# Asymptotic Methods in Probability and Statistics

## A Volume in Honour of Miklós Csörgő

**Edited by**

- B. Szyszkowicz, Carleton University, Dept. of Mathematics and Statistics, Ottawa, Canada

One of the aims of the conference on which this book is based, was to provide a platform for the exchange of recent findings and new ideas inspired by the so-called Hungarian construction and other approximate methodologies. This volume of 55 papers is dedicated to Miklós Csörgő a co-founder of the Hungarian construction school by the invited speakers and contributors to ICAMPS'97.

This excellent treatize reflects the many developments in this field, while pointing to new directions to be explored. An unequalled contribution to research in probability and statistics.

,

Published: November 1998

Imprint: North-holland

ISBN: 978-0-444-50083-0

## Contents

- Preface. List of Contributors.
**Part 1: Limit Theorems for variously mixing and quasi-associated random variables.**Rényi-mixing of occupation times (S. Csörgo). Limit theorems for maximal random sums (P. Kowalski, Z. Rychlik). Limit theorems for partial sums of quasi-associated random variables (T.M. Lewis). On the central limit theorem for triangular arrays of &fgr;-mixing sequences (M. Peligrad).**Part 2: Central limit theorems for logarithmic averages.**Results and problems related to the pointwise central limit theorem (I. Berkes). On two ergodic properties of self-similar processes (E. Csáki, A. Földes).**Part 3: Strong approximations, weighted approximations.**Jump diffusion approximation for a Markovian transport model (A.R. Dabrowski, H. Dehling). On the local oscillations of emperical and quantile processes (P. Deheuvels). Strong approximations in queueing theory (P.W. Glynn). Applications of weighted approximations via quantile inequalities (G.R. Shorack).**Part 4: Emperical distributions and processes.**Emperical processes based on pseudo-observations (K. Ghoudi, B. Remillard). A uniform Marcinkiewicz-Zygmund strong law of large numbers for emperical processes (P. Massart, E. Rio). On the comparison of theoretical and emperical distribution functions (L. Takács).**Part 5: Iterated random walks.**A random walk on a random walk path (K. Grill). Long excursions and iterated processes (P. Révész).**Part 6: Fine analytic path behaviour of the oscillations of stochastic processes.**Integral tests for some processes related to Brownian motion (S. Keprta). A lim inf result for the Brownian motion (W.V. Li). On the increments of l8-valued Gaussian processes (Z.Y. Lin, Y.C. Qin). A note on how small are the incrementals of a fractal Wiener process (C.-R. Lu, H. Yu). On a conjecture of Révész and its analogue for renewal processes (J. Steinebach). Asymptotic results for self-similar Markov processes (Y. Xiao).**Part 7: Multiparameter stochastic processes.**On Strassen's version of the law of the iterated logarithm for the two-parameter Wiener process (B. Chen). A maximal inequality and tightness for multiparameter stochastic processes (B.G. Ivanoff, N.C. Weber).**Part 8: Results related to studies of local time and hitting times of Bessel processes, a cautionary note on limiting sigma-algebras.**On asymptotic independence of partial sums (E. Csáki, A. Földes). Limiting sigma-algebras --- some counterexamples (D.L. Hanson). Convergence in law and convergence of moments: an example related to Bessel processes (Y. Hu, M. Yor).**Part 9: Large deviations, small ball problems, self normalization.**Analytic aspects of multilevel large deviations (D.A. Dawson, J. Gärtner). Large deviation upper bound and its application to measure values processes (S. Feng). Gaussian measure of a small ball and capacity in Wiener space (D. Khoshnevisan, Z. Shi). Recent developments on self-normalized limit theorems (Q.-M. Shao).**Part 10: Stochastic bifurcation.**On the time and direction of stochastic bifurcation (K. Burdzy, D.M. Frankel, A. Pauzner).**Part 11: Change-point analysis, u-statistics, non-smooth functions, comparison distributions.**Change point tests for randomly censored data (E-E. A.A. Aly). Weighted approximations of parameters-estimated empirical processes and changepoint analysis (J.A. Correa). The application of change point tests to data in fair hiring cases (B. Freidlin, J.L. Gastwirth). Parameter estimated standardized u-statistics (E. Gombay, L. Horváth). Remarks on test procedures for gradual changes (M. Hušková). Tests for constancy of a mean (F. Lombard). Non-parametric models for non-smooth functions (H.-G. Müller). Statistical methods mining, two sample data analysis, comparison distributions, and quantile limit theorems (E. Parzen).**Part 12: Emperical reliability, survival analysis.**On the Cox regression (L. Rejto, G. Tusnády). Some further results on the limiting proportion of types (M.D. Rothmann, R.P. Russo). Estimation of percentile residal lifetime processes for stationary observations (H. Yu). Ther Vervaat process (R. Zitikis).**Part 13: Gaussian bootstrap, Monte Carlo simulation.**A Gaussian bootstrap approach to estimation and tests (M.D. Burke). Tabulating weighted sup-norm functionals of Brownian bridges via Monte Carlo simulation(B.J. Eastwood, V.R. Eastwood).**Part 14: Autoregressive and moving average schemes.**Upper and lower bounds for the tail of the invariant distribution of some AR(1) processes (G. Haiman). Non Gaussian autoregressive and moving average schemes (M. Rosenblatt).**Part 15: Nonparametric curve estimations, regression diagnostics.**A regression residual process (R.J. Kulperger). A note on nonparametric estimations (P. Major, L. Rejto). Convergence rates for maximal score estimators in binary response regressions (S. Portnoy).**Part 16: Testing statistical hypotheses.**Applications of Hamming distance to the analysis of block designs (M. Alvo, P. Cabilio). Bayesian sequential and fixed sample testing of multihypotheses (J. Babb, A. Rogatko, S. Zacks). Vanishing shortcoming of data driven Neyman's tests (T. Inglot, W.C.M. Kallenberg, T. Ledwina).**Part 17: Tail index estimation, order statistics of order statistics.**Estimating the tail index (S. Csörgo, L. Viharos). Stability criteria for order statistics of order statistics (R.J. Tomkins).