347 research outputs found
An Empirical Process Central Limit Theorem for Multidimensional Dependent Data
Let be the empirical process associated to an
-valued stationary process . We give general conditions,
which only involve processes for a restricted class of
functions , under which weak convergence of can be
proved. This is particularly useful when dealing with data arising from
dynamical systems or functional of Markov chains. This result improves those of
[DDV09] and [DD11], where the technique was first introduced, and provides new
applications.Comment: to appear in Journal of Theoretical Probabilit
Using the Bootstrap to test for symmetry under unknown dependence
This paper considers tests for symmetry of the one-dimensional marginal distribution of fractionally integrated processes. The tests are implemented by using an autoregressive sieve bootstrap approximation to the null sampling distribution of the relevant test statistics. The sieve bootstrap allows inference on symmetry to be carried out without knowledge of either the memory parameter of the data or of the appropriate norming factor for the test statistic and its asymptotic distribution. The small-sample properties of the proposed method are examined by means of Monte Carlo experiments, and applications to real-world data are also presented
Sharp error terms for return time statistics under mixing conditions
We describe the statistics of repetition times of a string of symbols in a
stochastic process. Denote by T(A) the time elapsed until the process spells
the finite string A and by S(A) the number of consecutive repetitions of A. We
prove that, if the length of the string grows unbondedly, (1) the distribution
of T(A), when the process starts with A, is well aproximated by a certain
mixture of the point measure at the origin and an exponential law, and (2) S(A)
is approximately geometrically distributed. We provide sharp error terms for
each of these approximations. The errors we obtain are point-wise and allow to
get also approximations for all the moments of T(A) and S(A). To obtain (1) we
assume that the process is phi-mixing while to obtain (2) we assume the
convergence of certain contidional probabilities
Limiting distributions for explosive PAR(1) time series with strongly mixing innovation
This work deals with the limiting distribution of the least squares
estimators of the coefficients a r of an explosive periodic autoregressive of
order 1 (PAR(1)) time series X r = a r X r--1 +u r when the innovation {u k }
is strongly mixing. More precisely {a r } is a periodic sequence of real
numbers with period P \textgreater{} 0 and such that P r=1 |a r |
\textgreater{} 1. The time series {u r } is periodically distributed with the
same period P and satisfies the strong mixing property, so the random variables
u r can be correlated
A functional central limit theorem for interacting particle systems on transitive graphs
Abstract A nite range interacting particle system on a transitive graph is considered. Assuming that the dynamics and the initial measure are invariant, the normalized empirical distribution process converges in distribution to a centered diusion process. As an application, a central limit theorem for certain hitting times, interpreted as failure times of a coherent system in reliability, is derived
On the distribution of maximum value of the characteristic polynomial of GUE random matrices
Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random N×N matrices H from the Gaussian Unitary Ensemble (GUE), we consider the problem of characterising the distribution of the global maximum of DN(x):=−log|det(xI−H)| as N→∞ and x∈(−1,1). We arrive at an explicit expression for the asymptotic probability density of the (appropriately shifted) maximum by combining the rigorous Fisher-Hartwig asymptotics due to Krasovsky \cite{K07} with the heuristic {\it freezing transition} scenario for logarithmically correlated processes. Although the general idea behind the method is the same as for the earlier considered case of the Circular Unitary Ensemble, the present GUE case poses new challenges. In particular we show how the conjectured {\it self-duality} in the freezing scenario plays the crucial role in our selection of the form of the maximum distribution. Finally, we demonstrate a good agreement of the found probability density with the results of direct numerical simulations of the maxima of DN(x)
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