Weak convergence of mutually independent $Xₙ^B$ and $Xₙ^A$ under weak convergence of $Xₙ ≡ Xₙ^B - Xₙ^A$

W. Szczotka

Weak convergence of mutually independent

X ₙ^{B}

and

X ₙ^{A}

under weak convergence of

X ₙ \equiv X ₙ^{B} - X ₙ^{A}

W. Szczotka

Applicationes Mathematicae, Tome 33 (2006), p. 41-49 / Harvested from The Polish Digital Mathematics Library

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Résumé

For each n ≥ 1, let $v_{n, k}, k \geq 1$ and $u_{n, k}, k \geq 1$ 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 $X ₙ^{B} (t) = (1 / c ₙ) \sum_{j = 1}^{[n t]} (v_{n, j} - v ̅ ₙ)$ , $X ₙ^{A} (t) = (1 / c ₙ) \sum_{j = 1}^{[n t]} (u_{n, j} - u ̅ ̅ ₙ)$ , t ≥ 0, and $X ₙ = X ₙ^{B} - X ₙ^{A}$ . The main result gives conditions under which the weak convergence $X ₙ \overset{}{\to} X$ , where X is a Lévy process, implies $X ₙ^{B} \overset{}{\to} X^{B}$ and $X ₙ^{A} \overset{}{\to} X^{A}$ , where $X^{B}$ and $X^{A}$ are mutually independent Lévy processes and $X = X^{B} - X^{A}$ .

Publié le : 2006-01-01

Zbl 1105.60020

EUDML-ID : urn:eudml:doc:279572

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     author = {W. Szczotka},
     title = {Weak convergence of mutually independent $Xn^B$ and $Xn^A$ under weak convergence of $Xn [?] Xn^B - Xn^A$
            },
     journal = {Applicationes Mathematicae},
     volume = {33},
     year = {2006},
     pages = {41-49},
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     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am33-1-3}
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W. Szczotka. Weak convergence of mutually independent $Xₙ^B$ and $Xₙ^A$ under weak convergence of $Xₙ ≡ Xₙ^B - Xₙ^A$
            . Applicationes Mathematicae, Tome 33 (2006) pp. 41-49. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am33-1-3/