Limit Theorems and Wrapping Transforms in Bi-free Probability Theory

In this paper, we characterize idempotent distributions with respect to the bi-free multiplicative convolution on the bi-torus. Also, the bi-free analogous Levy triplet of an infinitely divisible distribution on the bi-torus without non-trivial idempotent factors is obtained. This triplet is unique and generates a homomorphism from the bi-free multiplicative semigroup of infinitely divisible distributions to the classical one. The relevances of the limit theorems associated with four convolutions, classical and bi-free additive convolutions and classical and bi-free multiplicative convolutions, are analyzed. The analysis relies on the convergence criteria for limit theorems and the use of push-forward measures induced by the wrapping map from the plane to the bi-torus. Different from the bi-free circumstance, the classical multiplicative L\'{e}vy triplet is not always unique. Due to this, some conditions are furnished to ensure uniqueness.


Introduction
The main aim of the present paper is to build the association among various limit theorems and their convergence criteria in classical and bi-free probability theories.
Bi-free probability theory, introduced by Voiculescu in [20], is an outspread research field of free probability theory, which grew out to intend to simultaneously study the left and right actions of algebras over reduced free product spaces.Since its creation, a great deal of research work has been conducted to better understand this theory and its connections to other parts of mathematics [17; 19; 21; 22].Aside from the combinatorial means, the utilization of analytic functions as transformations and the bond to classical probability theory also play crucial roles in the study and comprehension of this theory [12; 13].Especially, recent developments of bi-free harmonic analysis enable one to investigate bi-free limit theorems and other related topics from the probabilistic point of view [11].
To work in the probabilistic framework, we thereby consider the family P X of Borel probability measures on a complete separable metric space X and endow this family with a commutative and associative binary operation ♢.Classical and bi-free convolutions, respectively denoted by * and ⊞⊞, are two examples of such operations performed on P ‫ޒ‬ 2 .In probabilistic terms, µ 1 * µ 2 is the probability distribution of the sum of two independent bivariate random vectors respectively having distributions µ 1 and µ 2 .When restricted to compactly supported measures in P ‫ޒ‬ 2 , µ 1 ⊞⊞ µ 2 is the distribution of the sum of two bi-free bipartite self-adjoint pairs with distributions µ 1 and µ 2 , respectively [20].This new notion of convolution was later extended, without any limitation, to the whole class P ‫ޒ‬ 2 by the continuity theorem of transforms [11].The product of two independent random vectors having distributions on the bi-torus ‫ޔ‬ 2 gives rise to the classical multiplicative convolution ⊛, and the bi-free analog of multiplicative convolution ⊠⊠ is defined in a similar manner [22].
In (noncommutative) probability theory, the limit theorem and its related subject, the notion of infinite divisibility of distributions, have attracted much attention.By saying that a distribution in (P X , ♢) is infinitely divisible we mean that it can be expressed as the operation ♢ of an arbitrary number of copies of identical distributions from P X .The collection of measures having this infinitely divisible feature forms a semigroup and will be denoted by ID(X, ♢), or simply by ID(♢) if the identification of the metric space is unnecessary.Any measure satisfying µ = µ♢µ, known as idempotent, is an instance of infinitely divisible distributions.In the case of X = ‫,ޒ‬ these topics have been thoroughly studied in classical probability by the efforts of de Finetti, Kolmogorov, Lévy and Khintchine (see [16]), and the same themes in the free contexts have also been deeply explored in the literature [5].
Bi-free probability, as expected, also parallels perfectly aspects of classical and free probability theories [3].For example, the theory of bi-freely infinitely divisible distributions generalizes bi-free central limit theorem as they also serve as the limit laws for sums of bi-freely independent and identically distributed faces.Specifically, it was shown in [11] that for some infinitesimal triangular array {µ n,k } n≥1,1≤k≤n k ⊂ P ‫ޒ‬ 2 and sequence {v n } ⊂ ‫ޒ‬ 2 , the sequence (1-1) converges weakly if and only if so does the sequence The limiting distributions in (1-1) and (1-2) respectively belong to the semigroups ID( * ) and ID(⊞⊞), and their classical and bi-free Lévy triplets agree.This conformity consequently brings out an isomorphism between these two semigroups.Same tasks are performed in the case of bi-free multiplicative convolution in this paper.We determine ⊠⊠-idempotent elements and identify measures in P ‫ޔ‬ 2 bearing no nontrivial ⊠⊠-idempotent factors.Specifically, we demonstrate that ν ∈ ID(⊠⊠) has no nontrivial ⊠⊠-idempotent factor if and only if it belongs to P × ‫ޔ‬ 2 , the subcollection of P ‫ޔ‬ 2 with the attributes s j dν(s 1 , s 2 ) ̸ = 0, j = 1, 2.
