[PPT] - Complex Analytic Methods in Free vvb Probability Theory Hari PowerPoint Presentation

SLIDE 1

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Complex Analytic Methods in Free Probability Theory

Hari Bercovici BIRS, January 2008

SLIDE 2

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Random variables

(A, τ) a W*-probability space; A = L∞(τ)

SLIDE 3

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Random variables

(A, τ) a W*-probability space; A = L∞(τ)
L0(τ) = {y−1x : x, y ∈ A, y = 0 a.s.} arbitrary random

variables

SLIDE 4

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Random variables

(A, τ) a W*-probability space; A = L∞(τ)
L0(τ) = {y−1x : x, y ∈ A, y = 0 a.s.} arbitrary random

variables

This is still a *-algebra

SLIDE 5

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Random variables

(A, τ) a W*-probability space; A = L∞(τ)
L0(τ) = {y−1x : x, y ∈ A, y = 0 a.s.} arbitrary random

variables

This is still a *-algebra
x ∈ L0(τ), x = x∗ =

∞

−∞ t dex(t)

SLIDE 6

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Random variables

(A, τ) a W*-probability space; A = L∞(τ)
L0(τ) = {y−1x : x, y ∈ A, y = 0 a.s.} arbitrary random

variables

This is still a *-algebra
x ∈ L0(τ), x = x∗ =

∞

−∞ t dex(t)

µx(σ) = τ(ex(σ)) probability distribution of x

SLIDE 7

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Random variables

(A, τ) a W*-probability space; A = L∞(τ)
L0(τ) = {y−1x : x, y ∈ A, y = 0 a.s.} arbitrary random

variables

This is still a *-algebra
x ∈ L0(τ), x = x∗ =

∞

−∞ t dex(t)

µx(σ) = τ(ex(σ)) probability distribution of x
Ax w* closed algebra generated by

{ex((−∞, t)) : t ∈ R} (σ-algebra of x)

SLIDE 8

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Freeness

(xi)i∈I ⊂ L0(τ), xi = x∗

i

SLIDE 9

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Freeness

(xi)i∈I ⊂ L0(τ), xi = x∗

i

(xi)i∈I is free if (Axi)i∈I are free in (A, τ)

SLIDE 10

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Freeness

(xi)i∈I ⊂ L0(τ), xi = x∗

i

(xi)i∈I is free if (Axi)i∈I are free in (A, τ)
When x not selfadjoint, x = h + ik, Ax generated by

Ah ∪ Ak

SLIDE 11

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Freeness

(xi)i∈I ⊂ L0(τ), xi = x∗

i

(xi)i∈I is free if (Axi)i∈I are free in (A, τ)
When x not selfadjoint, x = h + ik, Ax generated by

Ah ∪ Ak

Corresponding notion: *-freeness

SLIDE 12

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Convolutions

x = x∗, y = y∗ free variables

SLIDE 13

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Convolutions

x = x∗, y = y∗ free variables
µx+y depends only on µx and µy

SLIDE 14

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Convolutions

x = x∗, y = y∗ free variables
µx+y depends only on µx and µy
µx+y = µx ⊞ µy free additive convolution

SLIDE 15

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Convolutions

x = x∗, y = y∗ free variables
µx+y depends only on µx and µy
µx+y = µx ⊞ µy free additive convolution
assume x ≥ 0

SLIDE 16

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Convolutions

x = x∗, y = y∗ free variables
µx+y depends only on µx and µy
µx+y = µx ⊞ µy free additive convolution
assume x ≥ 0
µ√xy√x depends only on µx and µy

SLIDE 17

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Convolutions

x = x∗, y = y∗ free variables
µx+y depends only on µx and µy
µx+y = µx ⊞ µy free additive convolution
assume x ≥ 0
µ√xy√x depends only on µx and µy
µ√xy√x = µx ⊠ µy free multiplicative convolution

SLIDE 18

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Convolutions

x = x∗, y = y∗ free variables
µx+y depends only on µx and µy
µx+y = µx ⊞ µy free additive convolution
assume x ≥ 0
µ√xy√x depends only on µx and µy
µ√xy√x = µx ⊠ µy free multiplicative convolution
x, y free unitary, µxy = µyx = µx ⊠ µy

