114 research outputs found
Invariance principle for stochastic processes with short memory
In this paper we give simple sufficient conditions for linear type processes
with short memory that imply the invariance principle. Various examples
including projective criterion are considered as applications. In particular,
we treat the weak invariance principle for partial sums of linear processes
with short memory. We prove that whenever the partial sums of innovations
satisfy the --invariance principle, then so does the partial sums of its
corresponding linear process.Comment: Published at http://dx.doi.org/10.1214/074921706000000734 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
An asymptotic variance of the self-intersections of random walks
We present a Darboux-Wiener type lemma and apply it to obtain an exact
asymptotic for the variance of the self-intersection of one and two-dimensional
random walks. As a corollary, we obtain a central limit theorem for random walk
in random scenery conjectured by Kesten and Spitzer in 1979
Variance of partial sums of stationary sequences
Let be a centred sequence of weakly stationary random
variables with spectral measure and partial sums . We
show that is regularly varying of index at
infinity, if and only if is regularly
varying of index at the origin ().Comment: Published in at http://dx.doi.org/10.1214/12-AOP772 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Asymptotic variance of stationary reversible and normal Markov processes
We obtain necessary and sufficient conditions for the regular variation of
the variance of partial sums of functionals of discrete and continuous-time
stationary Markov processes with normal transition operators. We also construct
a class of Metropolis-Hastings algorithms which satisfy a central limit theorem
and invariance principle when the variance is not linear in
Convergence of the Poincare Constant
The Poincare constant R(Y) of a random variable Y relates the L2 norm of a
function g and its derivative g'. Since R(Y) - Var(Y) is positive, with
equality if and only if Y is normal, it can be seen as a distance from the
normal distribution. In this paper we establish a best possible rate of
convergence of this distance in the Central Limit Theorem. Furthermore, we show
that R(Y) is finite for discrete mixtures of normals, allowing us to add rates
to the proof of the Central Limit Theorem in the sense of relative entropy.Comment: 11 page
Moderate deviations for stationary sequences of bounded random variables
In this paper we derive the moderate deviation principle for stationary
sequences of bounded random variables under martingale-type conditions.
Applications to functions of -mixing sequences, contracting Markov
chains, expanding maps of the interval, and symmetric random walks on the
circle are given
A sharp uniform bound for the distribution of sums of Bernoulli trials
In this note we establish a uniform bound for the distribution of a sum
of independent non-homogeneous Bernoulli trials.
Specifically, we prove that where
denotes the standard deviation of and is a universal
constant. We compute the best possible constant and we show
that the bound also holds for limits of sums and differences of Bernoullis,
including the Poisson laws which constitute the worst case and attain the
bound. We also investigate the optimal bounds for and fixed. An
application to estimate the rate of convergence of Mann's fixed point
iterations is presented.Comment: This paper is a revised version of a previous articl
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