One- and Two-Term Edgeworth Expansions for Finite Population Sample Mean. Exact Results. II

We prove the validity of one- and two-term Edgeworth expansions under optimal conditions (a Cramer-type smoothness condition and the minimal moment conditions) and provide precise bounds for the remainders of expansions. The bounds depend explicitly on the ratio p=N/n, where N denotes the sample size and n the population size, respectively.     

Degree distribution of a typical vertex in a general random intersection graph

For a general random intersection graph, we show an approximation of the vertex degree distribution by a Poisson mixture. Research supported by Lithuanian State Science and Studies Foundation Grant T-07149.     

Edgeworth Expansions for Studentized Versions of Symmetric Finite Population Statistics

We show the validity of the one-term Edgeworth expansion for Studentized asymptotically linear statistics based on samples drawn without replacement from finite populations. Replacing the moments defining the expansion by their estimators, we obtain an empirical Edgeworth expansion. We show the validity of the empirical Edgeworth expansion in probability.     

Empirical Edgeworth Expansion for Finite Population Statistics. I

For symmetric asymptotically linear statistics based on simple random samples, we construct the one-term empirical Edgeworth expansion, where the moments defining the true Edgeworth expansion are replaced by their jackknife estimators. In order to establish the validity of the empirical Edgeworth expansion (in probability), we prove the consistency of the jackknife estimators.     

Nonuniform estimate of the rate of convergence in the CLT with stable limit distribution

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. V. Kapsukas Vilnius State University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 1, pp. 14–26, January–March, 1989.     

On the rate of normal approximation inD[0, 1]

The paper gives estimates of the rate of convergence in the central limit theorems for stochastically continuous cadlag processes proved recently by Bézandry and Fernique (Ann. Inst H. Poincare,28) and Bloznelis and Paulauskas (to appear inStoch. Proc. Appl.). Research supported by the SFB 343 at Bielefeld, by a Grant from the Lithuanian Government, and by V.P. Grant 94. Vilnius University, Naugarduko 24; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 3, pp. 280–294, July–September, 1997.     

A note on Hamiltonicity of uniform random intersection graphs

We consider a collection of n independent random subsets of [m] = {1, 2, . . . , m} that are uniformly distributed in the class of subsets of size d, and call any two subsets adjacent whenever they intersect. This adjacency relation defines a graph called the uniform random intersection graph and denoted by G _n,m,d . We fix d = 2, 3, . . . and study when, as n,m → ∞, the graph G _n,m,d contains a Hamilton cycle (the event denoted \( {G_{n,m,d}} \in \mathcal{H} \)). We show that \( {\mathbf{P}}\left( {{G_{n,m,d}} \in \mathcal{H}} \right) = o(1) \) for d ² nm ^?1 ? lnm ? 2 ln lnm → ?∞ and \( {\mathbf{P}}\left( {{G_{n,m,d}} \in \mathcal{H}} \right) = 1 - o(1) \) for 2nm ^?1 ? lnm ? ln lnm → +∞.     

Fifty years in the field of probability: A conversation with professor Vygantas Paulauskas

Lithuanian Mathematical Journal - This year professor Vygantas Paulauskas is celebrating his 75 birthday. An outstanding probabilist he is well known for his research in the field of probability...