Weak convergence of mutually independent $X{ₙ}^B$ and $X{ₙ}^A$ under weak convergence of $Xₙ\equiv X{ₙ}^B-X{ₙ}^A$

W. Szczotka. "Weak convergence of mutually independent $Xₙ^B$ and $Xₙ^A$ under weak convergence of $Xₙ ≡ Xₙ^B - Xₙ^A$." Applicationes Mathematicae 33.1 (2006): 41-49. <http://eudml.org/doc/279572>.

@article{W2006,
abstract = {For each n ≥ 1, let $\{v_\{n,k\}, k ≥ 1\}$ and $\{u_\{n,k\}, k ≥ 1\}$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $Xₙ^B(t) = (1/cₙ) ∑_\{j=1\}^\{[nt]\}(v_\{n,j\} - v̅ₙ)$, $Xₙ^A(t) = (1/cₙ)∑_\{j=1\}^\{[nt]\}(u_\{n,j\}-u̅̅ₙ)$, t ≥ 0, and $Xₙ = Xₙ^B - Xₙ^A$. The main result gives conditions under which the weak convergence $Xₙ \mathrel \{\mathop \{\rightarrow \}\limits ^\{\}\}X$, where X is a Lévy process, implies $Xₙ^B \mathrel \{\mathop \{\rightarrow \}\limits ^\{\}\}X^B$ and $Xₙ^A\mathrel \{\mathop \{\rightarrow \}\limits ^\{\}\}X^A$, where $X^B$ and $ X^A$ are mutually independent Lévy processes and $X = X^B - X^A$.},
author = {W. Szczotka},
journal = {Applicationes Mathematicae},
keywords = {Lévy process},
language = {eng},
number = {1},
pages = {41-49},
title = {Weak convergence of mutually independent $Xₙ^B$ and $Xₙ^A$ under weak convergence of $Xₙ ≡ Xₙ^B - Xₙ^A$},
url = {http://eudml.org/doc/279572},
volume = {33},
year = {2006},
}

TY - JOUR
AU - W. Szczotka
TI - Weak convergence of mutually independent $Xₙ^B$ and $Xₙ^A$ under weak convergence of $Xₙ ≡ Xₙ^B - Xₙ^A$
JO - Applicationes Mathematicae
PY - 2006
VL - 33
IS - 1
SP - 41
EP - 49
AB - For each n ≥ 1, let ${v_{n,k}, k ≥ 1}$ and ${u_{n,k}, k ≥ 1}$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $Xₙ^B(t) = (1/cₙ) ∑_{j=1}^{[nt]}(v_{n,j} - v̅ₙ)$, $Xₙ^A(t) = (1/cₙ)∑_{j=1}^{[nt]}(u_{n,j}-u̅̅ₙ)$, t ≥ 0, and $Xₙ = Xₙ^B - Xₙ^A$. The main result gives conditions under which the weak convergence $Xₙ \mathrel {\mathop {\rightarrow }\limits ^{}}X$, where X is a Lévy process, implies $Xₙ^B \mathrel {\mathop {\rightarrow }\limits ^{}}X^B$ and $Xₙ^A\mathrel {\mathop {\rightarrow }\limits ^{}}X^A$, where $X^B$ and $ X^A$ are mutually independent Lévy processes and $X = X^B - X^A$.
LA - eng
KW - Lévy process
UR - http://eudml.org/doc/279572
ER -