Fix an infinitesimal triangular array {ν nk } n≥1,1≤k≤k n ⊂ P ‫ޔ‬ 2 and a sequence {ξ n } ⊂ ‫ޔ‬ 2 .We also manifest that the weak convergence of the sequence to some element in P × ‫ޔ‬ 2 yields the same property of the sequence (1-4) and that their limiting distributions are both infinitely divisible.This is done by distinct types of equivalent convergence criteria offered in the present paper.As in the case of addition, there exists a triplet concurrently serving as the classical and bifree multiplicative Lévy triplets of the limiting distributions in (1)(2)(3) and (1)(2)(3)(4).The consistency of their Lévy triplets, together with the description of ID(⊠⊠)\P × ‫ޔ‬ 2 , consequently produces a homomorphism from ID(⊠⊠) to ID(⊛).
Because of the nature of ID(⊠⊠)\P × ‫ޔ‬ 2 and that the limit in (1-4) may generally not have a unique Lévy measure, the homomorphism stated above is neither surjective nor injective.However, postulating the uniqueness of the Lévy measure, the weak convergence of (1-4) derives that of (1)(2)(3).
In addition to the previously mentioned conjunctions, what we would like to point out is that measures in P ‫ޒ‬ 2 and P ‫ޔ‬ 2 can be linked through the wrapping map W : → ‫ޔ‬ 2 , (x, y) → (e i x , e i y ).This wrapping map induces a map W * : P ‫ޒ‬ 2 → P ‫ޔ‬ 2 so that the measure ν nk = W * (µ nk ) = µ nk W −1 enjoys the property: the weak convergence of (1-1) or (1-2) yields the weak convergence of (1-3) and (1-4) with ξ n = W (v n ).Furthermore, the synchronous convergence allows one to construct a homomorphism W ⊞⊞ : ID(⊞⊞) → ID(⊠⊠) making the following diagram commute: (1-5) This diagram is a two-dimensional analog of [6, Theorem 1].The rest of the paper is organized as follows.In Section 2 we provide the necessary background in classical and noncommutative probability theories.In Section 3 we characterize ⊠⊠-idempotent distributions.In Section 4 we make comparisons of the convergence criteria of limit theorems, as well as those through wrapping transforms.Section 5 is devoted to offering bi-free multiplicative Lévy triplets of infinitely divisible distributions and investigating the relationships among limit theorems in additive and multiplicative cases.Section 6 provides the derivation of the diagram in (1-5).

Preliminary
2A. Convergence of measures.Let B X be the collection of Borel sets on a complete separable metric space (X, d).A point is selected from X and fixed, named the origin and denoted by x 0 in the following.In the present paper, we will be mostly concerned with the abelian groups X = ‫ޒ‬ d and X = ‫ޔ‬ d endowed with the relative topology from ‫ރ‬ d , where the origin is chosen to be the unit.They are respectively the d-dimensional Euclidean metric space and the d-dimensional torus (or the d-torus for short).The 1-torus is just the unit circle ‫ޔ‬ on the complex plane.A set contained in {x ∈ X : d(x, x 0 ) ≥ r } for some r > 0 is colloquially said to be bounded away from the origin.
Next, let us introduce several types of measures on X that will be discussed later.The first one is the collection M X of finite positive Borel measures on X.We shall also consider the set M x 0 X of all positive Borel measures that when confined to any Borel set bounded away from the origin yield a finite measure.Clearly, we have M X ⊂ M x 0 X .Another assortment concerned herein is the collection P X of elements in M X having unit total mass.
The set C b (X) of bounded continuous functions on X induces the weak topology on M X .Likewise, M x 0 X is equipped with the topology generated by C x 0 b (X), bounded continuous functions having support bounded away from the origin.Concretely, basic neighborhoods of a τ ∈ M x 0 X are of the form j=1,...,n τ ∈ M x 0 X : where ϵ > 0 and each We remark that τ is not unique as it may assign arbitrary mass to the origin.Nevertheless, any weak limit in M x 0 X that comes across in our discussions will serve as the so-called Lévy measure, which does not charge the origin.
