Universality of Asymptotically Ewens Measures on Partitions

We introduce a universality theorem for functionals of measures on partitions which"behave like"the Ewens measure. Various limit theorems for the Ewens measure, most notably the Poisson-Dirichlet limit for the longest parts, the functional central limit theorem for the number of parts, and the Erdos-Turan limit for the product of parts, extend to these asymptotically Ewens measures as easy corollaries. Our major contributions are: (1) extending the classes of measures for which these limit theorems hold; (2) characterising universality by a single, easily-checked criterion; and (3) greatly shortening the proofs of the limit theorems using the Feller coupling.


Introduction
Let P n be the partitions of n ∈ N, which we represent by α = (α 1 , . . . , α n ) ∈ P n , where α i is the number of parts of size i, so that α 1 + 2α 2 + · · · + nα n = n. Let P θ n (α) = n! θ(θ + 1) · · · (θ + n − 1) n i=1 θ αi i αi α i ! (1.1) be the Ewens (1972) measure on P n with parameter θ > 0. The Ewens measure should be interpreted as a random permutation weighted by the number of cycles, under which the α i are asymptotically independent Poisson with parameter θ/i. Any other probability measure P n on P n can be written as a perturbation of the Ewens measure with parameter θ > 0, that is, P n (α) = P θ,η n (α) = η(α) where η : P n → R + is (any multiple of) the Radon-Nikodym derivative of P n with respect to the Ewens measure P θ n , and Z θ,η n is a normalising constant. Since n, θ and η uniquely determine the measure, it will be convenient to use the notation P θ,η n to specify it. We will study partitions by their preimages under the Feller (1945) coupling, which we define as the map {0, 1} N → P n which sends a binary sequence (ξ 1 , ξ 2 , . . .) ∈ {0, 1} N to a partition (α 1 (n), . . . , α n (n)) ∈ P n , where is the number of gaps between 1s of length i − 1 in the string (ξ 1 , ξ 2 , . . . , ξ n , 1). It was shown by Arratia, Barbour and Tavaré (1992) that if P θ F is a measure on {0, 1} N under which the ξ i are independent Bernoulli variables with probability of success θ/(θ +i−1), then the measure induced on P n by the Feller coupling is the Ewens measure P θ n . In particular, we can embed P θ n within P θ F so that α i = α i (n), simultaneously for all n ∈ N. For any m, n ∈ N, (α 1 (n), . . . , α m (n)) is a partition of some integer k ≤ n. Then, for any sequence of measures P θ,η n on P n , n ∈ N, define the weights η n,m = η(α 1 (n), . . . , α m (n)), (1.4) where η is the (unscaled) Radon-Nikodym derivative of P θ,η k with respect to P θ k , for the appropriate choice of k = α 1 (n) + 2α 2 (n) + · · · + mα m (n). If we interpret the Radon-Nikodym derivative as weights, then η n,m should be interpreted as the weight of a partition of n considering only parts of size at most m. Note that since the weights are defined in terms of the α i (n), they are random variables under the Feller measure P θ F .
We call a sequence of measures P θ,η n asymptotically Ewens when: 1. The limits lim n→∞ η n,n and lim m→∞ lim n→∞ η n,m exist and agree in L 1 (P θ F ); and These measures are important in a variety of applications and include many extensively studied measures as special cases. Essentially, they generalise the logarithmic combinatorial structures of Arratia, Barbour and Tavaré (2000) by removing the conditioning relation. We will prove this claim in Section 4, while Section 2 contains background on these measures and the associated limit theorems.
We conclude the introduction by stating our main theorem, the proof of which is presented in Section 3, and showing how the Ewens measure limit theorems extend to asymptotically Ewens measures as easy corollaries.
(1.5) If X n d −→ X under the Ewens measure with parameter θ, then X n d −→ X under any asymptotically Ewens measure with parameter θ.
The criterion (1.5) has a very intuitive interpretation: if the functional does not depend on the first finitely many Feller variables, then it has a universal limit for any asymptotically Ewens measure, depending only on the parameter θ. Corollary 1.2 (Poisson-Dirichlet). Let L n,k be the length of the kth longest part of a partition of n, and let L n = (L n,1 , L n,2 , . . .). Under any asymptotically Ewens measure with parameter θ, L n /n converges in distribution in L 1 (N) to P D(θ), the Poisson-Dirchlet measure with parameter θ (Kingman, 1975(Kingman, , 1977. Proof. Consider strings (1, . . . , 1, ξ i+1 , . . . , ξ n ), for i = d, d − 1, . . . , 0. For each decrement of i, either the partition is unchanged, or a 1-part is deleted and some other part length increases by 1, which does not change the order of parts and thus changes L n by at most 2 in the L 1 (N) norm. Since there are d decrements from i = d to i = 0, max ξ1,...,ξn L n (ξ 1 , . . . , ξ n ) − L n (1, . . . , 1, ξ d+1 , . . . , ξ n ) 1 ≤ 2d.
(1.6) Hence, X n = L n /n satisfies the conditions of Theorem 1.1 applied to the normed space L 1 (N), so the result follows from the Poisson-Dirichlet limit for the Ewens measure (Kingman, 1975(Kingman, , 1977Watterson, 1976). Corollary 1.3 (CLT). Let ν n,t = α 1 + · · · + α n t , 0 ≤ t ≤ 1, be the number of parts of size at most n t in a partition of n. Under any asymptotically Ewens measure with parameter θ, (ν n,t − θt log n)/ √ θ log n converges in distribution in D[0, 1], the Skorohod space of right-continuous left-limit functions on [0, 1], to the standard Brownian motion W t .

