14 research outputs found
Random-step Markov processes
We explore two notions of stationary processes. The first is called a
random-step Markov process in which the stationary process of states, has a stationary coupling with an independent process on the
positive integers, of `random look-back distances'.
That is,
is independent of the `past states', , and for every
positive integer , the probability distribution on the `present', ,
conditioned on the event and on the past is the same as the
probability distribution on conditioned on the `-past', and . A random Markov process is a generalization of a
Markov chain of order and has the property that the distribution on the
present given the past can be uniformly approximated given the -past, for
sufficiently large. Processes with the latter property are called uniform
martingales, closely related to the notion of a `continuous -function'.
We show that every stationary process on a countable alphabet that is a
uniform martingale and is dominated by a finite measure is also a random Markov
process and that the random variables and associated
coupling can be chosen so that the distribution on the present given the
-past and the event is `deterministic': all probabilities are
in . In the case of finite alphabets, those random-step Markov
processes for which can be chosen with finite expected value are
characterized. For stationary processes on an uncountable alphabet, a stronger
condition is also considered which is sufficient to imply that a process is a
random Markov processes. In addition, a number of examples are given throughout
to show the sharpness of the results.Comment: 31 page
The Linus Sequence
Define the Linus sequence Ln for n ≥ 1 as a 0–1 sequence with L1 = 0, and Ln chosen so as to minimize the length of the longest immediately repeated block Ln−2r+1 Ln−r = Ln−r+1 Ln. Define the Sally sequence Sn as the length r of the longest repeated block that was avoided by the choice of Ln. We prove several results about these sequences, such as exponential decay of the frequency of highly periodic subwords of the Linus sequence, zero entropy of any stationary process obtained as a limit of word frequencies in the Linus sequence and infinite average value of the Sally sequence. In addition we make a number of conjectures about both sequences
An Outline of Ergodic Theory
An engaging introduction to ergodic theory for graduate students, and a useful reference for the professional mathematician