On the Role of the Law of Large Numbers in the Theory of Randomness

On the Role of the Law of Large Numbers in the Theory of Randomness In the first part of this article, we answer Kolmogorov's question (stated in 1963 in [1]) about exact conditions for the existence of random generators. Kolmogorov theory of complexity permits of a precise definition of the notion of randomness for an individual sequence. For infinite sequences, the property of randomness is a binary property, a sequence can be random or not. For finite sequences, we can solely speak about a continuous property, a measure of randomness. Is it possible to measure randomness of a sequence t by the extent to which the law of large numbers is satisfied in all subsequences of t obtained in an “admissible way”? The case of infinite sequences was studied in [2]. As a measure of randomness (or, more exactly, of nonrandomness) of a finite sequence, we consider the specific deficiency of randomness δ (Definition 5). In the second part of this paper, we prove that the function δ/ln(1/δ) characterizes the connections between randomness of a finite sequence and the extent to which the law of large numbers is satisfied. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

On the Role of the Law of Large Numbers in the Theory of Randomness

Loading next page...
 
/lp/springer_journal/on-the-role-of-the-law-of-large-numbers-in-the-theory-of-randomness-HZkoXtmGGG
Publisher
Kluwer Academic Publishers-Plenum Publishers
Copyright
Copyright © 2003 by MAIK “Nauka/Interperiodica”
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1023/A:1023638717091
Publisher site
See Article on Publisher Site

Abstract

In the first part of this article, we answer Kolmogorov's question (stated in 1963 in [1]) about exact conditions for the existence of random generators. Kolmogorov theory of complexity permits of a precise definition of the notion of randomness for an individual sequence. For infinite sequences, the property of randomness is a binary property, a sequence can be random or not. For finite sequences, we can solely speak about a continuous property, a measure of randomness. Is it possible to measure randomness of a sequence t by the extent to which the law of large numbers is satisfied in all subsequences of t obtained in an “admissible way”? The case of infinite sequences was studied in [2]. As a measure of randomness (or, more exactly, of nonrandomness) of a finite sequence, we consider the specific deficiency of randomness δ (Definition 5). In the second part of this paper, we prove that the function δ/ln(1/δ) characterizes the connections between randomness of a finite sequence and the extent to which the law of large numbers is satisfied.

Journal

Problems of Information TransmissionSpringer Journals

Published: Oct 4, 2004

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve Freelancer

DeepDyve Pro

Price
FREE
$49/month

$360/year
Save searches from
Google Scholar,
PubMed
Create lists to
organize your research
Export lists, citations
Read DeepDyve articles
Abstract access only
Unlimited access to over
18 million full-text articles
Print
20 pages/month
PDF Discount
20% off