Measures on Partitions
Measures on partitions arise naturally from combinatorial objects which consist of components of various sizes. For example, cycles of a random permutation, irreducible factors of a random polynomial or Jordan blocks of a random matrix are all described by measures on partitions when one cares about only the sizes of those components.
The most basic example is a uniformly random permutation, which corresponds to our P θ,η n when θ = 1 and η = 1 identically, and has been the subject of extensive study since the 19th century. The generalisation to θ > 0, still with η = 1 identically, was introduced by Ewens (1972) to model propagation of alleles in population genetics, and represents a random permutation weighted by the number of cycles, or perhaps more intuitively, a permutation formed in a Markov process where cycles are added at a rate θ (Hoppe, 1984).
Their logarithmic combinatorial structures generalise the decomposable combinatorial structures of Flajolet and Soria (1990), which are measures on partitions induced by the uniform measure on families of combinatorial objects determined by sizes of components. We will prove in Section 4 that logarithmic combinatorial structures are exactly asymptotically Ewens measures with weight function in the form above.
An important subclass of logarithmic combinatorial structures are measures with  Some examples of asympototically Ewens measures which are not logarithmic combinatorial structures include many restricted permutations, such as permutations with more even cycles than odd cycles, permutations whose squares have fixed points, or permutations with an even number of cycles; for any underlying measure that is a logarithmic combinatorial structure, these restrictions are asymptotically Ewens.
There are some examples of measures, such as the a-riffle shuffle measures of Diaconis, McGrath and Pitman (1995), and the restricted permutations studied by Lugo (2009), which are not asymptotically Ewens by our current definition, but behave similarly in the sense that they follow the Poisson-Dirichlet limit, as discussed in more detail in Section 2.2. Generalising our measures to include these examples would be an interesting direction for future work.
Finally, there are many measures which are not asymptotically Ewens in any sense, such as the uniform measure on partitions, Pitman's (1992) two-parameter family of measures (although this family includes the Ewens measure as a special case), and the induced measure on partitions from various measures on permutations such as the Plancherel measure and its generalisation, the Schur measures of Okounkov (2001).

The Poisson-Dirichlet Limit
The parameter θ in the Ewens measure corresponds to a rate of formation of new parts (Hoppe, 1984); indeed, θ is the global rate of mutation in Ewens' (1972) original genetic model. This insight extends to the asymptotically Ewens case, where the rate of formation of new parts, appropriately scaled, converges to the parameter θ. This is the intuitive reason why we expect the limit theorems to be universal: with new parts being added at the same rate, the relative sizes of parts should behave similarly.
The key notion that captures this behaviour is the Poisson-Dirichlet limit: the largest parts, normalised by 1 n , converge in distribution on L 1 (N) to a limit known as the Poisson-Dirichlet measure with parameter θ. This was first studied by Kingman (1975), who described it via the Dirichlet distribution on L 1 (N), and Watterson (1976), who found an explicit density.
Historically, Golomb (1964) was the first to calculate the expected value of the longest cycle of a uniformly random permutation, Shepp and Lloyd (1966) found the distributions of the kth longest cycles, and Kingman (1975Kingman ( , 1977 and Watterson (1976) found the joint distribution of longest cycles under the Ewens measure. Hansen (1994) proved the Poisson-Dirichlet limit for decomposable combinatorial structures, while the extension to logarithmic combinatorial structures was made by Arratia, Barbour and Tavaré (1999).
We further generalise the Poisson-Dirichlet limit to asymptotically Ewens measures. However, there are still many other measures which satisfy the Poisson-Dirichlet limit, such as the largest prime factors of a random integer (Knuth and Trabb Pardo, 1976), the a-riffle shuffle measures of Diaconis, McGrath and Pitman (1995), and the restricted permutations studied by Lugo (2009). We expect there to be a fundamental reason why we observe the same limit in these cases, although what that reason should be is currently beyond our grasp.
The Poisson-Dirichlet distribution has a two-parameter generalisation (Pitman and Yor, 1997), which is the limit of the ordered parts of Pitman's (1992) two-parameter family of measures on partitions. Since Pitman's measures are a direct generalisation of the Ewens measure, it seems plausible that our result could be extended in this direction.

