journal of Pure and Applied Mathematics},
year={2016},
volume={47},
pages={183-194}
}
• Published 30 January 2016
• Mathematics
• Indian Journal of Pure and Applied Mathematics
Lower bounds for persistence probabilities of stationary Gaussian processes in discrete time are obtained under various conditions on the spectral measure of the process. Examples are given to show that the persistence probability can decay faster than exponentially. It is shown that if the spectral measure is not singular, then the exponent in the persistence probability cannot grow faster than quadratically. An example that appears (from numerical evidence) to achieve this lower bound is… Expand
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#### References

SHOWING 1-10 OF 26 REFERENCES
Persistence in a stationary time series.
• Mathematics, Physics
• Physical review. E, Statistical, nonlinear, and soft matter physics
• 2001
This work studies the persistence in a class of continuous stochastic processes that are stationary only under integer shifts of time and constructs a specific sequence for which the persistence can be computed even though the sequence is non-Markovian. Expand
Persistence exponent for discrete-time, time-reversible processes
• Mathematics
• 2015
We study the persistence probability for some discrete-time, time-reversible processes. In particular, we deduce the persistence exponent in a number of examples: first, we deal with random walks inExpand
Long gaps between sign-changes of Gaussian Stationary Processes
• Mathematics
• 2013
We study the probability of a real-valued stationary process to be positive on a large interval $[0,N]$. We show that if in some neighborhood of the origin the spectral measure of the process hasExpand
Persistence exponents and the statistics of crossings and occupation times for Gaussian stationary processes.
• Mathematics, Physics
• Physical review. E, Statistical, nonlinear, and soft matter physics
• 2004
The correlator method is extended to calculate the occupation-time and crossing-number distributions, as well as their partial-survival distributions and the means and variances of the occupation time and number of crossings. Expand
Small ball probabilities of Gaussian fields
• Mathematics
• 1995
SummaryLower bounds on the small ball probability are given for Brownian sheet type Gaussian fields as well as for general Gaussian fields with stationary increments in ℝd. In particular, a sharpExpand
Persistence probabilities \& exponents
• Mathematics
• 2012
This article deals with the asymptotic behaviour as $t\to +\infty$ of the survival function $P[T > t],$ where $T$ is the first passage time above a non negative level of a random process startingExpand
The one-sided barrier problem for Gaussian noise
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.Expand
Persistence Probabilities and Exponents
• Mathematics
• 2015
This article deals with the asymptotic behavior as $$t \rightarrow +\infty$$ of the survival function $$\mathbb{P}[T > t]$$, where T is the first passage time above a non negative level of a randomExpand
RECENT DEVELOPMENTS ON LOWER TAIL PROBABILITIES FOR GAUSSIAN PROCESSES
• Mathematics
• 2005
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 andExpand
No zero-crossings for random polynomials and the heat equation
• Mathematics
• 2015
Consider random polynomial ∑ni=0aixi of independent mean-zero normal coefficients ai, whose variance is a regularly varying function (in i) of order α. We derive general criteria for continuity ofExpand