SLIDE 19

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Convolutions

x = x∗, y = y∗ free variables
µx+y depends only on µx and µy
µx+y = µx ⊞ µy free additive convolution
assume x ≥ 0
µ√xy√x depends only on µx and µy
µ√xy√x = µx ⊠ µy free multiplicative convolution
x, y free unitary, µxy = µyx = µx ⊠ µy
same notation, different semigroup

SLIDE 20

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Examples

δr ⊞ δs = δr+s; cr ⊞ cs = cr+s for Cauchy (arctangent)

dcr = r dt π(t2 + r 2)

SLIDE 21

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Examples

δr ⊞ δs = δr+s; cr ⊞ cs = cr+s for Cauchy (arctangent)

dcr = r dt π(t2 + r 2)

γr ⊞ γs = γr+s for semicircle of variance r

dγr = 1 2πr

4r − t2 dt

SLIDE 22

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Examples

δr ⊞ δs = δr+s; cr ⊞ cs = cr+s for Cauchy (arctangent)

dcr = r dt π(t2 + r 2)

γr ⊞ γs = γr+s for semicircle of variance r

dγr = 1 2πr

4r − t2 dt
µ = ν = (δ1 + δ−1)/2

SLIDE 23

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Examples

δr ⊞ δs = δr+s; cr ⊞ cs = cr+s for Cauchy (arctangent)

dcr = r dt π(t2 + r 2)