(2) For any f ∈ C b (X) and any B ⊂ B X , which is bounded away from the origin and satisfies τ (∂ B) = 0, we have (3) For every closed set F and open set G of X that are both bounded away from x 0 , we have and the set of discontinuities of h has τ -measure zero, then Finally, let us introduce the subset M x 0 X consisting of measures in M x 0 X that do not charge the origin x 0 .This set is metrizable and becomes a separable complete metric space [14,Theorem 2.2].In particular, the relative compactness of a subset Y of M x 0 X is equivalent to that any sequence of Y has a subsequence convergent in M x 0 X .We refer the reader to [14,Theorem 2.7] for an analog of Prokhorov's theorem, which characterizes the relative compactness of subsets in M x 0 X .

2B.
Notations.Below, we collect notations that will be commonly used in the sequel.The customary symbol arg s ∈ (−π, π] stands for the principal argument of a point s ∈ ‫,ޔ‬ while ℜs and ℑs respectively represent the real and imaginary parts of s.Here and elsewhere, points in a multidimensional space will be written in bold letters, for instance, s = (s 1 , . . ., s d ) ∈ ‫ޔ‬ Besides, we adopt the operational conventions in multidimensional spaces in the sequel, such as and e i s = (e is 1 , . . ., e is d ).The push-forward probabilities µ ( j)  = µπ −1 j , j = 1, . . ., d, on the real line induced by projections π j : ‫ޒ‬ d → ‫,ޒ‬ x → x j , are called marginals of µ ∈ P ‫ޒ‬ d .Marginals of probability measures on ‫ޔ‬ d are defined and displayed in the same way.On ‫ޔ‬ 2 , we shall also consider the (right) coordinate-flip transform h op : ‫ޔ‬ 2 → ‫ޔ‬ 2 defined as h op (s) = (s 1 , 1/s 2 ).Denote by s ⋆ = h op (s) and B ⋆ = {s ⋆ : s ∈ B} if s ∈ ‫ޔ‬ 2 and B ⊂ ‫ޔ‬ 2 .By the (right) coordinate-flip measure of ρ ∈ M 1 ‫ޔ‬ 2 , we mean the push-forward measure ρ ⋆ = ρh −1 op , alternatively defined as 2C. Free probability and bi-free probability.Aside from the classical convolution on P ‫ޒ‬ 2 , we shall also consider the bi-free convolution ⊞⊞, where the bi-free φtransform takes the place of Fourier transform [11]: for µ 1 , µ 2 ∈ P ‫ޒ‬ 2 , one has φ µ 1 ⊞⊞ µ 2 = φ µ 1 + φ µ 2 .All information about marginals of the bi-free convolution is carried over to the free convolution: 2 for j = 1, 2. Now, we turn to probability measures on the d-torus.The sequence is called the d-moment sequence of ν ∈ P ‫ޔ‬ d .In some circumstances, characteristic function and ν( p) are the precise terminology and notation used for this sequence.Owing to Stone-Weierstrass theorem, we have The bi-free multiplicative convolution of ν 1 , ν 2 ∈ P × ‫ޔ‬ 2 is determined by its marginals (ν 2 and the bi-free multiplicative formula for points (z, w) ∈ ‫ރ‬ 2 in a neighborhood of (0, 0) and (0, ∞).Here the free multiplicative convolution can be rephrased by means of the free -transform 2 valid in a neighborhood of the origin of the complex plane.The reader is referred to [4; 5; 12; 13; 17; 19; 21; 22] for more details along with properties of the transforms in (bi)-free probability theory.We remark that given a measure ν ∈ P × ‫ޔ‬ 2 , the transform ν is the identity map if and only if ν is a product measure, which leads to 2 ), In fact, (2-2) holds for any ν 1 , ν 2 ∈ P ‫ޔ‬ 2 by continuity arguments together with the facts that m p,q (ν 1 ⊠⊠ ν 2 ) can be expressed as a polynomial of m k,l (ν i ) for i = 1, 2, |k| ≤ | p|, |l| ≤ |q| and that ν ∈ P ‫ޔ‬ 2 is a product measure if and only if m p,q (ν) = m p (ν (1) ) m q (ν (2) ) for any p, q ∈ ‫.ޚ‬ Fix ν 1 , ν 2 ∈ P ‫ޔ‬ 2 , and let ν = ν 1 ⊠⊠ ν 2 .In order to analyze ν, it will be convenient to treat it as the distribution of a certain bipartite pair (u 1 u 2 , v 1 v 2 ), where (u 1 , v 1 ) and (u 2 , v 2 ) are bi-free bipartite unitary pairs in some C * -probability space having distributions ν 1 and ν 2 , respectively.Below, we briefly introduce the construction of such pairs carrying the mentioned properties.For more information, we refer the reader to [13; 20; 22].Associating each ν j with the Hilbert space H j = L 2 (ν j ) with specified unit vector ξ j , the constant function one in H j , consider the Hilbert space free product (H, ξ ) = * j=1,2 (H j , ξ j ).The left and right factorizations of H j from H can be respectively done via natural isomorphisms V j : H j ⊗ H(ℓ, j) → H and W j : H(r, j) ⊗ H j → H. Then for any T ∈ B(H j ), these isomorphisms induce the so-called left and right operators For any S j , T j ∈ B(H j ), pairs (λ 1 (S 1 ), ρ 1 (T 1 )) and (λ 2 (S 2 ), ρ 2 (T 2 )) are, by definition, bi-free in the C * -probability space (B(H), ϕ ξ ), where ϕ ξ ( • ) = ⟨• ξ, ξ ⟩.Particularly, the multiplication operators (S j f )(s, t) = s f (s, t) and (T j f )(s, t) = t f (s, t) for f ∈ H j furnish the desired pairs (u 1 , v 1 ) and (u 2 , v 2 ), where u j = λ j (S j ) and v j = ρ j (T j ).
