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Question 1. Let µ be any probability measure on [0, ), and let µ denote ∞ s the semicircle law, defined in (6). Under what conditions on µ, is µ ⊠ µs absolutely continuous? Our first result gives a sufficient condition on µ to answer Question 1. Theorem 1.1. Let µ be a probability measure on R such that µ([δ, )) = 1 ∞ for some δ > 0. Assume furthermore that µ has finite mean. Then, µ ⊠ µs is absolutely continuous. For stating the next question, we need to introduce a random matrix model. Let f be any non-negative integrable on [ π,π]2. Assume furthermore that f is even, that is, − (1) f( x, y)= f(x,y), π x,y π . − − − ≤ ≤ Then, there exists a mean zero stationary Gaussian process (Gi,j : i, j Z) such that ∈ π π (2) E (G G )= eι(ux+vy)f(x,y)dxdy, for all i, j, u, v Z , i,j i+u,j+v ∈ Z−π Z−π (1) ensuring that the right hand side is real. For N 1, let GN be the N N matrix defined by ≥ × (3) G (i, j) := (G + G )/√N, 1 i, j N . N i,j j,i ≤ ≤ Above and elsewhere, for any matrix H, H(i, j) denotes its (i, j)-th entry. It has been shown in Theorem 2.1 of Chakrabarty et al. [7] that there exists a (deterministic) probability measure νf such that (4) ESD(G ) ν , N → f weakly in probability, as N . ESD stands for the empirical spectral distribution for a symmetric →N ∞N random matrix H, which is a random probability measure on R defined× by N 1 (ESD(H)) ( ) := δ ( ) , · N λj · Xj=1 where λ1 . . . λN are the eigenvalues of H, counted with multiplicity. The second≤ question≤ that this paper attempts to answer is the following.

Question 2. Under what conditions on f, is νf absolutely continuous? The answer is provided by the following result. Theorem 1.2. If

ess inf(x,y)∈[−π,π]2 [f(x,y)+ f(y,x)] > 0 , then νf is absolutely continuous, where “ess inf” denotes the essential infi- mum. FREE MULTIPLICATIVE CONVOLUTION 3

While Questions 1 and 2 seem unrelated a priori, the reader will notice after seeing the proofs that they are not so. This is because random matrix theory is used as a tool for proving Theorem 1.1, Theorem 1.2 being anyway a question about a random matrix. It is shown in Proposition 2.1, which is a consequence of Theorem 1.2, that if a measure satisfies the conditions of Theorem 1.1 then, its free multiplicative convolution with a semicircle law is also the free additive convolution of another measure and a dilated semicircle law. The proofs are compiled in the following section. Many known facts are used, which are collected in Section 3 for the convenience of the reader.

2. Proofs For the proof of the results we shall refer to various known facts which are listed in the Appendix of this article. One of the main ingredients is the following fact which follows from Proposition 22.32, page 375 of Nica and Speicher [8]. The latter result reasserts the seminal discovery in Voiculescu [10] that the Wigner matrix is asymptotically freely independent of a deter- ministic matrix which has a compactly supported limiting spectral distribu- tion.

Fact 2.1. Assume that for each N, AN is a N N Gaussian Wigner matrix scaled by √N, that is, (A (i, j) : 1 i j × N) are i.i.d. normal ran- N ≤ ≤ ≤ dom variables with mean zero and variance 1/N, and AN (j, i) = AN (i, j). Suppose that B is a N N random matrix, such that as N , N × →∞ 1 k P k (5) Tr(BN ) x µ(dx), k 1 , N −→ R ≥ Z for some compactly supported (deterministic) probability measure µ. Fur- thermore, let the families (AN : N 1) and (BN : N 1) be independent. Then, as N , ≥ ≥ →∞ 1 k P k EF Tr (AN + BN ) x µ ⊞ µs(dx) for all k 1 , N −→ R ≥ h i Z where := σ(BN : N 1) and EF denotes the with respectF to . ≥ F We first proceed towards proving Theorem 1.2. The first step in that direction is Lemma 2.1 below. However, before stating that, we define a dilated semicircle law µs(t) for all t > 0. It is a probability measure on R given by

√4t x2 (6) (µ (t)) (dx)= − 1( x 2√t) dx, x R . s 2πt | |≤ ∈ In other words, it is the semicircle law with variance t. For t = 1, it equals the standard semicircle law, that is, µ µ (1). s ≡ s 4 A.CHAKRABARTYANDR.S.HAZRA