γr ⊞ γs = γr+s for semicircle of variance r

dγr = 1 2πr

4r − t2 dt
µ = ν = (δ1 + δ−1)/2
µ ⊞ ν =

dt π √ 4 − t2

SLIDE 24

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Cauchy transforms

µ probability distribution on R

SLIDE 25

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Cauchy transforms

µ probability distribution on R
z ∈ C+ upper half-plane

SLIDE 26

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Cauchy transforms

µ probability distribution on R
z ∈ C+ upper half-plane
Gµ(z) =

∞

−∞

dµ(t) z − t

SLIDE 27

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Cauchy transforms

µ probability distribution on R
z ∈ C+ upper half-plane
Gµ(z) =

∞

−∞

dµ(t) z − t

Fµ(z) = 1/Gµ(z); Fµ : C+ → C+

SLIDE 28

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Cauchy transforms

µ probability distribution on R
z ∈ C+ upper half-plane
Gµ(z) =

∞

−∞

dµ(t) z − t

Fµ(z) = 1/Gµ(z); Fµ : C+ → C+
Function inverse F <−1>

µ

defined in Dr,ε = {z : |z − ir| < (1 − ε)r} for large r

SLIDE 29

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Cauchy transforms

µ probability distribution on R
z ∈ C+ upper half-plane
Gµ(z) =

∞

−∞

dµ(t) z − t

Fµ(z) = 1/Gµ(z); Fµ : C+ → C+
Function inverse F <−1>

µ

defined in Dr,ε = {z : |z − ir| < (1 − ε)r} for large r

ϕµ(z) = Rµ(1/z) = F <−1>

µ

(z) − z V-transform of µ

SLIDE 30

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Cauchy transforms

µ probability distribution on R
z ∈ C+ upper half-plane
Gµ(z) =

∞

−∞

dµ(t) z − t

Fµ(z) = 1/Gµ(z); Fµ : C+ → C+
Function inverse F <−1>

µ

defined in Dr,ε = {z : |z − ir| < (1 − ε)r} for large r

ϕµ(z) = Rµ(1/z) = F <−1>

µ

(z) − z V-transform of µ

ϕµ⊞ν = ϕµ + ϕν

SLIDE 31

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Cauchy transforms

µ probability distribution on R
z ∈ C+ upper half-plane
Gµ(z) =

∞

−∞

dµ(t) z − t

Fµ(z) = 1/Gµ(z); Fµ : C+ → C+
Function inverse F <−1>

µ

defined in Dr,ε = {z : |z − ir| < (1 − ε)r} for large r

ϕµ(z) = Rµ(1/z) = F <−1>

µ

(z) − z V-transform of µ

ϕµ⊞ν = ϕµ + ϕν
There are corresponding results for ⊠

SLIDE 32

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

A basic tool

ϕµ(z) = F <−1>

µ

(z) − z

SLIDE 33

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

A basic tool

ϕµ(z) = F <−1>

µ

(z) − z

ϕµ(z) ≈ z − Fµ(z) in Dr,ε as r → ∞

SLIDE 34

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

A basic tool

ϕµ(z) = F <−1>

µ

(z) − z

ϕµ(z) ≈ z − Fµ(z) in Dr,ε as r → ∞
The approximation is uniformly better when µ is

concentrated near zero.

SLIDE 35

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

A basic tool

ϕµ(z) = F <−1>

µ

(z) − z

ϕµ(z) ≈ z − Fµ(z) in Dr,ε as r → ∞
The approximation is uniformly better when µ is

concentrated near zero.

meaning: ratio closer to one

SLIDE 36

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

A basic tool

ϕµ(z) = F <−1>

µ

(z) − z

ϕµ(z) ≈ z − Fµ(z) in Dr,ε as r → ∞
The approximation is uniformly better when µ is

concentrated near zero.

meaning: ratio closer to one
larger r
smaller ε

SLIDE 37

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Some results

µn → µ equivalent to ϕµn → ϕµ in Dr,ε, ε fixed, r large,

with some uniformity at ∞

SLIDE 38

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Some results

µn → µ equivalent to ϕµn → ϕµ in Dr,ε, ε fixed, r large,

with some uniformity at ∞

{µn}n≥1, νn = µ1 ⊞ µ2 ⊞ · · · ⊞ µn, ρn = µ1 ∗ µ2 ∗ · · · ∗ µn

SLIDE 39

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Some results

µn → µ equivalent to ϕµn → ϕµ in Dr,ε, ε fixed, r large,

with some uniformity at ∞

{µn}n≥1, νn = µ1 ⊞ µ2 ⊞ · · · ⊞ µn, ρn = µ1 ∗ µ2 ∗ · · · ∗ µn
νn → ν ⇔ ρn → ρ (three series theorem)

SLIDE 40

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Some results

µn → µ equivalent to ϕµn → ϕµ in Dr,ε, ε fixed, r large,

with some uniformity at ∞

{µn}n≥1, νn = µ1 ⊞ µ2 ⊞ · · · ⊞ µn, ρn = µ1 ∗ µ2 ∗ · · · ∗ µn
νn → ν ⇔ ρn → ρ (three series theorem)
n-divisibility: µ = ν ⊞ ν ⊞ · · · ⊞ ν
n times

SLIDE 41

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Some results

µn → µ equivalent to ϕµn → ϕµ in Dr,ε, ε fixed, r large,

with some uniformity at ∞

{µn}n≥1, νn = µ1 ⊞ µ2 ⊞ · · · ⊞ µn, ρn = µ1 ∗ µ2 ∗ · · · ∗ µn
νn → ν ⇔ ρn → ρ (three series theorem)
n-divisibility: µ = ν ⊞ ν ⊞ · · · ⊞ ν
n times
infinite-divisibility: all n

SLIDE 42

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Some results

µn → µ equivalent to ϕµn → ϕµ in Dr,ε, ε fixed, r large,

with some uniformity at ∞

{µn}n≥1, νn = µ1 ⊞ µ2 ⊞ · · · ⊞ µn, ρn = µ1 ∗ µ2 ∗ · · · ∗ µn
νn → ν ⇔ ρn → ρ (three series theorem)
n-divisibility: µ = ν ⊞ ν ⊞ · · · ⊞ ν
n times
infinite-divisibility: all n
µ is ∞-divisible ⇔ϕµ : C+ → C−

