Endre Csáki

Almost sure limit theorems for the maximum of stationary Gaussian sequences

We prove an almost sure limit theorem for the maxima of stationary Gaussian sequences with covariance rn under the condition rn log n(loglog n)1+[var epsilon]=O(1).

Strong invariance for local times

SummaryLet Y1, Y2, ... be a sequence of i.i.d. random variables with distribution P(Y1 = k) = pk (k = ±1, ±2,...), E(Y1) = 0, E(Y12) = σ2<∞. Put Tn = Y1+...+Yn and N(x,n) = # {k:0<k≦n, Tk = x}. Extending the result of Révész (1981) it is shown that for appropriate Skorohod construction we have 1

The local time of iterated Brownian motion

\mathop {{\text{sup}}}\limits_{x \in \mathbb{Z}} |{\text{L(}}x,n\sigma ^2 {\text{) - }}\sigma ^{\text{2}} N(x,n)| = o(n^{{1 \mathord{\left/ {\vphantom {1 {4 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {4 + \varepsilon }}} ){\text{a}}{\text{.s}}{\text{.}}

On infinite series of independent Ornstein-Uhlenbeck processes

provided all moments E(¦Y1¦m), m≧0 exists where L is the local time of a Wiener process. Certain rate of convergence is given also under weaker conditions and for ¦L(x,nσ2)-σ2N(x, n)¦ too, when x is fixed.

On the lower limits of maxima and minima of wiener process and partial sums

AbstractThis paper deals with the law of the iterated logarithm and its analogues for sup

Strong approximation of additive functionals

\mathop {\sup }\limits_{\left( x \right)} \left| {x - F_n \left( x \right)} \right|\left( {x\left( {1 - x} \right)} \right)^{ - \tfrac{1}{2}}