Recall from [13] that one can perform the opposite bi-free multiplicative convolution of ν 1 and ν 2 : Then ν 1 ⊠⊠ op ν 2 is the distribution of (u 1 u 2 , v 2 v 1 ), the pair obtained by performing the opposite multiplication on the right face The coordinate-flip map h op gives rise to a homeomorphism from the semigroup (P ‫ޔ‬ 2 , ⊠⊠) to another (P ‫ޔ‬ 2 , ⊠⊠ op ) satisfying which is the distribution of Passing to the transform op 2D. Limit theorem.Either in classical or in (bi-)free probability theory, one is concerned with the asymptotic behavior of the sequence where δ x is the Dirac measure concentrated at x ∈ X and {µ nk n } n≥1,1≤k≤k n is an infinitesimal triangular array in P X .The infinitesimality of {µ nk }, by definition, means that k 1 < k 2 < • • • and that for any ϵ > 0, we have One phenomenon related to equation (2)(3)(4) is the concept of infinite divisibility: µ ∈ (P X , ♢) is said to be infinitely divisible if for any n ∈ ‫,ގ‬ it coincides with the n-fold ♢-operation µ ♢n n of some µ n ∈ P X .
Commutative and associative binary operations to be considered throughout the paper are classical convolutions * and ⊛ on P ‫ޒ‬ d and P ‫ޔ‬ d , respectively, and bi-free additive and multiplicative convolutions ⊞⊞ and ⊠⊠ on P ‫ޒ‬ 2 and P ‫ޔ‬ 2 , respectively.The following convergence criteria play an essential role in the asymptotic analysis of limit theorems of P ‫ޒ‬ d .Condition 2.2.Let {τ n } be a sequence in M 0 ‫ޒ‬ d .(I) For j = 1, . . ., d, the sequence {σ n j } n≥1 defined by belongs to M ‫ޒ‬ d and converges weakly to some σ j ∈ M ‫ޒ‬ d .
(II) For j, ℓ = 1, . . ., d, the following limit exists in ‫:ޒ‬ For any vector u ∈ ‫ޒ‬ d , the following limits exist in ‫:ޒ‬ Although we describe the conditions in a higher dimension setup, the reader can effortlessly mimic the proof in [11] to obtain the equivalence of Conditions 2.2 and 2.3, and draw the following consequences: (1) The function Q( • ) = ⟨ A • , • ⟩ in (IV) defines a nonnegative quadratic form on ‫ޒ‬ d , where the matrix A = (a jℓ ) is given by In particular, a j j = σ j ({0}) for j = 1, . . ., d.