Lemma 2.1. Let α> 0. Denote (f + α)( , ) := f( , )+ α . · · · · Then, 2 (7) νf+α = νf ⊞ µs(8π α) . Proof. Let us first prove this for the case that f is a trigonometric polyno- mial, that is, of the form n (8) f(x,y)= a eι(jx+ky) 0 , j,k ≥ j,kX=−n for some finite n and real numbers aj,k, in addition to satisfying (1). By Fact 2 3.5, νf+α and νf have compact supports, and hence so does νf ⊞ µs(8π α). Therefore, it suffices to check that

(9) xkν (dx)= xk(ν ⊞ µ (8π2α))(dx) for all k 1 . f+α f s ≥ Z Z Let (G : i, j Z) be a mean zero stationary Gaussian process satisfying i,j ∈ (2). Let (H : i, j Z) be a family of i.i.d. N(0, 4π2α) random variables, i,j ∈ independent of (Gi,j : i, j Z). For N 1, let GN be as in (3), and further define the N N matrices∈ W and Z ≥by × N N W (i, j) := (H + H )/√N, 1 i, j N, N i,j j,i ≤ ≤ ZN := GN + WN . Fact 3.5 implies that

1 k P Tr G xkν (dx), k 1 , N N −→ f ≥   Z as N . This, along with Fact 2.1 and the observation that the upper → ∞ 2 triangular entries of WN are i.i.d. N(0, 8π α/N), implies that 1 (10) E Tr Zk P xk(ν ⊞ µ (8π2α))(dx), k 1 , N F N −→ f s ≥   Z where := σ(G : i, j Z). F i,j ∈ It is easy to see that (Gi,j + Hi,j : i, j Z) is a stationary mean zero Gaussian process whose spectral density is f∈+α, and hence Fact 3.5 implies that 1 1 E E Tr Zk =E Tr Zk xkν (dx) , N F N N N → f+α     Z and     1 1 Var E Tr Zk Var Tr Zk 0 , N F N ≤ N N →     as N , for all k 1. Combining  the above two limits and comparing with (→10) ∞ yields (9), and≥ completes the proof of (7) when f is a trigonometric polynomial. FREE MULTIPLICATIVE CONVOLUTION 5

Now suppose that f is just a non-negative integrable function satisfying (1). By considering the Fourier series of √f, one can construct non-negative even trigonometric polynomials fn such that f f in L1 . n → Since (7) holds with f replaced by fn, it follows that ⊞ 2 νfn+α = νfn µs(8π α) w ν ⊞ µ (8π2α) , −→ f s as n , the second line following from Fact 3.3 combined with Proposition →∞ 4.13 of Bercovici and Voiculescu [5]. Applying Fact 3.3 directly to νfn+α yields w ν ν , fn+α −→ f+α as n . This establishes the claimed identity and thus completes the proof.→ ∞  Proof of Theorem 1.2. Define 1 g(x,y) := [f(x,y)+ f(y,x)] , π x,y π . 2 − ≤ ≤ In view of Fact 3.4, it suffices to show that νg is absolutely continuous. The hypothesis implies that there exists α > 0 such that g α on [ π,π]2. Define ≥ − h( , ) := g( , ) α . · · · · − Notice that 2 νf = νg = νh ⊞ µs(8π α) , the two equalities following by Fact 3.4 and Lemma 2.1 respectively. Fact 3.1 completes the proof.  For proving Theorem 1.1, we prove the following result which is of some independent interest. This along with Fact 3.1 establishes Theorem 1.1. Proposition 2.1. If µ satisfies the hypothesis of Theorem 1.1, then there exists a probability measure η such that 2 µ ⊠ µs = η ⊞ µs(δ ) . Proof. Define a function r from [ π,π] to R by − 1 |x| 3 2 inf y R : µ( ,y] , 0 < x < π , r(x) := 2 / π ∈ π ≤ −∞ | | (0, n o otherwise . Let U be an Uniform( π,π) . Clearly, − (11) P (23/2πr(U) )= µ( ) , ∈ · · which implies that π ∞ r(x)dx = 2πE[r(U)] = 2−1/2 xµ(dx) < . ∞ Z−π Z0 6 A.CHAKRABARTYANDR.S.HAZRA

The hypothesis that µ( , δ) = 0 implies that −∞ δ r(x) , 0 < x < π . ≥ 23/2π | | Defining f(x,y) := r(x)r(y) , it follows that f is an even (as in (1)) non-negative integrable function bounded below by α almost everywhere, where δ2 α := . 8π2 Fact 3.2 and (11) imply that

µ ⊠ µs = νf 2 = νf−α ⊞ µs(8π α) , the second equality following from Lemma 2.1. Setting η := νf−α, this completes the proof. 

