Free powers of the free Poisson measure

Free powers of the free Poisson measure

COLLOQUIUM MATHEMATICUM
VOL. 123
2011
NO. 2
FREE POWERS OF THE FREE POISSON MEASURE
BY
MELANIE HINZ and WOJCIECH MŁOTKOWSKI (Wrocław)
Abstract. We compute moments of the measures ($p )t , where $ denotes the
free Poisson law, and and are the additive and multiplicative free convolutions. These
moments are expressed in terms of the Fuss–Narayana numbers.
1. Introduction. Free convolution is a binary operation on the class M
of probability measures on R, which corresponds to the notion of free independence in noncommutative probability (see [3, 8, 12]). Namely, if X, Y are
free noncommuting random variables, with distributions µ, ν ∈ M respectively, then the additive free convolution µ ν is the distribution of the sum
X + Y . Similarly, if moreover X ≥ 0 then the multiplicative
√ free
√ convolution
µ ν can be defined as the distribution of the product XY X.
Here we can confine ourselves to the class Mc of compactly supported
measures in M. Let Mc+ denote the class of those µ ∈ Mc \ {δ0 } with
support in [0, ∞). For µ ∈ Mc we define its moment generating function
∞
X
(1)
Mµ (z) :=
sm (µ)z m ,
m=0
defined in some neighborhood of 0, where
(2)
sm (µ) := xm dµ(x)
R
is the mth moment of µ. Then we define its R-transform Rµ (z) by the
equation
Mµ (z) = Rµ (zMµ (z)) + 1.
(3)
P∞
If Rµ (z) = m=1 rm (µ)z m then the numbers rm (µ) are called the free cumulants of µ. For µ, ν ∈ Mc we define the additive free convolution µ ν
and the additive free power µt by
(4)
Rµν (z) = Rµ (z) + Rν (z) and Rµt (z) = tRµ (z).
The latter is well defined at least for t ≥ 1.
2010 Mathematics Subject Classification: Primary 46L54; Secondary 44A60, 60C05.
Key words and phrases: additive and multiplicative free convolution, free Poisson distribution.
DOI: 10.4064/cm123-2-11
[285]
c Instytut Matematyczny PAN, 2011
286
M. HINZ AND W. MŁOTKOWSKI
The free S-transform (see [11]) of µ ∈ Mc+ is defined by the relation
Rµ (zSµ (z)) = z
(5)
or Mµ (z(1 + z)−1 Sµ (z)) = 1 + z.
Observe that
z
1
.
Sµt (z) = Sµ
t
t
(6)
If µ, ν ∈ Mc and µ has support contained in [0, ∞) then the multiplicative
free convolution µ ν and the multiplicative free powers µp are defined by
Sµν (z) := Sµ (z)Sν (z) and Sµp (z) = Sµ (z)p .
(7)
The powers are well defined at least for p ≥ 1.
For c ∈ R, c 6= 0, and µ ∈ M we define the dilation Dc µ ∈ M by
Dc µ(X) := µ(c−1 X) for every Borel subset X of R. Then we have
(8)
MDc µ (z) = Mµ (cz),
RDc µ (z) = Rµ (cz),
1
SDc µ (z) = Sµ (z).
c
The last formula, together with (6), leads to
Proposition 1.1. Assume that µ ∈ Mc+ , p, t > 0 and both the measures
(µ ) and (µp )t exist. Then
t p
Dtp−1 (µp )t = (µt )p .
Proof. If Sµ (z) is the free S-transform of µ then
p
p
1
z
1
z
Sµ
and p Sµ
t
t
t
t
are the free S-transforms of (µp )t and (µt )p respectively.
2. The free Poisson measure. Our aim is to study the additive and
multiplicative free powers of the free Poisson measure
r
4−x
1
dx on [0, 4].
$ :=
2π
x
It is known that $t is -infinitely divisible for t ≥ 1 and $p is -infinitely
divisible for p ≥ 1 (see [12, 8, 2, 7]). Therefore the double powers ($t )p
and ($p )t exist whenever p, t > 0 and max{p, t} ≥ 1.
The additive free powers $t , t > 0, are well known:
p
4t − (x − 1 − t)2
t
(9)
$ = max{1 − t, 0}δ0 +
dx
2πx
√
√
with the absolutely continuous part supported on [(1 − t)2 , (1 + t)2 ], as
FREE POISSON MEASURE
287
well as the corresponding functions:
2
(10)
M$t (z) =
(11)
tz
,
R$t (z) =
1−z
p
1 + (1 − t)z + (1 − (1 + t)z)2 − 4tz 2
k
∞
m X
X
m
m
t
m
=1+
,
z
k
k−1 m
m=1
k=1
S$t (z) =
1
.
t+z
For the multiplicative free powers $p , p > 0, it is known [2, 7] that
∞ X
m(p + 1) + 1
zm
(12)
M$p (z) =
,
m
m(p + 1) + 1
m=0
∞ X
mp + 1
zm
(13)
R$p (z) =
.
m
mp + 1
m=1
Explicit formulas for the measures $p are known only for natural p [9, 10].
Our aim now is to study the measures
$(p, t) := ($p )t ,
where p, t > 0 and max{p, t} ≥ 1. First observe that
∞ X
mp + 1
zm
(14)
R$(p,t) (z) = t
m
mp + 1
m=1
and
(15)
S$(p,t) (z) = tp−1 (t + z)−p .
Our previous remarks lead to the following
Proposition 2.1. Assume that p, t > 0 and max{p, t} ≥ 1.
• If p ≥ 1 then $(p, t) is -infinitely divisible.
• If t ≥ 1 then $(p, t) is -infinitely divisible.
