Extensions of results of Komlós, Major, and Tusnády to the multivariate case

Einmahl, Uwe

doi:10.1016/0047-259x(89)90097-3

Cited by 145 publications

(105 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The proof relies on a progressive blocking technique (see Bernstein [2]) coupled with a triadic Cantor like scheme and on the Komlós, Major and Tusnády approximation type results for independent r.v. 's (see [22], [10], [42]), which is in contrast to approaches usually based on martingale methods.…”

Section: Introductionmentioning

confidence: 77%

“…The rates of convergence in the weak invariance principle, for independent r.v. 's have been obtained in Prokhorov [31], Borovkov [3], Komlós, Major and Tusnády [22], Einmahl [10], Sakhanenko [34], [35], Zaitsev [42] among others. For δ ≤ 1 2 , in the case of martingale-differences, the rates are essentially the same as in the independent case (see, for instance Hall and Heyde [19], Kubilius [23], Grama [11]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

2014

View full text Add to dashboard Cite

Abstract. We prove an invariance principle for non-stationary random processes and establish a rate of convergence under a new type of mixing condition. The dependence is exponentially decaying in the gap between the past and the future and is controlled by an assumption on the characteristic function of the finite dimensional increments of the process. The distinct feature of the new mixing condition is that the dependence increases exponentially in the dimension of the increments. The proposed mixing property is particularly suited for processes whose behavior can be described in terms of spectral properties of some related family of operators. Several examples are discussed. We also work out explicit expressions for the constants involved in the bounds. When applied to Markov chains our result specifies the dependence of the constants on the properties of the underlying Banach space and on the initial state of the chain.

show abstract

Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

2014

View full text Add to dashboard Cite

show abstract

“…7] (see also the references therein). Our approach is to use Theorem 1.2 together with the strong approximation results of [15] and [20].…”

Section: Proof Of the Lattice Torus Covering Time Conjecturementioning

confidence: 99%

Cover times for Brownian motion and random walks in two dimensions

et al. 2004

View full text Add to dashboard Cite

Let T (x, ε) denote the first hitting time of the disc of radius ε centered at x for Brownian motion on the two dimensional torus T 2 . We prove that sup x∈T 2 T (x, ε)/| log ε| 2 → 2/π as ε → 0. The same applies to Brownian motion on any smooth, compact connected, two-dimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus Z 2 n is asymptotic to 4n 2 (log n) 2 /π. Determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete; so far, this structure was only studied nonrigorously in the physics literature. We also establish a conjecture, due to Kesten and Révész, that describes the asymptotics for the number of steps needed by simple random walk in Z 2 to cover the disc of radius n.

show abstract

“…Theorem 2.1 is related to the results of Einmahl (1987Einmahl ( , 1989 who obtained strong approximations for partial sums of independent and identically distributed random vectors with zero mean and with identity covariance matrix. In our setting, for any fixed d, the covariance matrix is not the identity, but this is not the central difficulty.…”

Section: Uniform Normal Approximationmentioning

confidence: 99%

“…In our setting, for any fixed d, the covariance matrix is not the identity, but this is not the central difficulty. The main value of Theorem 2.1 stems from the fact that it shows how the rate of the approximation depends on d; no such information is contained in the work of Einmahl (1987Einmahl ( , 1989, who did not need to consider the dependence on d. The explicit dependence of the right hand side of (2.5) on d is crucial in the applications presented in the following sections in which the dimension of the projection space depends on the sample size N. Very broadly speaking, Theorem 2.1 implies that in all reasonable statistics based on averaging the scores, even in those based on an increasing number of FPC's, the partial sums of scores can be replaced by Wiener processes to obtain a limit distribution. The right hand side of (2.5) allows us to derive assumptions on the eigenvalues required to obtain a specific result.…”

Section: Uniform Normal Approximationmentioning

confidence: 99%

Functional data analysis with increasing number of projections

Fremdt

Horváth

Kokoszka

et al. 2014

Journal of Multivariate Analysis

View full text Add to dashboard Cite

Abstract. Functional principal components (FPC's) provide the most important and most extensively used tool for dimension reduction and inference for functional data. The selection of the number, d, of the FPC's to be used in a specific procedure has attracted a fair amount of attention, and a number of reasonably effective approaches exist. Intuitively, they assume that the functional data can be sufficiently well approximated by a projection onto a finite-dimensional subspace, and the error resulting from such an approximation does not impact the conclusions. This has been shown to be a very effective approach, but it is desirable to understand the behavior of many inferential procedures by considering the projections on subspaces spanned by an increasing number of the FPC's. Such an approach reflects more fully the infinite-dimensional nature of functional data, and allows to derive procedures which are fairly insensitive to the selection of d. This is accomplished by considering limits as d → ∞ with the sample size.We propose a specific framework in which we let d → ∞ by deriving a normal approximation for the partial sum processwhere N is the sample size and ξ i,j is the score of the ith function with respect to the jth FPC. Our approximation can be used to derive statistics that use segments of observations and segments of the FPC's. We apply our general results to derive two inferential procedures for the mean function: a change-point test and a two-sample test. In addition to the asymptotic theory, the tests are assessed through a small simulation study and a data example.

show abstract

Extensions of results of Komlós, Major, and Tusnády to the multivariate case

Cited by 145 publications

References 7 publications

On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

Cover times for Brownian motion and random walks in two dimensions

Functional data analysis with increasing number of projections

Contact Info

Product

Resources

About