SLIDE 43

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Limit laws

{µnj : n ≥ 1, 1 ≤ j ≤ kn} distributions on R

SLIDE 44

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Limit laws

{µnj : n ≥ 1, 1 ≤ j ≤ kn} distributions on R
Infinitesimal if for all ε > 0

lim

n→∞ min 1≤j≤kn µnj((−ε, ε)) = 1

SLIDE 45

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Limit laws

{µnj : n ≥ 1, 1 ≤ j ≤ kn} distributions on R
Infinitesimal if for all ε > 0

lim

n→∞ min 1≤j≤kn µnj((−ε, ε)) = 1

µn = µn1 ⊞ µn2 ⊞ · · · ⊞ µnkn, νn = µn1 ∗ µn2 ∗ · · · ∗ µnkn

SLIDE 46

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Limit laws

{µnj : n ≥ 1, 1 ≤ j ≤ kn} distributions on R
Infinitesimal if for all ε > 0

lim

n→∞ min 1≤j≤kn µnj((−ε, ε)) = 1

µn = µn1 ⊞ µn2 ⊞ · · · ⊞ µnkn, νn = µn1 ∗ µn2 ∗ · · · ∗ µnkn
If µn → µ then µ is ∞-divisible

SLIDE 47

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Limit laws

{µnj : n ≥ 1, 1 ≤ j ≤ kn} distributions on R
Infinitesimal if for all ε > 0

lim

n→∞ min 1≤j≤kn µnj((−ε, ε)) = 1

µn = µn1 ⊞ µn2 ⊞ · · · ⊞ µnkn, νn = µn1 ∗ µn2 ∗ · · · ∗ µnkn
If µn → µ then µ is ∞-divisible
µn → µ ⇔ νn → ν, where ν is uniquely determined by

µ. (Ex.: µ semicircle corresponds with ν normal; CLT)

SLIDE 48

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Limit laws

{µnj : n ≥ 1, 1 ≤ j ≤ kn} distributions on R
Infinitesimal if for all ε > 0

lim

n→∞ min 1≤j≤kn µnj((−ε, ε)) = 1

µn = µn1 ⊞ µn2 ⊞ · · · ⊞ µnkn, νn = µn1 ∗ µn2 ∗ · · · ∗ µnkn
If µn → µ then µ is ∞-divisible
µn → µ ⇔ νn → ν, where ν is uniquely determined by

µ. (Ex.: µ semicircle corresponds with ν normal; CLT)

almost analogous results for ⊠

SLIDE 49

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Limit laws

{µnj : n ≥ 1, 1 ≤ j ≤ kn} distributions on R
Infinitesimal if for all ε > 0

lim

n→∞ min 1≤j≤kn µnj((−ε, ε)) = 1

µn = µn1 ⊞ µn2 ⊞ · · · ⊞ µnkn, νn = µn1 ∗ µn2 ∗ · · · ∗ µnkn
If µn → µ then µ is ∞-divisible
µn → µ ⇔ νn → ν, where ν is uniquely determined by

µ. (Ex.: µ semicircle corresponds with ν normal; CLT)

almost analogous results for ⊠
differences: the correspondence µ ↔ ν not bijective

SLIDE 50

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Limit laws

{µnj : n ≥ 1, 1 ≤ j ≤ kn} distributions on R
Infinitesimal if for all ε > 0

lim

n→∞ min 1≤j≤kn µnj((−ε, ε)) = 1

µn = µn1 ⊞ µn2 ⊞ · · · ⊞ µnkn, νn = µn1 ∗ µn2 ∗ · · · ∗ µnkn
If µn → µ then µ is ∞-divisible
µn → µ ⇔ νn → ν, where ν is uniquely determined by

µ. (Ex.: µ semicircle corresponds with ν normal; CLT)

almost analogous results for ⊠
differences: the correspondence µ ↔ ν not bijective
for the circle, there are no ⊠-idempotents

SLIDE 51

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Another basic tool

f, g : C+ → C analytic

SLIDE 52

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Another basic tool

f, g : C+ → C analytic
f ≺ g if f(z) = g(h(z)) for some h : C+ → C+ analytic

(Littlewood subordination)

SLIDE 53

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Another basic tool

f, g : C+ → C analytic
f ≺ g if f(z) = g(h(z)) for some h : C+ → C+ analytic

(Littlewood subordination)

