Abstract
Quasi-logarithmic combinatorial structures are a class of decomposable combinatorial structures which extend the logarithmic class considered by Arratia, Barbour and Tavaré (2003). In order to obtain asymptotic approximations to their component spectrum, it is necessary first to establish an approximation to the sum of an associated sequence of independent random variables in terms of the Dickman distribution. This in turn requires an argument that refines the Mineka coupling by incorporating a blocking construction, leading to exponentially sharper coupling rates for the sums in question. Applications include distributional limit theorems for the size of the largest component and for the vector of counts of the small components in a quasi-logarithmic combinatorial structure.
Highlights
Many of the classical random decomposable combinatorial structures, such as random permutations and random polynomials over a finite field, have component structure satisfying a conditioning relation: if Ci(n) denotes the number of components of size i, the distribution of the vector of component counts (C1(n), . . . , Cn(n)) of a structure of size n can be expressed asC1(n), . . . , Cn(n) = Z1, . . . , Zn T0,n = n, (1.1)where (Zi, i ≥ 1) is a fixed sequence of independent non-negative integer valued random variables, and Ta,n := n i=a+1 i Zi, ≤ a < n.If, as in the examples above, theZi satisfy iP[Zi = 1] → θ and iEZi → θ
We say that a decomposable combinatorial structure satisfies the quasi–logarithmic condition QLC if it satisfies the Conditioning Relation (1.1), if lim i→∞
We say that a decomposable combinatorial structure satisfies the quasi–logarithmic condition QLC2 if it satisfies the Conditioning Relation (1.1), if μi = O(i−α); δ(m, θ ) = O(m−β ) for some θ, α, β > 0, and if, for some r, s coprime, ψ > 0 and D defined in (2.5), (2.4) is satisfied with E for E
Summary
Many of the classical random decomposable combinatorial structures, such as random permutations and random polynomials over a finite field, have component structure satisfying a conditioning relation: if Ci(n) denotes the number of components of size i, the distribution of the vector of component counts (C1(n), . . . , Cn(n)) of a structure of size n can be expressed as. Knopfmacher (1979) introduced the notion of additive arithmetic semigroups, which give rise to decomposable combinatorial structures satisfying the conditioning relation, with negative binomially distributed Zi For these structures, iP[Zi = 1] ∼ iEZi = θi, where the θi do not always converge to a limit as i → ∞. Zan) can be bounded, even without assuming that the θj’s converge on average to any fixed θ , as long as they are bounded and bounded away from 0 (we do not require the latter condition) He considers only the case of Poisson distributed Zi, for which, inspecting the proof of Theorem 4.3, it is enough to obtain an estimate of the form n|P[Tan,n = n − k] − P[Tan,n = n − l]| ≤ C{|k − l|/n}γ,. Since we are interested in approximating the distribution of the largest components, for which some form of convergence to a θ seems necessary, we do not attempt this refinement
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