Characterizations of free Meixner
distributions
Michael Anshelevich
Texas A&M University
March 26, 2010
Michael Anshelevich
Characterizations of free Meixner distributions
Jacobi parameters.
Matrix

β0


1

J =
0

0

..
.
γ0
0
0
β1
γ1
0
1
β2
γ2
0
..
.
1
..
β3
..
.
.
..
.


.. 
.
.. 
.
;
.. 
.

..
.

mn
 ∗

Jn = 
 ∗
 ∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗

∗
∗

∗

∗
∗
A new sequence m1 , m2 , m3 , . . ..
Favard’s Theorem. [Stone 1932] All γi ≥ 0 ⇔ {mi } are
moments of a measure,
Z
mi = xi dµ(x).
Michael Anshelevich
Characterizations of free Meixner distributions
Definition.
b, c, t ∈ R, t, t + c ≥ 0.
Tridiagonal matrix

0
1

J =
0
0
0

t
0
0
0
b t+c
0
0 

1
b
t+c
0 

0
1
b
t + c
0
0
1
b
↔
µtb,c .
µtb,c = free Meixner distributions.
Bernstein-Szegő class: βi , γi eventually constant.
Michael Anshelevich
Characterizations of free Meixner distributions
Formula.
In terms of the Cauchy transform
Z
1
Gµ (z) =
dµ(x),
R z−x
dµ(x) = −
r
µtb,c =
1
·
2π
1
lim Im Gµ (x + iε) dx
π ε→0+
4(t + c) − (x − b)2
t + bx +
(c/t)x2
Michael Anshelevich
+
dx + 0, 1,or 2 atoms.
Characterizations of free Meixner distributions
Examples.
µ0,1 ,
−2
−1
0
1
µ0,0 ,
2
−2
−1
Michael Anshelevich
0
µ0,1/2 .
1
2
−2
−1
0
1
2
Characterizations of free Meixner distributions
Semicircle law.
b, c = 0.
µt0,0 =
1 p
4t − x2 .
2πt
Orthogonal polynomials: Chebyshev polynomials of the second
kind Un (x).
Generating function
∞
X
Un (x, t)wn =
n=0
Michael Anshelevich
1
.
1 − xw + tw2
Characterizations of free Meixner distributions
First characterization.
Theorem. [A. ’03]
Orthogonal polynomials Pn (x) for a measure µ have a generating function of the “resolvent-type” form
∞
X
Pn (x)wn = A(w)
n=0
1
1 − B(w)x
if and only if µ is a free Meixner distribution.
Michael Anshelevich
Characterizations of free Meixner distributions
Analogy: Meixner class.
Theorem. [Meixner 1934]
Orthogonal polynomials Pn (x) for a measure µ have a generating function of the exponential form
∞
X
1
Pn (x)wn = A(w)eB(w)x
n!
n=0
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.
Michael Anshelevich
Characterizations of free Meixner distributions
Lévy process.
Random variables
{Xt : t ∈ R},
Xt0 · · · · · · · · · · · · Xt1 · · · · · · Xt2 · · · · · · · · · · · · · · · · · · Xt3 .
Increments
X(t1 ) − X(t0 ),
X(t2 ) − X(t1 ), . . . ,
X(tk ) − X(tk−1 )
independent and stationary: (Xt − Xs ) ∼ Xt−s ∼ µt−s .
Example. Bt = Brownian motion.
1 −x2 /t
e
dx.
µt = √
2πt
Michael Anshelevich
Characterizations of free Meixner distributions
Conditional expectation.
E [·] = expectation.
E [·|X] = conditional expectation = projection.
E [f (X)|X] = f (X),
E [Y |X] = E [Y ] if Y, X independent.
Michael Anshelevich
Characterizations of free Meixner distributions
Exponential martingale.
Denote
h
iXt θ
Ft (θ) = E e
i
Z
=
eixθ dµt (x)
the Fourier transform.
h
i
E eiXt θ−log Ft (θ) |Xs = eiXs θ−log Fs (θ) .
t 7→ eiXt θ−log Ft (θ)
is a martingale.
Brownian motion
θ2
eiBt θ−t 2 .
Michael Anshelevich
Characterizations of free Meixner distributions
Other martingales.
t 7→ eiXt θ−log Ft (θ) .
Michael Anshelevich
Characterizations of free Meixner distributions
Proof.
For X, Y independent
FX+Y (θ) = FX (θ)FY (θ)
Fµt (θ) = F t (θ)
log Ft (θ) = t log F(θ).
h
iXt θ−log Ft (θ)
E e
i
h
i
i(Xt −Xs )θ iXs θ −t log F (θ)
|Xs = E e
e
e
|Xs
h
i
= E ei(Xt −Xs )θ eiXs θ e−t log F (θ)
= e(t−s) log F (θ) e−t log F (θ) eiXs θ = eiXs θ−log sF (θ)
Michael Anshelevich
Characterizations of free Meixner distributions
Martingale polynomials.
Generating function
exz−t log F (−iz) =
∞
X
1
Pn (x, t)z n .
n!
n=0
Each Pn = martingale polynomial.
Brownian motion: Hn (x, t) Hermite polynomials.
Hn (Bt , t) = martingale.
Meixner class: orthogonal martingale polynomials.
Michael Anshelevich
Characterizations of free Meixner distributions
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.
Michael Anshelevich
Characterizations of free Meixner distributions
