Frequency of symbol occurrences in bicomponent stochastic models

Analysis of Symbol Statistics in Bicomponent Rational Models

“…Under these assumptions it is known that if A 1 = 0 = B 1 then 0 < β 1 < 1, 0 < γ 1 and Yn−β 1 n √ γ 1 n converges in distribution to a normal r.v. of mean value 0 and variance 1 [8]. Here we show a Gaussian local limit law for {Y n } with a convergence rate O(n −1/2 ), under the further hypothesis that (A 1 , B 1 ) is aperiodic.…”

“…where c ∈ (0, π) is a constant satisfying relations (8) and q is an arbitrary value such that 1 3 < q < 1 2 . We get the following three propositions where we always assume the hypotheses of Theorem 1.…”

“…Note that simple changes may modify the limit distribution: for instance, setting to 3 the weight of transition 2 In this section we present a local limit theorem for {Y n } defined in an equipotent communicating bicomponent model with β 1 = β 2 and γ 1 = γ 2 . In this case, setting β = β 1 = β 2 and γ = γ 1 +γ 2 2 , it is proved that the distribution of Yn−βn √ γn converges to a mixture of Gaussian laws having mean 0 and variance uniformly distributed over an interval of extremes γ 1 γ and γ 2 γ [8]. Formally, we consider a r.v.…”

“…In this section we study the local limit properties of {Y n } assuming an equipotent communicating bicomponent model with β 1 = β 2 = β and γ 1 = γ 2 = γ. In this case, it is known [8] that Yn−βn √ γn converges in distribution to a Gaussian r.v. of mean 0 and variance 1 and here we present a local limit law with a convergence rate of the order O(n −1/2 ).…”

“…It is known that if A has a primitive transition matrix then Y n has a Gaussian limit distribution [4,16] and, under a suitable aperiodicity condition, it also satisfies a local limit theorem [4]. The limit distribution of Y n in the global sense is known also when the transition matrix of A consists of two primitive components [8].…”