Limit theorems and the Chernoff product
formula
Michael Anshelevich
(joint work with John Williams)
Texas A&M University
October 8, 2012
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Compositional square root.
Let
A = F : C+ → C+ analytic, F (z) ∼ z as z → ∞ .
Note A closed under composition.
Question. Given F ∈ A, does there exist g ∈ A such that
F = g ◦ g?
Answer. In general, no. (But, if it exists, it is unique!)
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Infinite divisibility.
Definition. If F ∈ A and for any n ∈ N, there exists gn ∈ A such
that
F = gn ◦ gn ◦ . . . ◦ gn ,
|
{z
}
n times
say F is infinitely divisible.
May write gn = F ◦(1/n) .
Theorem. (Belinschi 2005) If F ∈ A is infinitely divisible, then
F ◦t is defined for all t ≥ 0: there exist {Ft : t ≥ 0} which form a
semigroup under composition:
Ft ◦ Fs = Ft+s ,
Michael Anshelevich (joint work with John Williams)
F1 = F .
Limit theorems and the Chernoff product formula
Limit theorems.
Question. Given an infinitely divisible F ∈ A and an increasing
sequence (kn ). Under what conditions on {gn : n ∈ N} do
gn ◦ gn ◦ . . . ◦ gn → F
{z
}
|
kn
uniformly on compact sets?
Will answer, but the answer in terms of measures not functions.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Limit theorems in probability.
{µn : n ∈ N} = probability measures on R. Under what
conditions do
µ n ∗ µn ∗ · · · ∗ µn → ν
|
{z
}
kn
weakly?
Example. Central Limit Theorem: if
Z
supp(µn ) ↓ {0} ,
x dµn (x) = 0,
R
Z
n
x2 dµn (x) → 1,
R
then µ∗n
n → normal distribution.
Example. Poisson Limit Theorem: if npn → 1, then
((1 − pn )δ0 + pn δ1 )∗n → Poisson distribution.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Infinitely divisible measures.
Such limit measures are necessarily infinitely divisible:
ν = ν1 ,
{νt : t ≥ 0} ,
νt ∗ νs = νt+s .
Lévy-Hinchin formula.
Infinitely divisible measures are precisely those with Fourier
transforms of the form
Z
x2 + 1
γ,σ
iθx
dσ(x)
(Fν∗ )(θ) = exp iγθ + (e − 1 − iθx)
x2
R
for γ ∈ R and σ a finite Borel measure on R.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Limit theorems.
Theorem. (Hinchin, Gnedenko)
The sequence
µn ∗ µn ∗ · · · ∗ µn → ν∗γ,σ
{z
}
|
kn
weakly if and only if
kn
x2
dµn (x) → dσ(x)
x2 + 1
weakly, and
Z
kn
R
x2
x
dµn (x) → γ.
+1
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Other convolutions.
µ = probability measure on R. Its Cauchy transform
Z
1
Gµ (z) =
dµ(x), z ∈ C+ ,
R z−x
and
Fµ (z) =
1
.
Gµ (z)
Define three new convolutions , ], B. Binary operations on
measures.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Other convolutions.
Free convolution µ1 µ2 :
Fµ−1
(z) = Fµ−1
(z) + Fµ−1
(z) − z,
1
2
1 µ2
z in a domain.
Boolean convolution µ1 ] µ2 :
Fµ1 ]µ2 (z) = Fµ1 (z) + Fµ2 (z) − z,
z ∈ C+ .
Monotonic convolution µ1 B µ2 :
Fµ1 Bµ2 (z) = Fµ1 (Fµ2 (z)),
z ∈ C+ .
Theorems (Voiculescu, Maassen, Bercovici; Speicher, Woroudi;
Muraki, Franz): each operation preserves positivity. All come
from appropriate notions of independence.
Free convolution most complicated. But: monotonic convolution
non-commutative.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Infinitely divisible distributions.