(2) Measures τ and σ 1 , . . ., σ d are uniquely determined by the relations where {e j } is the standard basis of ‫ޒ‬ d . ( Now, let us briefly introduce the limit theorems of (1-1) and (1-2).Throughout our discussions in the paper, (2)(3)(4)(5) θ ∈ (0, 1) is an arbitrary but fixed quantity.To meet the purpose, consider the shifted triangular array μnk associated with an infinitesimal triangular array {µ nk } n≥1,1≤k≤k n ⊂ P ‫ޒ‬ d and the vector Due to lim n→∞ max k≤k n ∥v nk ∥ = 0, { μnk } so obtained is also infinitesimal.In conjunction with this centered triangular array, we focus on the positive measures It turns out that the sequence in (1-1) converges weakly to a certain µ * ∈ P ‫ޒ‬ d if and only if τ n defined in (2-7) meets Condition 2.3 (as well as Condition 2.2 since these two conditions are equivalent) and the limit exists in ‫ޒ‬ d .Additionally, µ * is * -infinitely divisible and possesses the characteristic function read as which is known as the Lévy-Khintchine representation.The limiting distribution is uniquely determined by the formula (2-9) and denoted by µ (v, A,τ ) * , and (v, A, τ ) is referred to as its Lévy triplet.The set ID( * ) is completely parameterized by the triplets (v, A, τ ), where As a matter of fact, when d = 2, the same convergence criteria are also necessary and sufficient to assure the weak convergence of (1-2).Paralleling to the classical case, the limiting distribution of (1-2) is ⊞⊞-infinitely divisible and owns the bi-free φ-transform, called bi-free Lévy-Khintchine representation, of the form Analogically, this limiting distribution is always expressed as µ (v, A,τ ) ⊞⊞ and said to own the bi-free Lévy triplet (v, A, τ ).Those triplets (v, A, τ ) satisfying (2-10) also give a complete parametrization of the set ID(⊞⊞), and therefore output a bijective homomorphism from ID( * ) onto ID(⊞⊞), sending an element µ (v, A,τ ) * in the first set to the distribution µ (v, A,τ ) ⊞⊞ in the second one.No matter in the classical or bi-free probability, * -and ⊞⊞-infinitely divisible distributions both appear as limiting distributions in the limit theorem.
Next, we turn our attention to the limit theorem on the d-torus, on which the Borel probability measures of interest are sometimes imposed the nonvanishing mean conditions: (2-11) For convenience, we adopt the symbol P × ‫ޔ‬ d to signify the collection of probability measures carrying such features.As will be shown in Theorem 3.12, when d = 2, these conditions (2-11) turn out to be necessary and sufficient for a ⊠⊠-infinitely divisible distribution to contain no nontrivial ⊠⊠-idempotent factors.We would also like to remind the reader that the symbol P × ‫ޔ‬ 2 introduced here is distinct from that in [13] as Theorem 3.10 of the present paper designates that the requirement m 1,1 (ν) ̸ = 0 in the limit theorem is redundant.

⊠⊠-Idempotent distributions
Particularly, µ is said to be ♢-idempotent when µ ′ = µ = µ ′′ .Idempotent distributions and other related subjects in classical probability have been extensively studied in [16].It is to questions of these sorts in the bi-free probability theory that the present section is devoted.
The normalized Lebesgue measure m = dθ/(2π ) on ‫ޔ‬ is the only ⊠-idempotent element except for the trivial one, the Dirac measure at 1. On ‫ޔ‬ 2 , the probability measure is ⊛-idempotent because m p,q (P) = 1 for p = q ∈ ‫ޚ‬ and zero otherwise.As a matter of fact, this singularly continuous measure is also ⊠⊠-idempotent proved below.
The following result is a direct consequence of Voiculescu's two-bands moment formula in [21, Lemma 2.1] and we provide its proof and notations for the later use.
Proof.Following the notations in Section 2C, let α j = ⟨S −1 j ξ j , ξ j ⟩, β j = ⟨T j ξ j , ξ j ⟩, h j = S −1 j ξ j − α j ξ j , and k j = T j ξ j − β j ξ j for j = 1, 2. Then On the other hand, we have u −1 Thus, the first desired result follows from the representation of m 1,1 (ν j ) given above and the computations Thanks to (2-3) and the first result, we obtain Results in Lemma 3.2 can also be easily derived by the momentcumulant formula and vanishing of bi-free mixed cumulants [8].
(4) P is a ⊠⊠-factor of ν if and only if where δ p,q is the Kronecker function of p and q.
A special case of (3-5) is the validity of for any c 1 , c 2 , d 1 , d 2 ∈ ‫ބ‬ ∪ ‫,ޔ‬ yielding the following results.

Equivalent conditions on limit theorems
This section is devoted to exploring the associations among the conditions introduced in Section 2D and the following one.(iii) There exists some ρ ∈ M 1 ‫ޔ‬ d with ρ({1}) = 0 (i.e., ρ ∈ M 1 ‫ޔ‬ d ) so that ρ n ⇒ 1 ρ.(iv) The following limits exist in ‫ޒ‬ for any p ∈ ‫ޚ‬ d : Condition 2.5 with d = 2 was used in [13,Theorem 3.4] to prove the limit theorem for the bi-free multiplicative convolution, while Condition 4.1 is beneficial for the corresponding classical limit theorem [10].More properties regarding these two conditions are presented below.d,  (4-1) and the quadratic form Q( • ) = ⟨ A • , • ⟩ on ‫ޚ‬ d is determined by the positive semidefinitive matrix A = (a jℓ ) whose entries are Moreover, a j j = 2λ j ({1}) for j = 1, . . ., d.