µ, ν distributions on R

SLIDE 54

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Another basic tool

f, g : C+ → C analytic
f ≺ g if f(z) = g(h(z)) for some h : C+ → C+ analytic

(Littlewood subordination)

µ, ν distributions on R
Then Fµ⊞ν ≺ Fµ (and Fµ⊞ν ≺ Fν)

SLIDE 55

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Another basic tool

f, g : C+ → C analytic
f ≺ g if f(z) = g(h(z)) for some h : C+ → C+ analytic

(Littlewood subordination)

µ, ν distributions on R
Then Fµ⊞ν ≺ Fµ (and Fµ⊞ν ≺ Fν)
Note: subordination functions F <−1>

µ

Fµ⊞ν obviously

exist at ∞; the important point is they continue to C+.

SLIDE 56

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Regularity consequences

If dµ/dt ∈ Lp, same is true for µ ⊞ ν

SLIDE 57

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Regularity consequences

If dµ/dt ∈ Lp, same is true for µ ⊞ ν
µ ⊞ ν has only finitely many atoms, with total mass < 1

SLIDE 58

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Regularity consequences

If dµ/dt ∈ Lp, same is true for µ ⊞ ν
µ ⊞ ν has only finitely many atoms, with total mass < 1
µ ⊞ ν has no singular continuous component

SLIDE 59

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Regularity consequences

If dµ/dt ∈ Lp, same is true for µ ⊞ ν
µ ⊞ ν has only finitely many atoms, with total mass < 1
µ ⊞ ν has no singular continuous component
the density of µ ⊞ ν is locally analytic a.e.

SLIDE 60

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Regularity consequences

If dµ/dt ∈ Lp, same is true for µ ⊞ ν
µ ⊞ ν has only finitely many atoms, with total mass < 1
µ ⊞ ν has no singular continuous component
the density of µ ⊞ ν is locally analytic a.e.
But: the density of µ ⊞ ν may have points of

nondifferentiability even when those of µ and ν don’t

SLIDE 61

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Ignorance

µ distribution of R

SLIDE 62

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Ignorance

µ distribution of R
ν symmetric to µ (µ = µx, ν = µ−x)

SLIDE 63

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Ignorance

µ distribution of R
ν symmetric to µ (µ = µx, ν = µ−x)
ρ = µ ∗ ν, λ = µ ⊞ ν; ρ and λ are symmetric

SLIDE 64

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Ignorance

µ distribution of R
ν symmetric to µ (µ = µx, ν = µ−x)
ρ = µ ∗ ν, λ = µ ⊞ ν; ρ and λ are symmetric
tails of ρ: 1 − ρ((−t, t)), t → ∞

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f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Ignorance

µ distribution of R
ν symmetric to µ (µ = µx, ν = µ−x)
ρ = µ ∗ ν, λ = µ ⊞ ν; ρ and λ are symmetric
tails of ρ: 1 − ρ((−t, t)), t → ∞
are comparable to those of µ

SLIDE 66

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Ignorance

µ distribution of R
ν symmetric to µ (µ = µx, ν = µ−x)
ρ = µ ∗ ν, λ = µ ⊞ ν; ρ and λ are symmetric
tails of ρ: 1 − ρ((−t, t)), t → ∞
are comparable to those of µ
what about λ?

SLIDE 67

f(z), µ ⊞ ν, etc.

H. Bercovici

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Operator-valued

A algebra, B ⊂ A unital subalgebra

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H. Bercovici

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Operator-valued

A algebra, B ⊂ A unital subalgebra
τ : A → B conditional expectation

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H. Bercovici

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Operator-valued

A algebra, B ⊂ A unital subalgebra
τ : A → B conditional expectation
projection so τ(bab′) = bτ(a)b′ for b, b′ ∈ B

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H. Bercovici

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Operator-valued

A algebra, B ⊂ A unital subalgebra
τ : A → B conditional expectation
projection so τ(bab′) = bτ(a)b′ for b, b′ ∈ B
Freeness can be defined relative to such τ

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H. Bercovici

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Operator-valued

A algebra, B ⊂ A unital subalgebra
τ : A → B conditional expectation
projection so τ(bab′) = bτ(a)b′ for b, b′ ∈ B
Freeness can be defined relative to such τ
x ∈ A has a combinatorial analogue of a distribution