ν -infinitely divisible = ν belongs to a -convolution
semigroup. Same for ] and B.
Theorem. (Bercovici, Voiculescu 1992, 1993; Speicher,
Woroudi 1997) -infinitely divisible measures are precisely
those ν with
Z
1 + xz
−1
Fν γ,σ (z) − z = γ +
dσ(x), z ∈ C+ ,
R z−x
]-infinitely divisible measures are precisely those ν with
Z
1 + xz
dσ(x), z ∈ C+ ,
z − Fν]γ,σ (z) = γ +
z
−
x
R
for γ ∈ R and σ a finite Borel measure on R.
γ,σ
Correspondences ν∗γ,σ ↔ ν
↔ ν]γ,σ = Bercovici-Pata
bijections.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Limit theorems.
Theorem.(Bercovici, Pata 1999).
The following are equivalent.
1
The sequence µn ∗ µn ∗ · · · ∗ µn → ν∗γ,σ weakly.
|
{z
}
kn
2
γ,σ
The sequence µn µn · · · µn → ν
weakly.
|
{z
}
kn
3
The sequence µn ] µn ] · · · ] µn → ν]γ,σ weakly.
|
{z
}
kn
4
kn
x2
x2 + 1
Z
dµn (x) → σ weakly, and kn
R
x
dµn (x) → γ.
x2 + 1
Note: monotone convolution missing.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Proof based on
kn log Fµn (θ) = log Fµn∗kn (θ) → log Fν∗γ,σ ,
−1
kn (Fµ−1
(z) − z) = (F −1
kn (z) − z) → Fν γ,σ (z),
n
µn
kn (Fµn (z) − z) = (Fµ]k
n (z) − z) → Fν γ,σ (z) − z.
]
n
But for monotonic convolution, want
Fµ◦kn n (z) = FµnBkn (z) → FνBγ,σ (z).
Much more complicated.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Chernoff product formula.
Chernoff product formula, discrete version.
Let {kn : n ∈ N} be an increasing sequence of positive integers,
and {Vn : n ∈ N} a family of contractions on a Banach space
X . Suppose A is an unbounded operator which generates a
strongly continuous semigroup of contractions {T (t) : t ≥ 0}, D
is a core for A, and for each f ∈ D,
lim kn (Vn − I)f → Af .
n→∞
Then for each f ∈ X ,
lim Vnkn f = T (1)f .
n→∞
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Classical convolution.
This approach used for classical convolution and limit theorems
by Feller (1971), Goldstein (1976).
But note: classical convolution operators all commute.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Monotone convolution semigroups.
z ∈ C+ . Thus
Recall Fµ1 Bµ2 (z) = Fµ1 (Fµ2 (z)),
B − convolution semigroups ↔
semigroups of analytic transformation of C+ under composition.
Results of Berkson and Porta (1978) imply: each monotonic
convolution semigroup {νt : t ≥ 0}, νt B νs = νt+s satisfies
∂Fνt
= Φ(Fνt ).
∂t
Moreover, by Nevanlinna-Pick theory
Z
1 + xz
−Φ(z) = γ +
dσ(x)
R z−x
for some γ, σ.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Monotone convolution semigroups.
Thus extend the Bercovici-Pata bijection: for γ, σ as before,
denote
Φγ,σ (z) = Fν]γ,σ (z) − z,
let {νt : t ≥ 0} be the monotone convolution semigroup it
generates, and define ν.γ,σ = ν1 .
Limit theorems? Vn = composition operator,
Vn f = f ◦ Fµn .
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Monotone limit theorem.
Theorem. (A, Williams)
The sequence
µn ] µn ] · · · ] µn → ν]γ,σ
|
{z
}
kn
weakly if and only if
γ,σ
µn B µn B · · · B µn → νB
{z
}
|
kn
weakly.
Known results: Muraki 2001 (CLT and Poisson limit for
compactly supported), Wang 2012 (full CLT, convergence to
stable laws).
