Characterizations of free Meixner distributions Texas A&M University March 26, 2010
Jacobi parameters. Matrix. β 0 γ 0 0 0... 1 β 1 γ 1 0.. m n J =. 0 1 β 2 γ.. 2 ; J n =. 0 0 1 β.. 3............... A new sequence m 1, m 2, m 3,.... Favard s Theorem. [Stone 1932] All γ i 0 {m i } are moments of a measure, m i = x i dµ(x).
Definition. b, c, t R, t, t + c 0. Tridiagonal matrix 0 t 0 0 0 1 b t + c 0 0 J = 0 1 b t + c 0 0 0 1 b t + c µt b,c. 0 0 0 1 b µ t b,c = free Meixner distributions. Bernstein-Szegő class: β i, γ i eventually constant.
Formula. In terms of the Cauchy transform 1 G µ (z) = z x dµ(x), R dµ(x) = 1 π lim ε 0 + Im G µ(x + iε) dx ( 4(t + c) (x b) 2 ) µ t b,c = 1 2π t + bx + (c/t)x 2 + dx + 0, 1,or 2 atoms.
2 1 0 1 2 2 1 0 1 2 2 1 0 1 2 Examples. µ 0,1, µ 0,0, µ 0,1/2.
Semicircle law. b, c = 0. µ t 0,0 = 1 4t x 2πt 2. Orthogonal polynomials: Chebyshev polynomials of the second kind U n (x). Generating function U n (x, t)w n = n=0 1 1 xw + tw 2.
First characterization. Theorem. [A. 03] Orthogonal polynomials P n (x) for a measure µ have a generating function of the resolvent-type form P n (x)w n 1 = A(w) 1 B(w)x n=0 if and only if µ is a free Meixner distribution.
Analogy: Meixner class. Theorem. [Meixner 1934] Orthogonal polynomials P n (x) for a measure µ have a generating function of the exponential form n=0 1 n! P n(x)w n = A(w)e B(w)x if and only if µ is a Meixner distribution: Normal, Gamma, hyperbolic secant. Binomial, Poisson, negative binomial. A lot more interesting and important class. Yet: have an analogy.
Lévy process. Random variables {X t : t R}, Increments X t0 X t1 X t2 X t3. X(t 1 ) X(t 0 ), X(t 2 ) X(t 1 ),..., X(t k ) X(t k 1 ) independent and stationary: (X t X s ) X t s µ t s. Example. B t = Brownian motion. µ t = 1 2πt e x2 /t dx.
Conditional expectation. E [ ] = expectation. E [ X] = conditional expectation = projection. E [f(x) X] = f(x), E [Y X] = E [Y ] if Y, X independent.
Exponential martingale. Denote [ F t (θ) = E e ixtθ] = e ixθ dµ t (x) the Fourier transform. ] E [e ixtθ log Ft(θ) X s = e ixsθ log Fs(θ). t eixtθ log Ft(θ) is Brownian motion a martingale. e ibtθ t θ2 2.
Other martingales. t 7 eixt θ log Ft (θ).
Proof. For X, Y independent F X+Y (θ) = F X (θ)f Y (θ) F µt (θ) = F t (θ) log F t (θ) = t log F(θ). ] ] E [e ixtθ log Ft(θ) X s = E [e i(xt Xs)θ e ixsθ e t log F(θ) X s [ = E e i(xt Xs)θ] e ixsθ t log F(θ) e = e (t s) log F(θ) e t log F(θ) e ixsθ ixsθ log sf(θ) = e
Martingale polynomials. Generating function e xz t log F( iz) = n=0 Each P n = martingale polynomial. 1 n! P n(x, t)z n. Brownian motion: H n (x, t) Hermite polynomials. H n (B t, t) = martingale. Meixner class: orthogonal martingale polynomials.
Free independence. Independent: E [f(x)g(y )] = E [f(x)] E [g(y )]. What if X, Y don t commute? What is the correct expression for E [f(x)g(y )h(x)] in terms of X and Y separately? Voiculescu s free independence.
Free probability (Voiculescu 1980s). Common structure in Operator Algebras Random Matrices Asymptotic Representation Theory Probability theory with (maximally) non-commuting random variables. Free versions of many probabilistic objects and theorems. (Free) independence, (free) product, (free) infinitely divisible distributions and limit theorems, (free) cumulants, (free) normal, Poisson, binomial distributions, etc.