τ(ab1ab2a · · · bn−1a)

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H. Bercovici

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Operator-valued

A algebra, B ⊂ A unital subalgebra
τ : A → B conditional expectation
projection so τ(bab′) = bτ(a)b′ for b, b′ ∈ B
Freeness can be defined relative to such τ
x ∈ A has a combinatorial analogue of a distribution

τ(ab1ab2a · · · bn−1a)

This is a “Fourier coefficient” of order n

SLIDE 73

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Operator-valued

A algebra, B ⊂ A unital subalgebra
τ : A → B conditional expectation
projection so τ(bab′) = bτ(a)b′ for b, b′ ∈ B
Freeness can be defined relative to such τ
x ∈ A has a combinatorial analogue of a distribution

τ(ab1ab2a · · · bn−1a)

This is a “Fourier coefficient” of order n
µx+y = µx ⊞ µy where ⊞ is defined combinatorially

SLIDE 74

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Operator-valued

A algebra, B ⊂ A unital subalgebra
τ : A → B conditional expectation
projection so τ(bab′) = bτ(a)b′ for b, b′ ∈ B
Freeness can be defined relative to such τ
x ∈ A has a combinatorial analogue of a distribution

τ(ab1ab2a · · · bn−1a)

This is a “Fourier coefficient” of order n
µx+y = µx ⊞ µy where ⊞ is defined combinatorially
Subordination survives

SLIDE 75

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Analytic despair

Gx(z) = τ((z − x)−1)

SLIDE 76

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Analytic despair

Gx(z) = τ((z − x)−1)
To be viewed as a function of z ∈ B. (If x = x∗, z can

be anything with positive invertible imaginary part; Siegel half-plane.)

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Analytic despair

Gx(z) = τ((z − x)−1)
To be viewed as a function of z ∈ B. (If x = x∗, z can

be anything with positive invertible imaginary part; Siegel half-plane.)

Function theory not sufficiently well developed to

undertake the analysis available when B = C.

SLIDE 78

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Analytic despair

Gx(z) = τ((z − x)−1)
To be viewed as a function of z ∈ B. (If x = x∗, z can

be anything with positive invertible imaginary part; Siegel half-plane.)

Function theory not sufficiently well developed to

undertake the analysis available when B = C.

Fully matricial analytic functions may be needed for full

understanding.

SLIDE 79

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

⊞λ

Xn(ω) an n × n random matrix, Xn = X ∗

n

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

⊞λ

Xn(ω) an n × n random matrix, Xn = X ∗

n

With E expected value, τn = Tr/n, Xn has distribution

given by GµXn(z) = Eτn((zI − Xn)−1) z ∈ C+

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

⊞λ

Xn(ω) an n × n random matrix, Xn = X ∗

n

With E expected value, τn = Tr/n, Xn has distribution

given by GµXn(z) = Eτn((zI − Xn)−1) z ∈ C+

Asymptotic (eigenvalue) distribution of Xn: limn µXn

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

⊞λ

Xn(ω) an n × n random matrix, Xn = X ∗

n

With E expected value, τn = Tr/n, Xn has distribution

given by GµXn(z) = Eτn((zI − Xn)−1) z ∈ C+

Asymptotic (eigenvalue) distribution of Xn: limn µXn
Xn, Yn independent (classically) with asymptotic

distributions µ, ν

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f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

⊞λ

Xn(ω) an n × n random matrix, Xn = X ∗

n

With E expected value, τn = Tr/n, Xn has distribution

given by GµXn(z) = Eτn((zI − Xn)−1) z ∈ C+

Asymptotic (eigenvalue) distribution of Xn: limn µXn
Xn, Yn independent (classically) with asymptotic

distributions µ, ν

Then (spoonful of salt here): Xn + Yn has asymptotic

distribution µ ⊞ ν.

