## The Annals of Probability

- Ann. Probab.
- Volume 8, Number 3 (1980), 584-599.

### Domains of Partial Attraction and Tightness Conditions

Naresh C. Jain and Steven Orey

#### Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent, identically distributed, random variables with a common distribution function $F. S_n$ denotes $X_1 + \cdots + X_n$. An increasing sequence of positive integers $(n_i)$ is defined to belong to $\mathscr{N}(F)$ if there exist normalizing sequences $(b_k)$ and $(a_k)$, with $a_k \rightarrow \infty$, so that every subsequence of $(a^{-1}_{n_i} S_{n_i} - b_{n_i})$ has a further subsequence converging in distribution to a nondegenerate limit. The main concern here is a description of $\mathscr{N}(F)$ in terms of $F$. This includes also conditions for $\mathscr{N}(F)$ to be void, as well as for $(1, 2, \cdots)\in \mathscr{N}(F)$, thus improving on some classical results of Doeblin. It is also shown that if there exists a unique type of laws so that $F$ is in the domain of partial attraction of a probability law if and only if the law belongs to that type, then in fact $F$ is in the domain of attraction of these laws.

#### Article information

**Source**

Ann. Probab., Volume 8, Number 3 (1980), 584-599.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994728

**Digital Object Identifier**

doi:10.1214/aop/1176994728

**Mathematical Reviews number (MathSciNet)**

MR573294

**Zentralblatt MATH identifier**

0442.60050

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

**Keywords**

Sums of independent random variables domain of partial attraction domain of attraction tight sequence set of uniform decrease

#### Citation

Jain, Naresh C.; Orey, Steven. Domains of Partial Attraction and Tightness Conditions. Ann. Probab. 8 (1980), no. 3, 584--599. doi:10.1214/aop/1176994728. https://projecteuclid.org/euclid.aop/1176994728