Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains

Abstract

The asymptotic behaviour of the sequence (𝒞 n (ω), wc,n (ω)/n), is studied where 𝒞 n (ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n ) until time n and wc,n (ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ω c ) which represents uniquely the chain (ξ n ) as a circuit chain, and ω c is given a probabilistic interpretation.

Cycle symmetries and circulation fluctuations for discrete-time and continuous-time Markov chains

“…The cycle representation theory of Markov chains [1][2][3][4][5][6][7] not only possesses rich theoretical contents, but has become a fundamental tool in dealing with nonequilibrium systems in natural sciences as well. We refer to two books [8,9] for the theoretical contents of the cycle representation theory and refer to two papers [10,11] for the applications of the cycle representation theory in physics, chemistry, and biology.…”

“…The earliest theoretical result of the cycle representation theory is probably the Kolmogorov's criterion for reversibility [12], which claims that a stationary Markov chain is reversible if and only if the product of transition probabilities (rates) along each cycle c and that along its reversed cycle c− are the same. Illuminated by the diagram method [13,14] developed by Hill in his study of cycle kinetics in biochemical systems, the Qians' [1][2][3][4][5] and Kalpazidou [6,9] introduced the important concept of circulations for Markov chains and further enriched the cycle representation theory. Let N c t denote the number of cycle c formed by a Markov chain up to time t. The circulation J c of cycle c is a nonnegative real number defined as the following almost sure limit: Recently, biophysicists have applied the cycle representation theory to study single-molecule enzyme kinetics and found an interesting relation named as the generalized Haldane equality [11,[15][16][17].…”

Orthogonal cycle transforms of stochastic matrices

“…The irreducibility of P means that the graph G is connected; that is, for any pair ( Let C denote the collection of all directed circuits of G. Then according to [2]-[41 and [6] the matrix P is decomposed by the circuits c 6 C as follows:…”

“…Equations (2) are called the cycle representation formulas for the transition probabilities pq, i, j ~ S.…”

“…As formal expressions, the cycle representation equations (2) lead to the question of whether or not these equations have a Hilbert space interpretation as decompositions on orthonormal collections. One approach to this question is given in the context of the theory of the unitary dilations of Riesz and Nagy (see [7] and [4]), Correspondingly, it is shown in [4] that the trigonometric functions may decompose P and the wc into classic Fourier series.…”