Proof.Suppose first that Condition 2.5 is satisfied.Then the relation guaranteed by item (i) of Condition 2.5 ensures that the measure is unambiguous and does not depend on j.In addition, it satisfies requirements ρ(‫ޔ‬ d \U ϵ ) < ∞ for any ϵ > 0 and (4-2).Hence the measure ρ that we just constructed belongs to M 1 ‫ޔ‬ d .To see ρ n ⇒ 1 ρ, pick a continuous function f on ‫ޔ‬ d with support contained within ‫ޔ‬ d \U δ for some δ > 0. Then this f produces d continuous functions on ‫ޔ‬ d , which are where These observations and the weak convergence λ n j ⇒ λ j then yield that Therefore, we have completed the verification of item (iii) of Condition 4.1.
We next demonstrate the validity of the following identities for 1 ≤ j, ℓ ≤ d, which confirms that of Condition 4.1(iv).To continue, observe that the mapping s → (ℑs) 2 /(1 − ℜs) is continuous on ‫ޔ‬ and at the origin, it takes value Then (4-2), (4-6), and the Hölder inequality imply that (ℑs j )(ℑs ℓ ) ∈ L 1 (ρ) for j, ℓ = 1, . . ., d.In order to get results (4-3) and (4-5), we examine the following differences which are related to them: which further splits into the sum of Apparently, we have lim ϵ→0 I 2 (ϵ) = 0 owing to (ℑs j )(ℑs ℓ ) ∈ L 1 (ρ).Next, take an ϵ ′ ∈ (ϵ, 2ϵ) and an ϵ ′′ ∈ ϵ 2 , ϵ with the attributes that ρ(∂U ϵ ′ ) = 0 and ρ(∂U ϵ ′′ ) = 0, the presence of which are insured by the finiteness of the measure 1 ‫ޔ‬ d \U ϵ/2 ρ on ‫ޔ‬ d .Then applying Proposition 2.1 to the established result On the other hand, working with the closed subset If ϵ ′ is also chosen so that λ j (∂U ϵ ′ ) = 0, then we draw once again from (4-6) that a j j = 2λ j ({1}) because lim sup This conclusion and (4-4) give (4-1).It is easy to see that the limits in (iv) of Condition 4.1 are equal to ⟨ A p, p⟩ for any p ∈ ‫ޚ‬ d (in fact, for any p ∈ ‫ޒ‬ d as well) with A = (a jℓ ) and a jℓ the value of the limit given in (4)(5).Also, it is clear that the quadratic form Q extends to ‫ޒ‬ d and is positive therein.Then the positivity of A ≥ 0 can be gained by that of Q on ‫ޒ‬ d .Next, we elaborate that Condition 4.1 implies Condition 2.5.Define λ j 's as in (4-1).These measures thus obtained are all in M ‫ޔ‬ d , and the arguments for this go as follows.Select a sequence ϵ m ↓ 0 as m → ∞ and ρ({∥arg s∥ = ϵ m }) = 0 for each m.Then (iv), along with Proposition 2.1, indicates that for any numbers m < m ′ both large enough, one has Thanks to monotone convergence theorem, (4-6), and the assumption ρ({1}) = 0, one further gets that for m large enough, (1 − ℜs j )1 U ϵm ∈ L 1 (ρ) for any j.This proves that λ j ‫ޔ(‬ d ) < ∞ and (ℑs j ) 2  ∈ L 1 (ρ) for any j.After the previous preparations, we are in a position to justify the weak convergence λ n j ⇒ λ j .Given a continuous function f on ‫ޔ‬ d , the difference is dominated by the sum of the following four terms: First, one can show that lim m→∞ lim sup n→∞ |D n2 (m)| = 0 by applying (4-6) and item (iv) to Similarly, one can show On the other hand, the finiteness of λ j ‫ޔ(‬ d ) leads to That we have lim n→∞ D n4 (m) = 0 for all m evidently follows from Condition 4.1(iii) and Proposition 2.1.Putting all these observations together illustrates λ n j ⇒ λ j .It remains to deal with (ii) of Condition 2.5, in which the integral is rewritten as For any j, ℓ, taking the operations lim m→∞ lim sup n→∞ and lim m→∞ lim inf n→∞ of the first integral gives the same value 1 2 [Q(e j +e ℓ )− Q(e j )− Q(e ℓ )], while doing the same thing to the second integral yields the value ‫ޔ‬ d (ℑs j )(ℑs ℓ ) dρ because of ρ n ⇒ 1 ρ and (ℑs j ) 2  +(ℑs ℓ ) 2 ∈ L 1 (ρ).This finishes the proof of the proposition.