Multivariate Stieltjes continued fractions
Michael Anshelevich
June 17, 2008
µ = probability measure on R, with all moments finite.
Identify with a functional µ on R[x].
Moments
µ[x], µ[x2], µ[x3], . . . .
Moment generating function
M µ(z) = 1 + µ[x]z + µ[x2]z 2 + µ[x3]z 3 + . . . .
1
Has a (Stieltjes) continued fractions expansion
M µ(z) = 1 + µ[x]z + µ[x2]z 2 + µ[x3]z 3 + . . .
1
.
=
2
γ1z
1 − β0 z −
γ2z 2
1 − β1 z −
γ3z 2
1 − β2 z −
1 − ...
Coefficients: Jacobi parameters.
For {Pn(x)} monic orthogonal polynomials for µ,
xPn(x) = Pn+1(x) + βnPn (x) + γnPn−1(x).
2
M ULTIVARIATE CASE
ϕ = positive linear functional on Rhx1, x2, . . . , xdi, polynomials in non-commuting
variables. Unital (= probability). A “state”.
3
M ULTIVARIATE CASE
ϕ = positive linear functional on Rhx1, x2, . . . , xdi, polynomials in non-commuting
variables. Unital (= probability). A “state”.
Where do these appear?
4
M ULTIVARIATE CASE
ϕ = positive linear functional on Rhx1, x2, . . . , xdi, polynomials in non-commuting
variables. Unital (= probability). A “state”.
Where do these appear?
Let X1, X2, . . . , Xd be symmetric (better, self-adjoint) operators on a Hilbert
space H, with common invariant dense domain containing a unit vector Ω.
Then their joint distribution in the vector state Ω is the functional
ϕ [P (x1, x2, . . . , xd)] = hΩ, P (X1 , X2, . . . , Xd)Ωi .
5
M ULTIVARIATE CASE
ϕ = positive linear functional on Rhx1, x2, . . . , xdi, polynomials in non-commuting
variables. Unital (= probability). A “state”.
Where do these appear?
Let X1, X2, . . . , Xd be symmetric (better, self-adjoint) operators on a Hilbert
space H, with common invariant dense domain containing a unit vector Ω.
Then their joint distribution in the vector state Ω is the functional
ϕ [P (x1, x2, . . . , xd)] = hΩ, P (X1 , X2, . . . , Xd)Ωi .
Moment generating function
ϕ
M (z1, z2, . . . , zd) = 1 +
d
X
i=1
ϕ [xi] zi +
d
X
i,j=1
h
i
ϕ xi xj z i z j + . . . .
6
Continued fraction expansion?
M µ(z) = 1 + µ[x]z + µ[x2]z 2 + µ[x3]z 3 + . . .
1
=
γ1z 2
1 − β0 z −
γ2z 2
1 − β1 z −
γ3z 2
1 − β2 z −
1 − ...
can be done by induction, breaks down if some γ = 0. Multivariate:
1+
X
aizi +
X
1
P
P
P
1 − bizi − bij zizj − bijk zizj zk . . .
1
6=
P
P
1 − bizi − bij ziFij zj
aij zizj + . . . =
if e.g. bij = 0, bikj 6= 0.
7
P RODUCT- TYPE EXAMPLES
Distributions with fixed marginals. For simplicity, restrict to the symmetric case:
µ(1), µ(2),
(i)
(i)
(i)
(i)
xiPn (xi) = Pn+1(xi) + γn Pn−1(xi),
i = 1, 2.
Commutative case: many measures µ on R2 with marginals µ(1), µ(2). Canonical one: product measure
µ[x1k x2n] = µ[x1k ]µ[x2n],
x1, x2 independent with respect to µ, orthogonal polynomials
(1)
Pi
(2)
(x1)Pj
(x2) .
8
Non-commutative case: want ϕ on Rhx1, x2i with
(1) n
ϕ [xn
=
µ
[x1 ],
]
1
(2) n
ϕ [xn
=
µ
[x2 ].
]
2
Again many choices, more than one canonical choice.
9
Non-commutative case: want ϕ on Rhx1, x2i with
(1) n
ϕ [xn
=
µ
[x1 ],
]
1
(2) n
ϕ [xn
=
µ
[x2 ].
]
2
Again many choices, more than one canonical choice.
Example.
1
1 + M (z1, z2) =
1−
γ11z12
γ21z12
1−
γ31z12
1−
1 − ...
−
γ12z22
γ22z22
1−
γ32z22
1−
1 − ...
10
Corresponds to the rule
h
ϕ x1
u(1)
x2
v(1)
x1
u(2)
x2
v(2)
i
h
. . . = ϕ x1
u(1)
i
h
ϕ x2
v(1)
i
h
ϕ x1
u(2)
i
h
ϕ x2
(recall x1, x2 do not commute).
Boolean product. Not very natural since very degenerate.
11
v(2)
i
...
Example. Free product, free independence. Rule for
u(1) v(1) u(2) v(2)
x2 x1 x2 . . .
ϕ x1
complicated, but appears in applications, e.g. random matrices.
12
1
1−
γ11 z1|z1
γ12 z2 |z2
γ21 z1|z1
−
1−
γ31 z1|z1 γ12 z2 |z2
γ21 z1|z1 γ22 z2|z2
1−
−
1−
−
1 − ...
1 − ...
1 − ...
1 − ...
−
γ12 z2 |z2
γ22
γ11 z1|z1
−
1−
γ21 z1 |z1 γ12z2 |z2
γ11 z1|
1−
−
1−
1 − ...
1 − ...
1−.
13
1
1−
γ11z12
γ21z12
1−
γ31z12
1−
1 − ...
−
γ12z22
γ22z22
1−
γ32z22
1−
1 − ...
14
For a special but large class of ϕ, have a matricial continued fraction expansion. Namely, when ϕ has monic multivariate orthogonal polynomials:
{P~u = x~u + . . .} ,
for~
u 6= ~v .
∗
ϕ P~u (x1, . . . , xd)P~v (x1, . . . , xd) = 0
Do not always have them. Existence equivalent to special (graded) Hilbert
space H and operators Xi.
15
Theorem. Let ϕ be a state with a monic orthogonal polynomials. There exist
matrices
C (k) = diagonal non-negative dk × dk matrix, k = 1, 2, . . .
and
(k)
Ti
= dk × dk matrix, k = 0, 1, . . . , i = 1, 2, . . . , d
such that
1 + M (z) =
1
.
P
P
(1)
| k1 Ek1 zk1
P
j1 zj1 Ej1 C
(0)
1 − i0 z i0 Ti
−
P
(2)| P E z
0
z
E
C
P
j2 j2 j2
k2 k2 k2
(1)
1 − i1 z i1 Ti
−
1
1 − ...
16
Here for matrices
A, B ∈ Mdk ×dk ≃ Md×d ⊗ Md×d ⊗ . . . ⊗ Md×d,
we use the notation
D
E
EiA|Ej
−1
= ei ⊗ I ⊗ . . . ⊗ I, AB (ej ⊗ I ⊗ . . . ⊗ I) ∈ Mdk−1×dk−1 .
B
Example.

∗
∗

E1 
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗


∗
∗
∗
∗


 E2 = 
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗

∗
∗

.
∗
∗
17