## Electronic Journal of Probability

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

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 46, 21 pp.

Dates
Accepted: 18 June 2012
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465062368

Digital Object Identifier
doi:10.1214/EJP.v17-2092

Mathematical Reviews number (MathSciNet)
MR2946153

Zentralblatt MATH identifier
1245.60028

Rights

#### Citation

Kevei, Peter; Mason, David. The asymptotic distribution of randomly weighted sums and self-normalized sums. Electron. J. Probab. 17 (2012), paper no. 46, 21 pp. doi:10.1214/EJP.v17-2092. https://projecteuclid.org/euclid.ejp/1465062368

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