□ An intuitive thought is that measures on ‫ޔ‬ d obtained by rotating measures within controllable angles maintain the same structural properties, such as Condition 4.1, as the original ones.The statement and its rigorous proof are given below.Proof.First of all, (4-8) reveals that lim n max k ∥θ nk ∥ = 0. We now argue that ρn ⇒ 1 ρ as well by using Proposition 2.1.To do so, pick a closed subset F ⊂ ‫ޔ‬ d \U r for some r > 0. Since ρ(F) < ∞, it follows that given any δ > 0, there exists a closed set F ′ ⊂ ‫ޔ‬ d \U r/2 such that e iθ nk F ⊂ F ′ for all sufficiently large n and for all 1 ≤ k ≤ k n , and ρ(F ′ \F) < δ.Then implies that lim sup n→∞ ρn (F) ≤ lim sup n→∞ ρ n (F ′ ) ≤ ρ(F ′ ) ≤ ρ(F) + δ.Consequently, we arrive at the inequality lim sup n→∞ ρn (F) ≤ ρ(F).In the same vein, one can show that lim inf n→∞ ρn (G) ≥ ρ(G) for any set G which is open and bounded away from 1. Hence ρn ⇒ 1 ρ by Proposition 2.1.Next, we turn to demonstrate that both ρ n and ρn bring out the tantamount quantities in (4-5), which asserts that the quadratic form in (iv) output by them is unchanged on ‫ޚ‬ d .Any index n considered below is always sufficiently large.In the case j = ℓ, we have the estimate where we express θ nk = (θ nk1 , . . ., θ nkd ).The inequality will help us to continue with the arguments.Consideration given to the first term on the right-hand side of (4-9) gives by the hypothesis, while analyzing the second term results in These estimates then lead to lim ϵ→0 lim sup n→∞ U ϵ (ℑs j ) 2 d ρn (s) ≤ a j j .Employing the opposite inclusion U ϵ/2 ⊂ e −iθ nk U ϵ and inequality allows us to obtain lim ϵ→0 lim inf n→∞ U ϵ (ℑs j ) 2 dρ n (s) ≥ a j j .Now we deal with the situation j ̸ = ℓ in (4-5).After careful consideration of all available information, the focus is only needed on the summand Recall from (2-1) that the push-forward measure τ A useful and frequently used result regarding W is the change-of-variables formula stating that a Borel function f on ‫ޔ‬ d belongs to L 1 (τ W −1 ) if and only if the function x → f (e i x ) lies in L 1 (τ ), and the equation (4-12) holds in either case.In the following, we will translate conditions introduced in Section 2D accordingly via the wrapping map W .
Proof.Suppose that Condition 2.3 holds for τ n and τ , and let A = (a jℓ ) represent the matrix produced by these measures in (IV).According to Proposition 4.2, we shall only elaborate that Condition 4.1 is applicable to ρ n and ρ.
That ρ n ⇒ 1 ρ is clearly valid according to the continuous mapping theorem, Proposition 2.1.It remains to argue that in Condition 4.1(iv), ρ n also outputs A. The simple observation that e i x ∈ U ϵ if and only if x belongs to the set (4-13) and formula (4-12) help us to establish that for j, ℓ = 1, . . ., d, Observe next that we have The same arguments also elaborate the identity Apparently, the selection of ϵ does not vary the validity of these identities, and so we have established that ρ n generates the matrix A in (iv) as well.□ Measures in M 0 ‫ޒ‬ d can be wrapped either clockwise or counterclockwise (see equation (4)(5)(6)(7)(8)(9)(10)(11)) in all variables, and consequences, such as Proposition 4.4, are not affected at all by this slight change.As a matter of fact, it is also the case when one wraps some variables counterclockwise and others clockwise.Without loss of generality, we shall use the simplest circumstance, the 2-dimensional opposite wrapping map W ⋆ 2 : ‫ޒ‬ 2 → ‫ޔ‬ 2 , (x 1 , x 2 ) → (e i x 1 , e −i x 2 ), to illustrate these features.The following result is merely an easy consequence of the continuous mapping theorem, the relations We add one remark on item (2) of the preceding proposition: if Q( p) = ⟨ A p, p⟩, then Q( p ⋆ ) = ⟨ A op p, p⟩, where the (i, j)-entry of A op is (−1) i+ j A i j .for measures in ID(⊠⊠) ∩ P × ‫ޔ‬ 2 , one may take another parametrization (γ , A, ρ) (with the same γ ) having the following properties with d = 2: (5-1) γ ∈ ‫ޔ‬ d , A is a positive semidefinite d × d symmetric matrix, and ρ is a positive measure on ‫ޔ‬ d so that ρ({1}) = 0 and ∥1 − ℜs∥ ∈ L 1 (ρ).
