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The asymptotic distribution of randomly weighted sums and self-normalized sums

   
@article{EJP2092,
	author = {Peter Kevei and David Mason},
	title = {The asymptotic distribution of randomly weighted sums and self-normalized sums},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {17},
	year = {2012},
	keywords = {Self-normalized sums; Feller class; stable distributions},
	abstract = {We consider the self-normalized sums $T_{n}=\sum_{i=1}^{n}X_{i}Y_{i}/\sum_{i=1}^{n}Y_{i}$, where $\{ Y_{i} : i\geq 1 \}$ are non-negative i.i.d.~random variables, and $\{ X_{i} : i\geq 1 \}  $ are i.i.d. random variables, independent of $\{ Y_{i} : i \geq 1 \}$. The main result of the paper is that each subsequential limit law of $T_n$ is continuous for any non-degenerate $X_1$ with finite expectation, if and only if $Y_1$ is in the centered Feller class.},
	pages = {no. 46, 1-21},
	issn = {1083-6489},
	doi = {10.1214/EJP.v17-2092},    
        url = {http://ejp.ejpecp.org/article/view/2092}}