The Classical Central Limit Theorem

In this chapter, the first approach is made to establishing the convergence of scaled random sums, considering independent sequences. The classic Lindeberg–Lévy, Khinchine, Lindeberg–Feller, and Liapunov theorems are proved. The main focus is on the treatment of heterogeneous summands, applying the Lindeberg condition, and extensions are given to allow trending (growing or shrinking) variances. The final sections review cases of the central limit theorem under non-standard conditions and α‎-stable convergence.

VICTORIA transform, RESPECT and REFORM methods for the proof of the G-permanent pencil law under G-Lindeberg condition for some random matrices from G-elliptic ensemble

The G-pencil law under the G-Lindeberg condition for a random matrix is proven.

Donsker’s fuzzy invariance principle under the Lindeberg condition

We prove an analogue of the Donsker theorem under the Lindeberg condition in a fuzzy setting. Specifically, we consider a certain triangular system of d-dimensional fuzzy random variables { X n , i ∗ } , $\begin{array}{} \{X_{n,i}^*\}, \end{array}$ n ∈ ℕ and i = 1, 2, …, kn , which take as their values fuzzy vectors of compact and convex α-cuts. We show that an appropriately normalized and interpolated sequence of partial sums of the system may be associated with a time-continuous process defined on the unit interval t ∈ [0, 1] which, under the assumption of the Lindeberg condition, tends in distribution to a standard Brownian motion in the space of support functions.

A Dependent Lindeberg Central Limit Theorem for Cluster Functionals on Stationary Random Fields

In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes (Zn(f))f∈F whose index set F is a family of cluster functionals valued on blocks of values of a stationary random field. The practicality and applicability of the result depend mainly on the usual Lindeberg condition and on a sequence Tn which summarizes the dependence between the blocks of the random field values. Finally, in application, we use the previous result in order to show the Gaussian asymptotic behavior of the proposed iso-extremogram estimator.

VICTORIA transform, RESPECT and REFORM methods for the proof of the G-Elliptic Law under G-Lindeberg condition and twice stochastic condition for the variances and covariances of the entries of some random matrices

The G-Elliptic law under the G-Lindeberg condition for the independent pairs of the entries of a random matrix is proven.

𝑉-law for random block matrices under the generalized Lindeberg condition

The V-law under generalized Lindeberg condition for the independent blocks of random matrices having double stochastic matrix of covariances and different expectations of their array is proven.

V-density for eigenvalues of random block matrices with independent blocks whose entries have different variances and expectations

V-density under Lindeberg condition for the independent blocks of random matrices having different variances and expectations is found.

On Lindeberg-Feller Limit Theorem

In the Lindeberg-Feller theorem, the Lindeberg condition is present. The fulfillment of this condition must be checked for any ε > 0. We formulae a new condition in terms of some generalization of moments of order 2 + α, which does not depend on ε, and show that this condition is equivalent to the Lindeberg condition, and if this condition is valid for some α > 0 then it is valid for any α > 0. In the non-classical setting (in the absence of conditions of a uniform infinitely smallness) V.I. Rotar formulated an analogue of the Lindeberg condition in terms of the second pseudo-moments. The paper presents the same modification of Rotar`s condition, which does not depend on ε. In addition, we discuss variants of the simple proofs of theorems of Lindeberg-Feller and Rotar and some related inequalities.

The limit G-Law for the solutions of systems of linear algebraic equations with independent random coefficients under the G-Lindeberg condition

The limit G-Law for the solutions of the systems of linear algebraic equations with independent random coefficients is proven under the G-Lindeberg condition.