## The Annals of Statistics

- Ann. Statist.
- Volume 3, Number 4 (1975), 939-946.

### A Finite Memory Test of the Irrationality of the Parameter of a Coin

Patrick Hirschler and Thomas M. Cover

#### Abstract

Let $X_1,X_2,...$ be a Bernoulli sequence with parameter p. An algorithm $T_{n=1}=\\f(T_n,X_n,n)$; $d_n = d(T_n); \\f:\{1,2,\1dots,8\} \times \{0,1\} \times \{0,1, \1dots}\rightarrow \{1, \1dots, 8\}; d:\{1,2,\dots,8\} \rigtharrow \{H_0,H_1\}$; is found such that $d(T_n)= H_0$ all but a finite number of times with probability one if p is rational, and $d(T_n)= H_1$ all but a finite number of times with probability one if p is irrational (and not in a given null set of irrationals). Thus, an 8-state memory with a time-varying algorithm makes only a finite number of mistakes with probability one on determining the rationality of the parameter of a coin. Thus, determining the rationality of the Bernoulli parameter p does not depend on infinite memory of the data.

#### Article information

**Source**

Ann. Statist., Volume 3, Number 4 (1975), 939-946.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343194

**Digital Object Identifier**

doi:10.1214/aos/1176343194

**Mathematical Reviews number (MathSciNet)**

MR388695

**Zentralblatt MATH identifier**

0325.62010

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62C99: None of the above, but in this section

**Keywords**

Finite memory coin hypothesis testing rationals

#### Citation

Hirschler, Patrick; Cover, Thomas M. A Finite Memory Test of the Irrationality of the Parameter of a Coin. Ann. Statist. 3 (1975), no. 4, 939--946. doi:10.1214/aos/1176343194. https://projecteuclid.org/euclid.aos/1176343194