# Long range dependent models in information theory

### Barlas Oguz

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2012-204

October 23, 2012

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-204.pdf

Long range dependence refers to stochastic processes for which correlations persist at much longer time scales as compared to traditional models. For such processes the central limit theorem does not in general hold, and the smoothing effect of the law of large numbers takes more time to settle in. Such phenomena have been observed in many different fields including financial time series, DNA sequences, network traffic and variable bit-rate video. The bursty nature and persistent correlation structure of long range dependent processes make them tough to control and predict in practice, and tough to analyze in theory. In this thesis we look at the origins of long range dependence through the use of Markov models.

We first introduce a model of long range dependence using countable state Markov chains. A positive recurrent, aperiodic Markov chain is said to be long range dependent (LRD) when the indicator function of a particular state is LRD. This happens if and only if the return time distribution for that state has infinite variance. We investigate the question of whether other instantaneous functions of the Markov chain also inherit this property. We provide conditions under which the function has the same degree of long range dependence as the chain itself. We illustrate our results through three examples in diverse fields: queuing networks, source compression, and finance. We then prove information-theoretic pointwise lossless source coding theorems for a class of sources constructed from this model. We are able to show that the code length process at the output of an encoder inherits the long range dependent nature of the source irrespective of the coding algorithm chosen. We extend our results to lossy source coding under suitable conditions, demonstrating quite generally the information-theoretic relevance of long range dependence.

**Advisor:** Venkat Anantharam

BibTeX citation:

@phdthesis{Oguz:EECS-2012-204, Author = {Oguz, Barlas}, Title = {Long range dependent models in information theory}, School = {EECS Department, University of California, Berkeley}, Year = {2012}, Month = {Oct}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-204.html}, Number = {UCB/EECS-2012-204}, Abstract = {Long range dependence refers to stochastic processes for which correlations persist at much longer time scales as compared to traditional models. For such processes the central limit theorem does not in general hold, and the smoothing effect of the law of large numbers takes more time to settle in. Such phenomena have been observed in many different fields including financial time series, DNA sequences, network traffic and variable bit-rate video. The bursty nature and persistent correlation structure of long range dependent processes make them tough to control and predict in practice, and tough to analyze in theory. In this thesis we look at the origins of long range dependence through the use of Markov models. We first introduce a model of long range dependence using countable state Markov chains. A positive recurrent, aperiodic Markov chain is said to be long range dependent (LRD) when the indicator function of a particular state is LRD. This happens if and only if the return time distribution for that state has infinite variance. We investigate the question of whether other instantaneous functions of the Markov chain also inherit this property. We provide conditions under which the function has the same degree of long range dependence as the chain itself. We illustrate our results through three examples in diverse fields: queuing networks, source compression, and finance. We then prove information-theoretic pointwise lossless source coding theorems for a class of sources constructed from this model. We are able to show that the code length process at the output of an encoder inherits the long range dependent nature of the source irrespective of the coding algorithm chosen. We extend our results to lossy source coding under suitable conditions, demonstrating quite generally the information-theoretic relevance of long range dependence.} }

EndNote citation:

%0 Thesis %A Oguz, Barlas %T Long range dependent models in information theory %I EECS Department, University of California, Berkeley %D 2012 %8 October 23 %@ UCB/EECS-2012-204 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-204.html %F Oguz:EECS-2012-204