Proof of Theorem 1.1. Follows from Proposition 2.1 and Fact 3.1.  Remark 1. Theorem 1.1 is similar in spirit to Theorem 4.1 of [2]. Remark 2. Although finiteness of the mean of µ is used in proving Propo- sition 2.1 and thereby Theorem 1.1, the authors conjecture that the results are true without that assumption.

3. Appendix In this section, we collect the various facts that have been used in the proofs in Section 2. The following fact is Corollary 2 of Biane [6].

Fact 3.1. For any probability measure µ, µ ⊞ µs is absolutely continuous with respect to the . The remaining facts are all quoted from Chakrabarty et al. [7]. Fact 3.2 (Theorem 2.4, Chakrabarty et al. [7]). Let r be a non-negative integrable even function defined on [ π,π], and − f(x,y) := r(x)r(y), π x,y π . − ≤ ≤ Then,

νf = µr ⊠ µs , 3/2 where νf is as in (4), µr is the law of 2 πr(U), and U is a Uniform( π,π) random variable. − FREE MULTIPLICATIVE CONVOLUTION 7

Fact 3.3 (Lemma 3.3, Chakrabarty et al. [7]). Suppose that for all 1 n 2 ≤ ≤ , gn is a non-negative, integrable and even function on [ π,π] . By even, it∞ is meant that g ( x, y)= g (x,y) for all x,y. If − n − − n g g in L1 as n , n → ∞ →∞ then w ν ν ∞ . gn −→ g Fact 3.4 (Lemma 3.5, Chakrabarty et al. [7]). If f is a non-negative inte- grable even function on [ π,π]2, and − 1 g(x,y) := [f(x,y)+ f(y,x)] , 2 then νf = νg . The following fact has been proved in the course of proving Proposition 3.1 of Chakrabarty et al. [7]; see (3.16) and (3.17) therein. Fact 3.5. Let f be a non-negative even trigonometric polynomial on [ π,π]2 − as in (8). Let the matrix GN be constructed as in (3) using the random variables (Gi,j) which are as in (2). Then, νf has compact support, and for all k 1, ≥ 1 k k lim E Tr(GN ) = x νf (dx) , N→∞ N R   Z 1 k and lim Var Tr(GN ) = 0 . N→∞ N   Acknowledgement The authors are grateful to Manjunath Krishnapur for helpful discussions and an anonymous referee for constructive comments.

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[5] H. Bercovici and D. Voiculescu. Free convolution of measures with unbounded support. Indiana University Mathematics Journal, 42:733– 773, 1993. [6] P. Biane. On the free convolution with a semi-circular distribution. Indiana University Mathematics Journal, 46(3):705–718, 1997. [7] A. Chakrabarty, R. S. Hazra, and D. Sarkar. From ran- dom matrices to long range dependence. Available at http://arxiv.org/pdf/1401.0780.pdf, 2014. [8] A. Nica and R. Speicher. Lectures on the Combinatorics of Free Prob- ability. Cambridge University Press, New York, 2006. [9] V. P´erez-Abreu and N. Sakuma. Free infinite divisibility of free mul- tiplicative mixtures of the Wigner distribution. Journal of Theoretical Probability, 25(1):100–121, 2012. [10] D. Voiculescu. Limit laws for random matrices and free products. In- ventiones mathematicae, 104(1):201–220, 1991. [11] D. Voiculescu. The analogues of entropy and of Fisher’s information measure in free probability theory, i. Communications in mathematical physics, 155(1):71–92, 1993. [12] P. Zhong. On the free convolution with a free multiplicative analogue of the normal distribution. To appear in Journal of Theoretical Probability, DOI: 10.1007/s10959-014-0556-x, 2013. Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, New Delhi E-mail address: [email protected]

Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata E-mail address: [email protected]