I. K. Matsak

Some remarks on the ordinal strong law of large numbers

We prove that the ordinal law of large numbers and the law of large numbers in the norm are equivalent for Banach lattices that do not contain uniformly the space ln 1 .

Limit theorems for the maximal residuals in linear and nonlinear regression models

Limit theorems for the maximal residuals in linear and nonlinear regression models are obtained in the paper. An application of the main result for constructing a regression model adequacy test is given. Classical results of the regression analysis describe various properties of residual sums of squares errors (see, for example, [1, 2]). In other words, the classical results deal with the properties of the sum of squares of deviations between observations and a regression function where the least squares estimator is substituted in place of the unknown parameter. The current paper deals with the asymptotic behavior of the maximal residual. We prove the convergence of distributions of the normalized appropriately maximal residual to a limit distribution of the normalized maximum as the size of a sample grows to infinity of identically distributed errors of observations. We consider both cases, the linear and nonlinear regression models. We apply the asymptotic results to construct some regression model adequacy test (see Section 3). 1. Linear regression model Consider the following linear regression model

The asymptotic stability of the maximum of independent random elements in function Banach lattices

We generalize some well-known results on the asymptotic stability of the maximum of independent random variables in R1 to the case of q-concave Banach ideal spaces. A theorem on the relative asymptotic stability of the maximum of independent random elements in function Banach lattices is proved.

The law of large numbers for the max-scheme in Banach lattices

We prove that the law of large numbers for the max-scheme in Banach lattices is equivalent to the condition that E ‖X‖ < ∞. Some generalizations of this proposition are considered.

Asymptotic stability of the maximum of normal stochastic processes

Under quite general conditions, we prove that the maximum of a sequence of normal stochastic processes in the space C[0,1] is asymptotically stable almost surely. The scheme of the maximum of independent random variables always attracts the attention of mathematicians because of various applications. Gnedenko [1] was the first to study degenerate limit laws for a sequence of extremes although the case of the limit normal distribution was known earlier. Galambos [2] gives the main asymptotic results on the convergence to degenerate laws along with the classical theory of extremal values. He also provides several results on almost sure convergence. Let (ξn) be a sequence of independent identically distributed random variables and let zn = max1≤i≤n ξi. We say that a sequence (zn) is relatively stable in probability if there exists a sequence of numbers (an) such that (1) zn an P −→ 1 as n → ∞. We also say that (zn) is stable in probability if zn − an P −→ 0 (2) as n → ∞. Gnedenko [1] obtains the criteria for relations (1) and (2). If the almost sure convergence is substituted for the convergence in probability in relations (1) and (2), then we say that a sequence (zn) is relatively stable or stable almost surely. Some important results on the almost sure convergence are obtained by Barndorff-Nielsen [3]. Below is one of the classical results on almost sure convergence. Let γ1, γ2, . . . be a sequence of independent normal random variables such that E γi = 0 and E γ 2 i = 1. Put zn = max1≤i≤n γi and (3) bn = { (2 lnn), n > 1, 1, n = 1. Then lim n→∞ zn bn = 1 a.s., (4) 2000 Mathematics Subject Classification. Primary 60B12.

Some applications of the Gnedenko-Korolyuk method to empirical distributions

A new proof of the Kolmogorov theorem on the asymptotic behavior of the deviation between a theoretical and an empirical distribution function is pre- sented. We use the Gnedenko-Korolyuk approach based on some combinatorial prop- erties of the merged sample constructed from two other independent samples. Some statistical applications of the Gnedenko-Korolyuk theorem are discussed.

Ordinal law of the iterated logarithm in Banach lattices and some applications

Necessary and sufficient conditions are found for the ordinal law of the iterated logarithm in Banach lattices of type Lp. As a corollary of our general results, we obtain a new law of the iterated logarithm for empirical processes in the spaces Lp(−∞,∞).