Strong Invariance Principle Research Articles

How Big are the Increments of a Wiener Process?

Let $\beta_T = (2a_T\lbrack\log(T/a_T) + \log\log T\rbrack)^{-\frac{1}{2}}, 0 < a_T \leqslant T < \infty$ and $\{W(t); 0 \leqslant t < \infty\}$ be a standard Wiener process. This paper studies the almost sure limiting behaviour of $\sup_{0\leqslant t\leqslant T-a_T} \beta_T|W(t + a_T) - W(t)|$ as $T \rightarrow \infty$ under varying conditions on $a_T = c \log T, c > 0$, the Erdos-Renyi law of large numbers for the Wiener process. A number of other results for the Wiener process also follow via choosing $a_T$ appropriately. Connections with strong invariance principles and the P. Levy modulus of continuity for $W(t)$ are also established.

Strong approximations of the Hoeffding, Blum, Kiefer, Rosenblatt multivariate empirical process

Hoeffding ( Ann. Math. Statist. 1948) and Blum, Kiefer and Rosenblatt ( Ann. Math. Statist. 1961) constructed distribution free tests of independence based on a multivariate empirical process. We establish strong invariance principles for the latter and also for appropriate functionals of it.

Stationary solutions of reaction‐diffusion equations

AbstractGiven a semilinear reaction‐diffusion equation. If the corresponding ordinary differential equation admits a convex compact positively invariant set and the boundary data assume their values in this set then the first and third boundary value problem have stationary solutions. The proofs are based on Weinberger's strong invariance principle, some related tools and the Leray‐Schauder degree. The theorem is applied to several equations from theoretical biology, also in the case of distinct diffusion rates.

Limit theorems for weighted sums and stochastic approximation processes

By establishing certain limit theorems for martingale transforms and weighted sums, a Marcinkiewicz-Zygmund strong law together with weak and strong invariance principles is developed for stochastic approximation processes, under minimal conditions on the errors.

On $r$-Quick Convergence and a Conjecture of Strassen

In this paper, we prove a conjecture of Strassen on the set of $r$-quick limit points of the normalized linearly interpolated sample sum process in $C\lbrack 0, 1 \rbrack$. We give the best possible moment conditions for this conjecture to hold by finding the $r$-quick analogue of the classical law of the iterated logarithm and its converse. The proof is based on an $r$-quick version of Strassen's strong invariance principle and a theorem on the $r$-quick limit set of a semi-stable Gaussian process. Some applications of Strassen's conjecture are given. We also consider the notion of $r$-quick convergence related to the law of large numbers and outline some statistical applications to indicate the usefulness of this concept.

Skorohod embedding of multivariate RV's, and the sample DF

The main purpose of this paper is to study certain representations of sums of iid k-vector rv's as embeddings in k-dimensional Brownian motion by vectors of stopping times, in extension of Skorohod's scheme [20], and consequent error estimates for weak and strong invariance principles. In particular, letting k→∞ we embed the sample df in the Gaussian process with 2-dimensional time to which it has long been known to converge weakly. We discuss previous sample df embeddings, which have yielded related results; while some of our estimates are slight improvements, the emphasis here will be on the naturality of the embedding per se (although it will be indicated why it is probably far from the final word on the subject.