Other Limit Theorems
It is classical that the number of i-cycles in a uniformly random permutation are asymptotically independent Poisson with parameter 1/i. The usual proof is by generating functions, which also works for multiplicative weights η(α) = i ζ αi i , where the parameter becomes θζ i /i. Such a proof is given by Betz, Ueltschi and Velenik (2011); see also the book of Arratia, Barbour and Tavaré (2003) for a thorough treatment of generating function techniques in this setting. For logarithmic combinatorial structures, Arratia, Barbour and Tavaré (2000) prove that the number of parts of size i are asymptotically independent, although this result is in some sense one of the defining assumptions of logarithmic combinatorial structures.
The total number of parts was first studied by Goncharov (1942), who found a central limit theorem for the number of cycles in a uniformly random permutation. The functional central limit theorem as seen in Corollary 1.3 was first proved by DeLaurentis and Pittel (1985) for the uniform permutation case, and extended to the Ewens measure by Hansen (1990). Flajolet and Soria (1990) proved a central limit theorem for decomposable combinatorial structures, and the two theorems were unified by Arratia, Barbour and Tavaré (2000), who proved a functional central limit theorem for logarithmic combinatorial structures.
The asymptotic moments of the smallest parts were derived by Shepp and Lloyd (1966). They have not been the subject of extensive study; some facts which are known about them are listed in the book of Arratia, Barbour and Tavaré (2003). The shortest parts depend heavily on the first few Feller variables, and do not fall under the scope of our universality theorem.
It is also possible to canonically order the parts under the Ewens measure by the order in which they appear in the Chinese restaurant coupling. This limit is called the Griffiths-Engen-McCloskey measure (Griffiths, 1979), and behaves similarly to the longest parts; in fact its order statistics exactly follow the Poisson-Dirichlet measure. This limit theorem does not generalise to asymptotically Ewens measures due to the a lack of a canonical order.
The lowest common multiple of parts is a statistic of interest when the partition is induced by a permutation, as it is the group order of the permutation. Turán (1965, 1967) found a central limit theorem for the logarithm of the lowest commmon multiple in the uniform permutation case. Their proof, and all subsequent proofs, worked via the product of parts, in particular showing that the product satisfies the same central limit theorem. The generalisation to the Ewens measure was proved by Barbour and Tavaré (1994), who also proved the functional form in Corollary 1.4, while the extension to logarithmic combinatorial structures was made by Arratia, Barbour and Tavaré (2000).

Universality
For logarithmic combinatorial structures, Arratia, Barbour and Tavaré (2000) prove lim n→∞ L θ,η n (α dn , . . . , α n ) − L θ n (α dn , . . . , α n ) T V = 0, where L θ,η n and L θ n are the laws of (α dn , . . . , α n ) under P θ,η n and P θ n respectively, and d n is any sequence satisfying d n → ∞ and d n /n → 0. They also proved the other limit theorems above, but most of their proofs ran in parallel to (2.1) instead of directly invoking it.
This left open the question of a simple criterion to determine whether a functional is universal, as well as the question of whether asymptotic independence of the α i is a necessary condition. Our model answers both of these questions, removing the requirement for the α i to be asymptotically independent, and also giving an easilychecked criterion for universality.

Logarithmic Combinatorial Structures
A uniform logarithmic combinatorial structure (Arratia, Barbour and Tavaré, 2000) is a sequence of measures P n on P n , n ∈ N, such that: 1. (Conditioning Relation) There is some sequence of independent random variables Y 1 , Y 2 , . . . such that P n (α) = P ∀ i ≤ n, Y i = α i i≤n iY i = n ; and 2. (Uniform Logarithmic Condition) Each variable Y i satisfies iP[Y i = 1] − θ ≤ e i and iP[Y i = ] ≤ e i c for ≥ 2, where e i and c are vanishing sequences such that e i /i and c are summable.
We will prove η n,m and η n,n are uniformly integrable and converge in probability to η ∞ = i ζ i (α i (∞)), which has positive and finite expectation, by defining several intermediate weights and proving a sequence of asymptotic equivalences between them.
For brevity, all limits are implicitly n → ∞ with m fixed, then m → ∞. We also omit writing the measure explicitly; the only measure used is P θ F .

Conclusion
We studied measures on partitions in terms of perturbations of the Ewens measure, and showed that under a certain condition, many limit theorems for the Ewens measure are universal. This unifies the proofs of the limit theorems under a single universality theorem, while simultaneously extending the class of measures for which they hold beyond the previous frontier of logarithmic combinatorial structures. An interesting direction for future work is to better understand the asymptotically Ewens condition, which will perhaps allow more cases to be unified under the universality theorem.