SLIDE 84

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

⊞λ

Xn(ω) an n × n random matrix, Xn = X ∗

n

With E expected value, τn = Tr/n, Xn has distribution

given by GµXn(z) = Eτn((zI − Xn)−1) z ∈ C+

Asymptotic (eigenvalue) distribution of Xn: limn µXn
Xn, Yn independent (classically) with asymptotic

distributions µ, ν

Then (spoonful of salt here): Xn + Yn has asymptotic

distribution µ ⊞ ν.

Assume now Xn, Yn are n × λn matrices, and

|Xn| = (X ∗

n Xn)1/2, |Yn| have asymptotic distributions µ, ν

SLIDE 85

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

⊞λ

Xn(ω) an n × n random matrix, Xn = X ∗

n

With E expected value, τn = Tr/n, Xn has distribution

given by GµXn(z) = Eτn((zI − Xn)−1) z ∈ C+

Asymptotic (eigenvalue) distribution of Xn: limn µXn
Xn, Yn independent (classically) with asymptotic

distributions µ, ν

Then (spoonful of salt here): Xn + Yn has asymptotic

distribution µ ⊞ ν.

Assume now Xn, Yn are n × λn matrices, and

|Xn| = (X ∗

n Xn)1/2, |Yn| have asymptotic distributions µ, ν

Then: |Xn + Yn| has asymptotic distribution µ ⊞λ ν

SLIDE 86

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

⊞λ

Xn(ω) an n × n random matrix, Xn = X ∗

n

With E expected value, τn = Tr/n, Xn has distribution

given by GµXn(z) = Eτn((zI − Xn)−1) z ∈ C+

Asymptotic (eigenvalue) distribution of Xn: limn µXn
Xn, Yn independent (classically) with asymptotic

distributions µ, ν

Then (spoonful of salt here): Xn + Yn has asymptotic

distribution µ ⊞ ν.

Assume now Xn, Yn are n × λn matrices, and

|Xn| = (X ∗

n Xn)1/2, |Yn| have asymptotic distributions µ, ν

Then: |Xn + Yn| has asymptotic distribution µ ⊞λ ν
many results extend to this operation, questions remain

SLIDE 87

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Boolean

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

SLIDE 88

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Boolean

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are Boolean independent if for bj ∈ B, cj ∈ C,

τ(·b1c1b2c2 · · · bncn·) = ·τ(b1)τ(c1) · · · τ(bn)τ(cn)·

SLIDE 89

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Boolean

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are Boolean independent if for bj ∈ B, cj ∈ C,

τ(·b1c1b2c2 · · · bncn·) = ·τ(b1)τ(c1) · · · τ(bn)τ(cn)·

µb+c = µb ⊎ µc for b ∈ B, c ∈ C

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f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Boolean

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are Boolean independent if for bj ∈ B, cj ∈ C,

τ(·b1c1b2c2 · · · bncn·) = ·τ(b1)τ(c1) · · · τ(bn)τ(cn)·

µb+c = µb ⊎ µc for b ∈ B, c ∈ C
analytic calculation:

Fµ⊎ν(z) − z = [Fµ(z) − z] + [Fν(z) − z]

SLIDE 91

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Boolean

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are Boolean independent if for bj ∈ B, cj ∈ C,

τ(·b1c1b2c2 · · · bncn·) = ·τ(b1)τ(c1) · · · τ(bn)τ(cn)·

µb+c = µb ⊎ µc for b ∈ B, c ∈ C
analytic calculation:

Fµ⊎ν(z) − z = [Fµ(z) − z] + [Fν(z) − z]

much of the limit theory is preserved, but

SLIDE 92

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Boolean

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are Boolean independent if for bj ∈ B, cj ∈ C,

τ(·b1c1b2c2 · · · bncn·) = ·τ(b1)τ(c1) · · · τ(bn)τ(cn)·

µb+c = µb ⊎ µc for b ∈ B, c ∈ C
analytic calculation:

Fµ⊎ν(z) − z = [Fµ(z) − z] + [Fν(z) − z]

much of the limit theory is preserved, but
generally δs ⊎ δt = δs+t

SLIDE 93

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Boolean

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are Boolean independent if for bj ∈ B, cj ∈ C,

