## The Annals of Probability

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

T. P. Hill

#### 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

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