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.
Michael Anshelevich
Characterizations of free Meixner distributions
Examples.
Free central limit theorem.
Limit = semicircular distribution = µ0,0 .
Free Poisson limit theorem.
Limit = free Poisson distribution = µ1,0 .
Binomial distribution = sum of independent coin tosses.
Coin toss = projection.
Sum of freely independent projections = free binomial distribution = µt0,−1 , t = number of tosses. In the free case, t ≥ 1 real!
Michael Anshelevich
Characterizations of free Meixner distributions
Processes with free increments.
Xt = process with freely independent increments.
Theorem. [Biane 1998]. {Xt } is a Markov processes.
Equivalently:
t 7→
1
1 − Xt z + tR(z)
is a martingale.
Here R(z) = R-transform (Voiculescu), analog of log F(iθ).
Free Meixner processes have orthogonal martingale
polynomials.
Michael Anshelevich
Characterizations of free Meixner distributions
Reverse martingales.
{Xt } a process with freely independent increments.
{Pn (x, t)} polynomials, Pn of degree n.
Pn a martingale if for s < t,
E [Pn (Xt , t)|Xs ] = Pn (Xs , s).
Pn a reverse martingale if for s < t,
E [Pn (Xs , s)|Xt ] =
Michael Anshelevich
f (s)
· Pn (Xt , t).
f (t)
Characterizations of free Meixner distributions
Characterization: reverse martingales.
Theorem. [Laha, Lukacs 1960]
Their result implies:
A Lévy process {Xt } has a family of polynomials Pn which are
both martingales and reverse martingales
if and only if each Xs has a Meixner distribution.
Theorem. [Bożejko, Bryc ’06]
Their result implies:
A free Lévy process {Xt } has a family of polynomials Pn which
are both martingales and reverse martingales
if and only if each Xs has a free Meixner distribution.
Michael Anshelevich
Characterizations of free Meixner distributions
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), . . .
Michael Anshelevich
Characterizations of free Meixner distributions
Motivation: Sturm-Liouville operators.
Operator
y 7→ (py 0 )0 + qy 0 .
DpD + qD :
Symmetric in L2 (w(x) dx) for
w0
q
= .
w
p
With appropriate boundary conditions, self-adjoint.
Has orthogonal eigenfunctions.
Michael Anshelevich
Characterizations of free Meixner distributions
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
w0
linear
=
,
w
quadratic
two more distributions (Bessel polynomials, t-distribution).
Easy (calculus).
Michael Anshelevich
Characterizations of free Meixner distributions
Operator Lµ .
Replace D with Lµ .
Definition. For µ a measure,
Z
f (x) − f (y)
dµ(y) = (I ⊗ µ)∂[f ].
Lµ [f ] =
x−y
R
Lµ 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.
Michael Anshelevich
Characterizations of free Meixner distributions
Bochner-Pearson type characterization.
Theorem. [A. ’09]
pL2µ + qLµ
has polynomial eigenfunctions if and only if µ is a free Meixner
distribution.
Michael Anshelevich
Characterizations of free Meixner distributions
More on free Meixner distributions.
The Cauchy transform
p
−linear − quadratic
.
Gµ (z) =
quadratic
dµ(x) = −
1
lim Im G(x + iε) dx.
π ε→0+
But also the Hilbert transform
H[µ](x) =
1
linear
lim Re G(x + iε) dx =
.
+
π ε→0
quadratic
In fact for Hµ = 2πH[µ],
q
−Hµ = .
p
Michael Anshelevich
Characterizations of free Meixner distributions
Classical-free correspondence.
DpD + qD ↔ pL2µ + qLµ .
w0
q
↔ ↔ −Hµ .
w
p
Meaning of Hµ : free conjugate variable.
Random matrix picture: if
w(x) dx = e−V (x) dx,
then
1 −trV (M )
e
dM → µ
Z
for
−
w0
= V 0 = Hµ .
w
Michael Anshelevich
Characterizations of free Meixner distributions
Random matrix picture.
Gaussian Unitary ensemble:
e−x
2 /2
→ Wigner law (semicircular).
Wishart ensemble:
xα−1 e−x 10≤x → Marchenko-Pastur law (free Poisson).
Jacobi ensemble:
(1 − x)α−1 xβ−1 10≤x≤1 → . . . (free binomial).
Michael Anshelevich
Characterizations of free Meixner distributions
Summary.
d + ex
w0
↔
↔ −Hµ .
w
a + bx + cx2
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
?
Michael Anshelevich
Characterizations of free Meixner distributions