scholarly journals Almost Sure Limit Theorems for Multivariate General Standard Normal Sequences and Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Zhicheng Chen ◽  
Xinsheng Liu
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Almost Sure Convergence ◽  
Central Limit Theorems ◽  
Random Vectors ◽  
Standard Normal ◽  
Almost Sure Limit Theorems ◽  
General Standard

Under suitable conditions, the almost sure central limit theorems for the maximum of general standard normal sequences of random vectors are proved. The simulation of the almost sure convergence for the maximum is firstly performed, which helps to visually understand the theorems by applying to two new examples.

scholarly journals On central limit theorems for branching processes with dependent immigration

2020 ◽  
pp. 7-15
Author(s):  
V. Golomoziy ◽  
S. Sharipov
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Branching Processes ◽  
Central Limit Theorems ◽  
Standard Normal Distribution ◽  
Immigration Process ◽  
Standard Normal ◽  
Mean And Variance ◽  
The Mean ◽  
Regularly Varying

In this paper we consider subcritical and supercritical discrete time branching processes with generation dependent immigration. We prove central limit theorems for fluctuation of branching processes with immigration when the mean of immigrating individuals tends to infinity with the generation number and immigration process is m−dependent. The first result states on weak convergence of the fluctuation subcritical branching processes with m−dependent immigration to standard normal distribution. In this case, we do not assume that the mean and variance of immigration process are regularly varying at infinity. In contrast, in Theorem 3.2, we suppose that the mean and variance are to be regularly varying at infinity. The proofs are based on direct analytic method of probability theory.

scholarly journals On the random functional central limit theorems with almost sure convergence for subsequences

2012 ◽  
Vol 45 (2) ◽  
Author(s):  
Zdzisław Rychlik ◽  
Konrad S. Szuster
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Central Limit Theorems ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Random Sum ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

AbstractIn this paper we present functional random-sum central limit theorems with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related ‘logarithmic’ limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random-sum central limit theorems for subsequences.

scholarly journals Almost sure convergence for self-normalized products of sums of partial sums of ρ¯-mixing sequences

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2471-2488
Author(s):  
Qunying Wu ◽  
Yuanying Jiang
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Almost Sure Convergence ◽  
Stationary Sequence ◽  
Central Limit Theorems ◽  
The Self ◽  
Partial Sums ◽  
Independent Case ◽  
Products Of Sums ◽  
Mixing Sequences

Let X,X1,X2,... be a stationary sequence of ??-mixing positive random variables. A universal result in the area of almost sure central limit theorems for the self-normalized products of sums of partial sums (?kj =1(Tj/(j(j+1)?/2)))?=(?Vk) is established, where: Tj = ?ji=1 Si,Si = ?i k=1 Xk,Vk = ??ki=1 X2i,? = EX, ? > 0. Our results generalize and improve those on almost sure central limit theorems obtained by previous authors from the independent case to ??-mixing sequences and from partial sums case to self-normalized products of sums of partial sums.

scholarly journals High-dimensional limit theorems for random vectors in ℓpn-balls

2019 ◽  
Vol 21 (01) ◽  
pp. 1750092 ◽  
Author(s):  
Zakhar Kabluchko ◽  
Joscha Prochno ◽  
Christoph Thäle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Large Deviation ◽  
Rate Function ◽  
Central Limit Theorems ◽  
High Dimensional ◽  
Random Vectors ◽  
Random Direction

In this paper, we prove a multivariate central limit theorem for [Formula: see text]-norms of high-dimensional random vectors that are chosen uniformly at random in an [Formula: see text]-ball. As a consequence, we provide several applications on the intersections of [Formula: see text]-balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an [Formula: see text]-ball onto a line spanned by a random direction [Formula: see text]. The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreover, we complement our central limit theorems by providing a complete description of the large deviation behavior, which covers fluctuations far beyond the Gaussian scale. In the regime [Formula: see text] this displays in speed and rate function deviations of the [Formula: see text]-norm on an [Formula: see text]-ball obtained by Schechtman and Zinn, but we obtain explicit constants.

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