We show that the accumulated gain Sn and the maximal gain Mn in n St. Petersburg games satisfy almost sure limit theorems with nondegenerate limits, even though ordinary asymptotic distributions do not exist for Sn and Mn with any numerical centering and norming sequences.

The discovery of the almost sure central limit theorem (Brosamler, Math. Proc. Cambridge Philos. Soc. 104 (1988) 561-574; Schatte, Math. Nachr. 137 (1988) 249-256) revealed a new phenomenon in classical central limit theory and has led to an extensive literature in the past decade. In particular,...

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>

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.

We consider polynomial variables which define an important subclass of Duan's augmented processes. We prove functional central limit theorems for the observations as well as for the volatility process under the assumption of finite second moments. The results imply the convergence of CUSUM,...