In order to compute moments of $(p, t) we will use the Lagrange inversion formula which says that if a function z = f (w) is analytic at the point
w = a and f 0 (a) 6= 0, f (a) =: b, then for the inverse function w = g(z) we
have
∞
X
dm−1
w − a m (z − b)m
.
(16)
g(z) = a +
dwm−1 f (w) − b
m!
w=a
m=1
Now we are ready to prove (see [6] for the special case p = 2)
288
M. HINZ AND W. MŁOTKOWSKI
Theorem 2.2. For p, t > 0 with max{p, t} ≥ 1, define $(p, t) :=
($p )t . Then
k
∞
m X
X
m
mp
t
(17)
M$(p,t) (z) = 1 +
zm
.
k−1 m−k m
m=1
k=1
Proof. Putting in (16)
f (w) := tp−1 w(1 + w)−1 (t + w)−p
and a = b = 0 we have, in view of (5) and (15), M$(p,t) (z) = 1 + g(z).
Therefore
∞
X
zm
dm−1 m(1−p)
m
mp (t
(1
+
w)
(t
+
w)
)
.
M$(p,t) (z) = 1 +
dwm−1
w=0 m!
m=1
But now
m
(1 + w) (t + w)
mp
k X
X
∞ m mp
w
m k
w
=t
k
k
t
k=0
k=0
k ∞
X
X
m
mp i−k
k
mp
w
t .
=t
i
k−i
mp
k=0
i=0
Therefore
m−1
X m mp dm−1
m
mp mp
((1 + w) (t + w) ) = t (m − 1)!
ti−m+1
dwm−1
i
m
−
1
−
i
w=0
i=0
m X
m
mp
mp
= t (m − 1)!
tk−m ,
k−1 m−k
k=1
which leads to our statement.
Note that in view of Proposition 1.1 there is no point to study powers
like
((($p1 )t1 )p2 )t2 ...
because all of them are dilations of some of $(p, t).
It would be interesting to verify the following
Conjecture. Assume that p, t > 0. Then the sequence {sm (p, t)}∞
m=0
defined by: s0 (p, t) := 1 and
k
m X
m
mp
t
sm (p, t) :=
k−1 m−k m
k=1
for m ≥ 1, is positive definite if and only if max{p, t} ≥ 1.
The coefficients at (12) and (17) have a remarkable combinatorial meaning, which was found by Edelman [4]. Namely, fix m, p ∈ N and let NC(p) (m)
FREE POISSON MEASURE
289
denote the set of all noncrossing partitions π of {1, . . . , mp} such that p divides the cardinality of every block of π. Then the cardinality of NC(p) (m)
is expressed as the Fuss–Catalan number :
m(p + 1) + 1
1
(p)
|NC (m)| =
.
m
m(p + 1) + 1
For other applications of these numbers we refer to [5].
For π ∈ NC(p) (m) we define its rank rk(π) := m − |π|. The elements of
fixed rank are counted by the Fuss–Narayana numbers:
m
mp
1
(p)
|{π ∈ NC (m) : rk(π) = k − 1}| =
.
k−1 m−k m
There is a natural partial order on NC(p) (m). Namely, we say that π ∈
NC(p) (m) is finer than σ ∈ NC(p) (m), and write π σ, if every block of π is
contained in a block of σ. Then NC(p) (m) equipped with and rk becomes
a graded partially ordered set (which means that for any π, σ ∈ NC(p) (m)
with π σ, every unrefinable chain π = π0 ≺ π1 ≺ · · · ≺ πr = σ from π
to σ has the same length r = rk(σ) − rk(π)) and a join-semilattice (i.e. any
two elements in NC(p) (m) have a least upper bound).
More general structures, noncrossing partitions on Coxeter groups, were
studied in [1].
3. Symmetrization of ($p )t . For µ ∈ M concentrated
on [0, ∞),
s
2
s
we define its symmetrization µ by R f (x ) dµ (x) = R f (x) dµ(x) for every
compactly supported continuous function f : R → R. If Mµ (z) is the moment
generating function of µ then Mµs (z) = Mµ (z 2 ), which means that s2m (µs ) =
sm (µ) and odd moments of µs are zero.
Now we will compute the free cumulants for the symmetrization of $(p, t).
Theorem 3.1. Assume that p, t > 0 with max{p, t} ≥ 1. Then for the
symmetrization $(p, t)s of $(p, t) we have
k
∞
m X
X
−m
mp
t
2m
(18)
R$(p,t)s (z) =
z
k−1 m−k m
m=1
k=1
∞
m X
X
m+k−2
mp (−t)k
.
=−
z 2m
k−1
m−k
m
m=1
k=1
Proof. The cumulant generating function R(z) for $(p, t)s satisfies
(19)
tp−1 R(z)(1 + R(z)) = z 2 (R(z) + t)p .
To check this, it is sufficient to substitute z 7→ zM (z 2 ) and compare with
(5) and (15). Therefore R(z) = R0 (z 2 ), where R0 satisfies
(20)
tp−1 R0 (z)(1 + R0 (z)) = z(R0 (z) + t)p .
290
M. HINZ AND W. MŁOTKOWSKI
To conclude, we use the Lagrange inversion formula as in the proof of
Theorem 2.2 putting f (w) := tp−1 w(1 + w)(t + w)−p , a = b = 0 to get
R0 (z) = g(z).
Acknowledgments. This research was supported by MNiSW grant
N N201 364436.
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Melanie Hinz, Wojciech Młotkowski
Mathematical Institute
University of Wrocław
Pl. Grunwaldzki 2/4
50-384 Wrocław, Poland
E-mail: [email protected]
[email protected]
Received 19 February 2011;
revised 29 April 2011
(5470)