τ(·b1c1b2c2 · · · bncn·) = ·τ(b1)τ(c1) · · · τ(bn)τ(cn)·

µb+c = µb ⊎ µc for b ∈ B, c ∈ C
analytic calculation:

Fµ⊎ν(z) − z = [Fµ(z) − z] + [Fν(z) − z]

much of the limit theory is preserved, but
generally δs ⊎ δt = δs+t
There is a multiplicative analogue

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Monotonic

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Monotonic

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are monotone independent if for bj ∈ B, cj ∈ C,

τ(b1c1) = τ(c1b1) = τ(b1)τ(c1), τ(c1b1c2) = τ(c1)τ(b1)τ(c2), and b1c1b2 = τ(c1)b1b2,

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Monotonic

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are monotone independent if for bj ∈ B, cj ∈ C,

τ(b1c1) = τ(c1b1) = τ(b1)τ(c1), τ(c1b1c2) = τ(c1)τ(b1)τ(c2), and b1c1b2 = τ(c1)b1b2,

µb+c = µb ⊲ µc b ∈ B, c ∈ C

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Monotonic

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are monotone independent if for bj ∈ B, cj ∈ C,

τ(b1c1) = τ(c1b1) = τ(b1)τ(c1), τ(c1b1c2) = τ(c1)τ(b1)τ(c2), and b1c1b2 = τ(c1)b1b2,

µb+c = µb ⊲ µc b ∈ B, c ∈ C
analytic calculation: Fµ⊲ν = Fµ ◦ Fν

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Monotonic

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are monotone independent if for bj ∈ B, cj ∈ C,

τ(b1c1) = τ(c1b1) = τ(b1)τ(c1), τ(c1b1c2) = τ(c1)τ(b1)τ(c2), and b1c1b2 = τ(c1)b1b2,

µb+c = µb ⊲ µc b ∈ B, c ∈ C
analytic calculation: Fµ⊲ν = Fµ ◦ Fν
some of the limit theory is preserved

SLIDE 99

f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Monotonic

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are monotone independent if for bj ∈ B, cj ∈ C,

τ(b1c1) = τ(c1b1) = τ(b1)τ(c1), τ(c1b1c2) = τ(c1)τ(b1)τ(c2), and b1c1b2 = τ(c1)b1b2,

µb+c = µb ⊲ µc b ∈ B, c ∈ C
analytic calculation: Fµ⊲ν = Fµ ◦ Fν
some of the limit theory is preserved
δs ⊲ δt = δs+t

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f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Monotonic

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are monotone independent if for bj ∈ B, cj ∈ C,

τ(b1c1) = τ(c1b1) = τ(b1)τ(c1), τ(c1b1c2) = τ(c1)τ(b1)τ(c2), and b1c1b2 = τ(c1)b1b2,

µb+c = µb ⊲ µc b ∈ B, c ∈ C
analytic calculation: Fµ⊲ν = Fµ ◦ Fν
some of the limit theory is preserved
δs ⊲ δt = δs+t
there is a multiplicative version of the convolution

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f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Monotonic

(A, τ) probability space, B, C ⊂ A subalgebras not

containing the unit.

B and C are monotone independent if for bj ∈ B, cj ∈ C,

τ(b1c1) = τ(c1b1) = τ(b1)τ(c1), τ(c1b1c2) = τ(c1)τ(b1)τ(c2), and b1c1b2 = τ(c1)b1b2,

µb+c = µb ⊲ µc b ∈ B, c ∈ C
analytic calculation: Fµ⊲ν = Fµ ◦ Fν
some of the limit theory is preserved
δs ⊲ δt = δs+t
there is a multiplicative version of the convolution
there is an operator-valued version as well

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H. Bercovici

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Omitted

c-freeness

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H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Omitted

c-freeness
rates of convergence

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f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Omitted

c-freeness
rates of convergence
convolution powers

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f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Omitted

c-freeness
rates of convergence
convolution powers
much more

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f(z), µ ⊞ ν, etc.

H. Bercovici

Free convolutions Analytic apparatus Limit theorems Regularity Extensions Omissions vvb

Omitted

c-freeness
rates of convergence
convolution powers
much more
credits

SLIDE 107