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Proof (sketch)
Step I. Assume
n
µ]k
→ ν]γ,σ
n
where {µn : n ∈ N} , ν]γ,σ all have uniformly bounded variance.
Let C+
1 = {z ∈ C : =z ≥ 1}, D = Span ({Gρ : ρ ∈ M2 }), and X
its completion with respect to the norm
kf k = sup |f (z)|.
z∈C+
1
Then each composition operator by Fµ
is a contraction on X
leaves D invariant.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Proof in the finite variance case.
The operator
Ah = Φγ,σ h0
generates the semigroup of composition operators
corresponding to γ, σ
has D as a core
for h ∈ D,
kn (h ◦ Fµn − h) − Φγ,σ h0 → 0
So the result follows from the Chernoff product formula.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Proof of the general result.
Step II.
n
µ]k
→ ν]γ,σ
n
⇒
n
µ.k
→ ν.γ,σ
n
in general.
Cutoff the measures, and approximate (carefully).
Step III. The converse:
n
µ.k
→ ν.γ,σ
n
⇒
n
µ]k
→ ν]γ,σ .
n
In general, the converse to the Chernoff theorem is false. True
in our case.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Key lemma.
Let µn B µn · · · B µn → ν weakly. Then
|
{z
}
kn
1
The family of measures
n
o
B
µn j : n ∈ N, j = 1, . . . , kn
is tight.
2
For every > 0 there exist α, β > 0 such that
kn =(Fµn (z) − z) ≤ 2=(Fν (z) − z) ≤ |z|
for large n ∈ N and z ∈ Γα,β .
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Compositional limit theorem.
Recall A = {F : C+ → C+ analytic, F (z) ∼ z as z → ∞}.
Theorem. (A, Williams)
For {gn : n ∈ N} , F ∈ A, F infinitely divisible, and an increasing
sequence (kn ),
gn ◦ gn ◦ . . . ◦ gn → F
{z
}
|
kn
uniformly on compact sets if and only if
kn (gn (z) − z) → Φ,
where F = F1 ,
∂Ft
∂t
= Φ(Ft ).
Equivalent conditions on µn but not on gn .
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Composition in the unit disk.
Similar but different results for the unit disk. Straightforward
equivalence is false:
ηµn (z)
z
kn
→
ην∪x
z
6⇔
ηµ◦knn (z) → ην (z).
But true under additional assumptions.
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Non-identically distributed arrays.
Let {µnk : n ∈ N, 1 ≤ k ≤ kn } be an infinitesimal array, that is,
for every ε > 0
lim max µnk ({x ∈ R : |x| ≥ ε}) = 0.
n→∞ 1≤k≤kn
Denote
Z
ank =
x dµnk (x),
dµ◦nk (x) = dµnk (x + ank ).
(−1,1)
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula
Non-identically distributed arrays.
Theorem.(Gnedenko; Chistyakov, Götze; Bercovici, Wang;
Wang). The following are equivalent.
1
The sequence µn1 ∗ µn2 ∗ · · · ∗ µnkn → ν∗γ,σ weakly.
2
γ,σ
The sequence µn1 µn2 · · · µnkn → ν
weakly.
3
The sequence µn1 ] µn2 ] · · · ] µnkn → ν]γ,σ weakly.
4
kn
X
k=1
x2
dµ◦ (x) → σ
x2 + 1 nk
weakly, and
Z
kn X
ank +
k=1
R
Michael Anshelevich (joint work with John Williams)
x
◦
dµ (x) → γ.
x2 + 1 nk
Limit theorems and the Chernoff product formula
Generalization of Chernoff formula.
Corresponding Chernoff-type result: if cnk uniformly small and
kn
X
cnk → a,
k=1
then
kn
Y
(1 + cnk ) → ea .
k=1
True for numbers, for commuting bounded operators. For
matrices?
Michael Anshelevich (joint work with John Williams)
Limit theorems and the Chernoff product formula