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What Is a Random Sequence?Author(s): Benjamin WeissSource: Advances in Applied Probability, Vol. 9, No. 2 (Jun., 1977), p. 224Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426368 .
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224 6TH CONFERENCE ON STOCHASTIC PROCESSES AND THEIR APPLICATIONS 224 6TH CONFERENCE ON STOCHASTIC PROCESSES AND THEIR APPLICATIONS 224 6TH CONFERENCE ON STOCHASTIC PROCESSES AND THEIR APPLICATIONS
of B to those of B. Each sequence is decomposed, in a nested way, into n-blocks. This is done with the aid of marker strings. Then n-blocks are mapped into n-blocks in a consistent way.
The codes constructed have finite expected length. This means that using such a code, we can get from a typical sequence of B a typical sequence of B where each finite part of the sequence of B is determined after a time length which is
proportional to its length.
What is a random sequence?
BENJAMIN WEISS, Institute for Advanced Studies, Jerusalem
In its simplest form the title question is: when should we consider a sequence (xi, x2,. * * ) of zeroes and ones to be 'completely random', that is to say a 'typical' outcome of an infinite sequence of independent coin-tossings with equally likely probabilities of zero and one. The answer given by R. von Mises was that a random sequence, or kollektiv, is one such that for any subsequence determined
by a 'selection rule' the asymptotic frequency of zero exists and equals ?. Ergodic theory suggests that we consider a sequence to be 'typical' if it is a generic point for the corresponding probability measure, or in our case-a sequence that
corresponds to a normal number in E. Borel's terminology. Adopting this as a definition we investigate the class of selection rules that operate, i.e. select out from any random sequence a new random sequence. Among the results that we will describe are (1) a complete characterization of the constant selection rules that operate, and (2) a class of recurrent selection rules that operate-this class contains the rules given by finite automata.
II E. Markov and Semi-Markov Processes
On a-recurrent semi-Markov processes and a-invariant measures
E. ARJAS, University of Oulu
An a-recurrent Markov process on a countable state-space I and with
transition probabilities P,i(t) is known to possess an (essentially unique) a- invariant measure 7r = (Ti ) satisfying S, ir Pj(t) = e-"'rr for all j E I, t 0.
According to Cheong (1970) this does not generalize to a-recurrent semi-
Markov processes {X(t)} on I. A generalization is achieved, however, by
considering the bivariate process {X(t), V-(t)}, where V-(t) is the backward
of B to those of B. Each sequence is decomposed, in a nested way, into n-blocks. This is done with the aid of marker strings. Then n-blocks are mapped into n-blocks in a consistent way.
The codes constructed have finite expected length. This means that using such a code, we can get from a typical sequence of B a typical sequence of B where each finite part of the sequence of B is determined after a time length which is
proportional to its length.
What is a random sequence?
BENJAMIN WEISS, Institute for Advanced Studies, Jerusalem
In its simplest form the title question is: when should we consider a sequence (xi, x2,. * * ) of zeroes and ones to be 'completely random', that is to say a 'typical' outcome of an infinite sequence of independent coin-tossings with equally likely probabilities of zero and one. The answer given by R. von Mises was that a random sequence, or kollektiv, is one such that for any subsequence determined
by a 'selection rule' the asymptotic frequency of zero exists and equals ?. Ergodic theory suggests that we consider a sequence to be 'typical' if it is a generic point for the corresponding probability measure, or in our case-a sequence that
corresponds to a normal number in E. Borel's terminology. Adopting this as a definition we investigate the class of selection rules that operate, i.e. select out from any random sequence a new random sequence. Among the results that we will describe are (1) a complete characterization of the constant selection rules that operate, and (2) a class of recurrent selection rules that operate-this class contains the rules given by finite automata.
II E. Markov and Semi-Markov Processes
On a-recurrent semi-Markov processes and a-invariant measures
E. ARJAS, University of Oulu
An a-recurrent Markov process on a countable state-space I and with
transition probabilities P,i(t) is known to possess an (essentially unique) a- invariant measure 7r = (Ti ) satisfying S, ir Pj(t) = e-"'rr for all j E I, t 0.
According to Cheong (1970) this does not generalize to a-recurrent semi-
Markov processes {X(t)} on I. A generalization is achieved, however, by
considering the bivariate process {X(t), V-(t)}, where V-(t) is the backward
of B to those of B. Each sequence is decomposed, in a nested way, into n-blocks. This is done with the aid of marker strings. Then n-blocks are mapped into n-blocks in a consistent way.
The codes constructed have finite expected length. This means that using such a code, we can get from a typical sequence of B a typical sequence of B where each finite part of the sequence of B is determined after a time length which is
proportional to its length.
What is a random sequence?
BENJAMIN WEISS, Institute for Advanced Studies, Jerusalem
In its simplest form the title question is: when should we consider a sequence (xi, x2,. * * ) of zeroes and ones to be 'completely random', that is to say a 'typical' outcome of an infinite sequence of independent coin-tossings with equally likely probabilities of zero and one. The answer given by R. von Mises was that a random sequence, or kollektiv, is one such that for any subsequence determined
by a 'selection rule' the asymptotic frequency of zero exists and equals ?. Ergodic theory suggests that we consider a sequence to be 'typical' if it is a generic point for the corresponding probability measure, or in our case-a sequence that
corresponds to a normal number in E. Borel's terminology. Adopting this as a definition we investigate the class of selection rules that operate, i.e. select out from any random sequence a new random sequence. Among the results that we will describe are (1) a complete characterization of the constant selection rules that operate, and (2) a class of recurrent selection rules that operate-this class contains the rules given by finite automata.
II E. Markov and Semi-Markov Processes
On a-recurrent semi-Markov processes and a-invariant measures
E. ARJAS, University of Oulu
An a-recurrent Markov process on a countable state-space I and with
transition probabilities P,i(t) is known to possess an (essentially unique) a- invariant measure 7r = (Ti ) satisfying S, ir Pj(t) = e-"'rr for all j E I, t 0.
According to Cheong (1970) this does not generalize to a-recurrent semi-
Markov processes {X(t)} on I. A generalization is achieved, however, by
considering the bivariate process {X(t), V-(t)}, where V-(t) is the backward
This content downloaded from 91.229.229.49 on Sat, 14 Jun 2014 03:46:34 AMAll use subject to JSTOR Terms and Conditions
