Remarks on the functional central limit theorem for martingales

Summary It is shown that filling a gap in the proof of Lemma 3 in Rootzén (1977) and using an appropriate truncation procedure one obtains general necessary conditions for the functional CLT for martingale difference arrays leading simultaneously to a solution of a remaining problem posed by Rootzén (1977). As to sufficient conditions which are weaker than previous ones in the literature Rootzén and the authors arrived independently at the result stated as Theorem 1 in the present paper.Dedicated to Professor Leopold Schmetterer on the occasion of his 60th birthday     

A central limit theorem for martingales and an application to branching processes

A functional central limit theorem is obtained for martingales which are not uniformly asymptotically negligible but grow at a geometric rate. The function space is not the usual C[0,1] or D[0,1] but R^N, the space of all real sequences and the metric used leads to a non-separable metric space.The main theorem is applied to a martingale obtained from a supercritical Galton-Watson branching process and as simple corollaries the already known central limit theorems for the Harris and Lotka-Nagaev estimators of the mean of the offspring distribution, are obtained.     

Uniform bounds in the central limit theorem for C(S) valued martingales

Vilnius University. Published in Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 31, No. 1, pp. 146–165, January–March, 1991.     

A functional central limit theorem for ϱ-mixing sequences

In this note a functional central limit theorem for ?-mixing sequences of I. A. Ibragimov (Theory Probab. Appl.20 (1975), 135–141) is generalized to nonstationary sequences (X_n)_n ∈ $\text{N}$, satisfying some assumptions on the variances and the moment condition $\text{E |X}{}_{\mathrm{n}}\text{|}{}^{\mathrm{2\; +\; b}}\text{= O(n}{}^{\mathrm{<mtext>b</mtext><mtext>2</mtext><mtext>??</mtext>}}\text{)}$ for some b > 0, ? > 0.     

A FUNCTIONAL CENTRAL LIMIT THEOREMFOR NEGATIVELY ASSOCIATED SEQUENCE

Let(x1,j≥1)be a sequence of negatively associated random variables with ex1=o,ex^21＜∞.in this paper a functional central limit theorem for negatively associated random variables under some conditions withbout stationarity is proved which is the same as the results for positively associated random variables.     

On the remained term for the central limit theorem

Summary An upper bound for the remainder term of the Edgeworth expansion for the distribution of the normalized sum of independent and identically distributed random variables is given in terms of 3rd and 4th order moments, together with the total variation of the probability density function of the underlying distribution. The Institute of Statistical mathematics     

On the central limit theorem for lacunary trigonometric series

Хорошо известно, что в ероятностное поведе ние лакунарного тригоно метрического ряда {cos 2πn _kx} тесно связ ано с «критическим» у словием лакунарности (*) $$\frac{{n_{k + 1} }}{{n_k }} \geqq 1 + \frac{{c_k }}{{\sqrt k }},c_k \to \infty $$ . Например, если выполн ено условие (*), то последовательность {cos2πn _kx} удовлетворяет центральной предель ной теореме, и при этом условие (*) не может быть ослабле но. Для последовательносте й, удовлетворяющих (*), и звестны и другие результаты по добного рода, в то время как для более медленно расту щих последовательносте й {n_k} не известно, по-видимому, ничего. В с татье развит метод, ко торый при помощи мартингально й техники позволяет проводить исследование систем {cos 2πn_kx} для последовательно стей, не удовлетворяю щих условию (*). Получено про стое объяснение условия (*), изучено, как «пропа-дает» центральная предель ная теорема при посте пенном ослаблении условия (*) и дока-заны некоторые центральн ые предельные теорем ы в отсутствие этого усл овия. Получены другие предельные те оремы для {cos 2πn_kx}, напри мер, закон повторного лог арифма и принципы инвариантн ости.     

A functional central limit theorem for a generalized occupancy problem

A functional central limit theorem is proved for a class of finitely exchangeable random variables which are based on an occupancy scheme.