We obtain a strong approximation for the logarithmic average of sample extremes. The central limit theorem and laws of the iterated logarithm are immediate consequences.

We study the asymptotic behavior of the empirical distribution function and the empirical process of squared residuals. We prove the Glivenko-Cantelli theorem for the empirical distribution function. We show that the two-parameter empirical process converges to a Gaussian process.

Motivated by problems in functional data analysis, in this paper we prove the weak convergence of normalized partial sums of dependent random functions exhibiting a Bernoulli shift structure.

Trimming is a standard method to decrease the effect of large sample elements in statistical procedures, used, e.g., for constructing robust estimators and tests. Trimming also provides a profound insight into the partial sum behavior of i.i.d. sequences. There is a wide and nearly complete...

We prove a functional central limit theorem for modulus trimmed i.i.d. variables in the domain of attraction of a nonnormal stable law. In contrast to the corresponding result under ordinary trimming, our CLT contains a random centering factor which is inevitable in the nonsymmetric case. The...

We investigate the estimation of parameters in the random coefficient autoregressive (RCA) model X_k = (ϕ + b_k)X_k - 1 + e_k, where (ϕ, omega-super-2, σ-super-2) is the parameter of the process, , . We consider a nonstationary RCA process satisfying E log |ϕ + b_0| = 0 and show that σ-super-2...

Principal component analysis has become a fundamental tool of functional data analysis. It represents the functional data as "X"_"i"("t")&equals;"μ"("t")&plus;Σ_{1≤"l"&infin ;}"η"_"i", "l"&plus; "v"_"l"("t "), where "μ" is the common mean, "v"_"l" are the eigenfunctions of the covariance operator and the...</sub>

Let X1, X2,... be independent, identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn = [summation operator]k[less-than-or-equals, slant]n Xk. We give nearly optimal criteria for an (unbounded) measurable function f to satisfy the a.s. central limit theorem, i.e., a.s., where...

We obtain an approximation for the logarithmic averages of I{k1/2a(k) [less-than-or-equals, slant] S(k) [less-than-or-equals, slant] k1/2b(k)}, where a(k) --> 0, b(k) --> 0 (k --> [infinity]) and S(k) is partial sum of independent, identically distributed random variables.