Zipf and non-Zipf Laws for Homogeneous Markov Chain
Let us consider a homogeneous Markov chain with discrete time and with a finite set of states $E_0,\ldots,E_n$ such that the state $E_0$ is absorbing, states $E_1,\ldots,E_n$ are nonrecurrent. The goal of this work is to study frequencies of trajectories in this chain, i.e., "words" compos...
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Zusammenfassung: | Let us consider a homogeneous Markov chain with discrete time and with a
finite set of states $E_0,\ldots,E_n$ such that the state $E_0$ is absorbing,
states $E_1,\ldots,E_n$ are nonrecurrent. The goal of this work is to study
frequencies of trajectories in this chain, i.e., "words" composed of symbols
$E_1,\ldots,E_n$ ending with the "space" $E_0$.
Let us order words according to their probabilities; denote by $p(t)$ the
probability of the $t$th word in this list. In this paper we prove that in a
typical case the asymptotics of the function $p(t)$ has a power character, and
define its exponent from the matrix of transition probabilities. If this matrix
is block-diagonal, then with some specific values of transition probabilities
the power asymptotics gets (logarithmic) addends. But if this matrix is rather
sparse, then probabilities quickly decrease; namely, the rate of asymptotics is
greater than that of the power one, but not greater than that of the
exponential one. We also establish necessary and sufficient conditions for the
exponential order of decrease and obtain a formula for determining the exponent
from the transition probability matrix and the initial distribution vector. |
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DOI: | 10.48550/arxiv.1207.1872 |