The following corollary, derived from Theorem 2.4 and [10], supplies the link between classical and bi-free limit theorems on the bi-torus.The attentive reader can also notice that the hypothesis L(ρ) = {ρ} is redundant in the implication (2) ⇒ (1).
The goal of this section is to provide an alternative description for the -transform of a measure in ID(⊠⊠) ∩ P × ‫ޔ‬ 2 in terms of its bi-free multiplicative Lévy triplets.To achieve this, we need some basics.For any p ∈ ‫,ގ‬ the function By induction, this finishes the proof of the first assertion.To prove the second assertion, it suffices to show π −π (1 − cos( pθ ))/(1 − cos θ ) dθ = 2 pπ, which can be easily obtained by using (5-4) again.□ Fix a measure ν ∈ P × ‫ޔ‬ 2 ∩ ID(⊠⊠), and suppose that its (bi-)free -transforms are given as in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16).Due to the integral representations, both u 1 and u 2 are analytic in = ‫)ޔ\ރ(‬ ∪ {∞} and u is analytic in 2 .Hence the function which is also an analytic function in 2 .When ν ∈ ID(⊠⊠ op )∩P × ‫ޔ‬ 2 , one can obtain an equivalent formula for U ν in terms of the bi-free multiplicative Lévy triplet, which we call the bi-free multiplicative Lévy-Khintchine representation.Note that we acquire the following proof with the help of limit theorems, in spite of the algebraic nature of the statement.Also, it is simpler even though there exists an algebraic proof.
Proof.Before carrying out the main proof, let us record some properties instantly inferred from the hypotheses for the later utilization.Because the index n goes to infinity ultimately, it is always big enough whenever mentioned in the proof.
For the second term in (5-10), λ n j = (1 − ℜs j ) ρn ⇒ λ j ∈ M ‫ޔ‬ 2 yields that As noted above that ρ n meets Condition 2.5 if and only if so does ρ ′ n and that lim n→∞ k n |θ n j | 2 = 0 in either case.Consequently, we have shown lim n→∞ E n j = 0 for j = 1, 2 and arrived at γ = lim n→∞ γ n if (1) or (3) holds.□ Remark 5.13.In spite of δ ⊠⊠2n −1 = δ 1 , 2nδ −1 fails to converge in M 1 ‫ޔ‬ 2 .This example demonstrates that in Theorem 5.12, the rotated probabilities νn are a necessary medium in the convergence criteria of the bi-free multiplicative limit theorem.For the same inference, the converse statement of Proposition 5.11 does not hold, yet it does in the additive setting [11,Theorem 5.6].
4) of Proposition 3.4 can be strengthened as that P is the only nontrivial ⊠⊠-idempotent factor of ν ∈ P ‫ޔ‬ 2 if and only if m 1,1 (ν) ̸ = 0 and (3-3) holds.Remark 3.7.The notions of bi-R-diagonality and Haar bi-unitary elements were first introduced in [18, Example 4.7] and [7, Definition 10.1.2],respectively.A Haar bi-unitary element is a bipartite pair having distribution P ⋆ [15, Definition 2.15].The opposite multiplication plays a key role when characterizing bi-R-diagonal pairs in terms of Haar bi-unitary elements [15, Theorem 4.4].Moreover, measures ν ∈ P ‫ޔ‬ 2 satisfying (3-4) are bi-R-diagonal because of ν = ν ⊠⊠ op P ⋆ according to Proposition 3.4 and because of [15, Theorem 4.4].For any c ∈ ‫,ބ‬ define dκ c (s) = 1 − |c| 2 |1 − cs| 2 dm(s), which is the probability measure on ‫ޔ‬ induced by the Poisson kernel.It is the normalized Haar measure on ‫ޔ‬ in case c = 0.By taking the weak limit we define κ c = δ c for c ∈ ‫.ޔ‬Alternatively, κ c with c ∈ ‫ބ‬ ∪ ‫ޔ‬ is the unique probability measure on ‫ޔ‬ determined by the requirement m p (κ c ) = c p for p ∈ ‫.ގ‬ Also, we have m p (κ c ) = c|p| for p ∈ ‫.ގ−‬ Observe that for any c, d ∈ ‫ބ‬ ∪ ‫,ޔ‬ we have

Proposition 4 . 2 .
Condition 2.5 is equivalent to Condition 4.1, in which