# Lithuanian Mathematical Journal

Lithuanian Mathematical Journal > 2016 > 56 > 1 > 77-106

*G*:= (

*X*

_{1},…,

*X*

_{ n }) be a vector of standard Gaussian random variables. For a function

*H*: ℝ → ℝ, we consider the normalized weighted

*H*-sum of

*G*, and under suitable hypotheses on

*G*and

*H*, we prove a Berry–Esséentype bound for it. We also prove a functional central limit theorem for a partial-sum process corresponding to such sums. The proof of the former theorem is based on a bound of the Kolmogorov...

Lithuanian Mathematical Journal > 2016 > 56 > 1 > 107-113

*A*-Darboux functions, namely, the class of functions

*f*: ℝ

*→*ℝ such that for all

*a, b ∈*ℝ with

*a < b*and each

*y*between

*f*(

*a*) and

*f*(

*b*)

*,*there is a point

*x*

_{0}

*∈*(

*a, b*)

*∩ A*(where

*A*is a nonempty fixed subset of ℝ) such that

*f*(

*x*

_{0}) =

*y*. Furthermore, we generalize the notion of the

*A*-Darboux property for functions mapping a topological space into a topological space.

Lithuanian Mathematical Journal > 2016 > 56 > 1 > 1-15

Lithuanian Mathematical Journal > 2016 > 56 > 1 > 16-31

Lithuanian Mathematical Journal > 2016 > 56 > 1 > 127-132

*L*-function to study the mean value properties of the Cochrane sums and give an interesting mean square value formula for it.

Lithuanian Mathematical Journal > 2016 > 56 > 1 > 49-59

*X*

_{1},…,

*X*

_{ n }with corresponding distributions

*F*

_{1},…,

*F*

_{ n }such that

*X*

_{1},…,

*X*

_{ n }admit some dependence structure and

*n*

^{−1}(

*F*

_{1}+· · ·+

*F*

_{ n }) belongs to the class of dominatedly varyingtailed distributions. We establish weak equivalence relations among

**P**(

*S*

_{ n }>

*x*),

**P**(max{

*X*

_{1},…,

*X*

_{ n }} >

*x*),

**P**(max{

*S*

_{1},…,

*S*

_{ n }} >

*x*), and ∑ k = 1 n F k ¯ ...

Lithuanian Mathematical Journal > 2016 > 56 > 1 > 133-141

Lithuanian Mathematical Journal > 2016 > 56 > 1 > 72-76

Lithuanian Mathematical Journal > 2016 > 56 > 1 > 114-126

*L*

_{ s }norms (1 ≪

*s*≪

*∞*) of the normal approximation of random variables the characteristic functions of which satisfy some linear homogeneous differential equation. Note that this differential equation is satisfied, for example, for Poisson, Gamma, binomial, negative binomial random variables, Poisson random sums, binomial random sums, negative binomial random sums, and...

Lithuanian Mathematical Journal > 2016 > 56 > 1 > 32-48

Lithuanian Mathematical Journal > 2016 > 56 > 1 > 60-71