Examples. Free central limit theorem. Limit = semicircular distribution = µ 0,0. Free Poisson limit theorem. Limit = free Poisson distribution = µ 1,0. Binomial distribution = sum of independent coin tosses. Coin toss = projection. Sum of freely independent projections = free binomial distribution = µ t 0, 1, t = number of tosses. In the free case, t 1 real!
Processes with free increments. X t = process with freely independent increments. Theorem. [Biane 1998]. {X t } is a Markov processes. Equivalently: is a martingale. t 1 1 X t z + tr(z) Here R(z) = R-transform (Voiculescu), analog of log F(iθ). Free Meixner processes have orthogonal martingale polynomials.
Reverse martingales. {X t } a process with freely independent increments. {P n (x, t)} polynomials, P n of degree n. P n a martingale if for s < t, E [P n (X t, t) X s ] = P n (X s, s). P n a reverse martingale if for s < t, E [P n (X s, s) X t ] = f(s) f(t) P n(x t, t).
Characterization: reverse martingales. Theorem. [Laha, Lukacs 1960] Their result implies: A Lévy process {X t } has a family of polynomials P n which are both martingales and reverse martingales if and only if each X s has a Meixner distribution. Theorem. [Bożejko, Bryc 06] Their result implies: A free Lévy process {X t } has a family of polynomials P n which are both martingales and reverse martingales if and only if each X s has a free Meixner distribution.
Further characterizations. Algebraic Riccati equation (A. 07) Free Jacobi fields (Bożejko, Lytvynov 09) Free quadratic exponential families (Bryc 09) Free quadratic harnesses (Bryc, Wesołowski 05) Etc. Other appearances: Szegö (1922), Bernstein (1930), Boas & Buck (1956), Carlin & McGregor (1957), Geronimus (1961), Allaway (1972), Askey & Ismail (1983), Cohen & Trenholme (1984), Kato (1986), Freeman (1998), Saitoh & Yoshida (2001), Kubo, Kuo & Namli (2006), Belinschi & Nica (2007),...
Motivation: Sturm-Liouville operators. Operator Symmetric in L 2 (w(x) dx) for DpD + qd : y (py ) + qy. w w = q p. With appropriate boundary conditions, self-adjoint. Has orthogonal eigenfunctions.
Bochner s Theorem. Theorem. [Bochner 1929] DpD + qd has (orthogonal) polynomial eigenfunctions if and only if deg p 2, deg q 1 and w(x) dx = normal distribution (Hermite polynomials) w(x) dx = Gamma distribution (Legendre polynomials) w(x) dx = Beta distribution (Jacobi polynomials) Pearson (1924): for w w = linear quadratic, two more distributions (Bessel polynomials, t-distribution). Easy (calculus).
Operator L µ. Replace D with L µ. Definition. For µ a measure, f(x) f(y) L µ [f] = dµ(y) = (I µ) [f]. x y R L µ maps polynomials to polynomials, lowers degree by one. Origin: Maps orthogonal polynomials to associated orthogonal polynomials. Related to the generator of the free Brownian motion.
Bochner-Pearson type characterization. Theorem. [A. 09] pl 2 µ + ql µ has polynomial eigenfunctions if and only if µ is a free Meixner distribution.
More on free Meixner distributions. The Cauchy transform But also the Hilbert transform G µ (z) = linear quadratic. quadratic dµ(x) = 1 π lim Im G(x + iε) dx. ε 0 + H[µ](x) = 1 π lim Re G(x + iε) dx = linear ε 0 + In fact for H µ = 2πH[µ], H µ = q p. quadratic.
Classical-free correspondence. DpD + qd pl 2 µ + ql µ. w w q p H µ. Meaning of H µ : free conjugate variable. Random matrix picture: if then for w(x) dx = e V (x) dx, 1 Z e trv (M) dm µ w w = V = H µ.
Random matrix picture. Gaussian Unitary ensemble: e x2 /2 Wigner law (semicircular). Wishart ensemble: x α 1 e x 1 0 x Marchenko-Pastur law (free Poisson). Jacobi ensemble: (1 x) α 1 x β 1 1 0 x 1... (free binomial).
Summary. w w d + ex a + bx + cx 2 H µ. Correspondence between parameters not precise. normal Poisson binomial semicircular Marchenko-Pastur free binomial normal gamma beta hyperbolic secant gamma negative binomial free h.s. free g. free n.b. Bessel t-distribution?