## The Annals of Probability

- Ann. Probab.
- Volume 10, Number 3 (1982), 828-830.

### Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely

#### Abstract

Suppose that for every independent sequence of random variables satisfying some hypothesis condition $H$, it follows that the partial sums converge almost surely. Then it is shown that for every arbitrarily-dependent sequence of random variables, the partial sums converge almost surely on the event where the conditional distributions (given the past) satisfy precisely the same condition $H$. Thus many strong laws for independent sequences may be immediately generalized into conditional results for arbitrarily-dependent sequences.

#### Article information

**Source**

Ann. Probab., Volume 10, Number 3 (1982), 828-830.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993792

**Digital Object Identifier**

doi:10.1214/aop/1176993792

**Mathematical Reviews number (MathSciNet)**

MR659552

**Zentralblatt MATH identifier**

0486.60028

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60G45

**Keywords**

Conditional Borel-Cantelli Lemma conditional three-series theorem conditional strong laws martingales arbitrarily-dependent sequences of random variables almost sure convergence of partial sums

#### Citation

Hill, T. P. Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely. Ann. Probab. 10 (1982), no. 3, 828--830. doi:10.1214/aop/1176993792. https://projecteuclid.org/euclid.aop/1176993792