A penalised model reproducing the mod-Poisson fluctuations in the Sathe-Selberg theorem

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by  Yacine Barhoumi-Andréani

A PENALISED MODEL REPRODUCING THE MOD-POISSON
FLUCTUATIONS IN THE SATHÉ-SELBERG THEOREM
YACINE BARHOUMI-ANDRÉANI
Abstract. We construct a probabilistic model for the number of divisors of a random
7
uniform integer that converges in the mod-Poisson sense to the same limiting function as
1
0 its original counterpart, the one arising in the Sathé-Selberg theorem. This construction
2 involvesaconditioningandgivesanalternativeperspectivetotheusualparadigmof“hybrid
product” modelsdevelopedbyGonek,HughesandKeatinginthecaseoftheRiemannZeta
n
function.
a
J

2
1
1. Introduction

]
R The Erdös-Kac theorem in probabilistic number theory concerns the Gaussian fluctuations
P of the number of prime divisors of a random integer : if P denotes the set of prime numbers,
h. let ω(N) be the number of prime divisors of N ∈ N defined by
t
a ω(N) := (cid:88)1
m {p|N}
p∈P
[
and let Un be a random variable uniformly distributed in {1,...,n}. The Erdös-Kac theorem
1
writes (see [4, 5])
v
32 sup(cid:12)(cid:12)(cid:12)P(cid:18)ω(U√n)−loglogn (cid:54) x(cid:19)−(cid:90) x e−u2/2√du (cid:12)(cid:12)(cid:12) −−−−→ 0 (1)
4 x∈R(cid:12) loglogn −∞ 2π(cid:12) n→+∞
3
The key understanding of this theorem is the following : the random variables
0
1. B(n) := 1
p {p|Un}
0
7 are {0,1}-Bernoulli random variables that are weakly correlated, and their approximation by
1 a sequence of independent random variables is accurate at the the level of this Central Limit
:
v Theorem (CLT). Concretely, one can perform the approximation
i
X ω(U ) = (cid:88)1 = (cid:88)B(n) ≈ (cid:88) B(∞)
r n {p|Un} p n→+∞ p
a p∈P p∈P p∈P,p(cid:54)n
(∞)
the B ’s being independent Bernoulli random variables such that
p
(cid:16) (cid:17) 1 (cid:16) (cid:17)
P B(∞) = 1 = = 1−P B(∞) = 0
p p p
To measure the accuracy of this last approximation, we introduce the independent model
(cid:88)
Ω := B(∞)
n p
p∈P,p(cid:54)n
Date: January 13, 2017.
2010 Mathematics Subject Classification. 60E10, 60E05, 60F05, 60G50, 60F99, 11K99, 11K65.
1
2 Y.BARHOUMI-ANDRÉANI
A model of a random or a deterministic sequence is a random sequence that can be sub-
stituted to its orginal in a prescribed framework while still capturing its main properties, for
instance a particular type of convergence.
At the order of renormalisation of the CLT given by (1), the independent model is accurate
since one can write
ω(U√n)−loglogn ≈ Ωn−(cid:80)p∈P,p(cid:54)n p1 −−−L−→ N (0,1)
(cid:113)
loglogn (cid:80) 1 n→+∞
p∈P,p(cid:54)n p
Here, we have used the well-known estimate for the prime harmonic sum
(cid:88) 1
H(P) := = loglogn+O(1) (2)
n p
p∈P,p(cid:54)n
This model is interesting to understand the Erdös-Kac CLT, but it hides a certain amount
(n)
of information since at the second order of renormalisation the dependency of the B ’s
p
re-appears : one has the following result due to Selberg [17] improving a result of Sathé [18]
(cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
E zω(Un) = e(z−1)(loglogn+κ)Φ (z) 1+O (logn)Re(z−2) (3)
ω
where, for R > 0, the O is uniform for |z| (cid:54) R, where κ is an absolute positive constant and
(cid:18) (cid:19) (cid:18) (cid:19)
Φω(z) := (cid:89) 1+ z−1 e−z−k1 (cid:89) 1+ z−1 e−z−p1 (4)
k p
k∈N∗ p∈P
But one has
(cid:16) (cid:16) (cid:17)(cid:17)
E(cid:0)zΩn(cid:1) = e(z−1)(loglogn+κ(cid:48))Φ (z) 1+O (logn)Re(z−2) (5)
Ω
with κ(cid:48) another absolute constant and
(cid:18) (cid:19)
(cid:89) z−1 −z−1
ΦΩ(z) := 1+ e p
p
p∈P
leading to a corrective factor ΦC(z) := (cid:81)k∈N∗(cid:0)1+ z−k1(cid:1)e−z−k1 such that Φω(z)= ΦΩ(z)ΦC(z).
This factor is easily seen to be the limiting function of the random variable
n
(cid:88) (∞)
C := B
n k
k=1
which is equal in distribution to the number of cycles C(σ ) of a random uniform permutation
n
σ ∈ S (see [1, 12]), hence the name.
n n
Let (X ) be a sequence of random variables and let P be a Poisson-distributed random
n γn
variable of parameter γ ∈ R . When there exists a continuous function Φ : A ⊆ C → C
n +
satisfying some technical conditions (see definition 2.1) such that the following convergence
holds locally uniformly in z ∈ A
E(cid:0)zXn(cid:1)
−−−−→ Φ(z)
E(zPγn) n→+∞
one says that (X ,γ ) converges in the mod-Poisson sense to Φ.
n n n
This particular type of convergence was introduced in [12] following a similar development
in the Gaussian setting in [9]. It is unusual in Probability theory where the Fourier-Laplace
A MODEL REPRODUCING THE MOD-POISSON FLUCTUATIONS IN THE SELBERG-SATHÉ THEOREM 3
transform is not often renormalised ; it implies the usual CLT by a change of renormalisation.
This is the mode of convergence underlying equations (3) and (5) since
E(cid:0)zPγn(cid:1) = eγn(z−1)
A natural question arises from the last computations :
Question 1.1. How to refine the independent model Ω to get a model that would reproduce
n
the mod-Poisson fluctuations, i.e. a model that would converge in the mod-Poisson sense to
the function Φ ?
ω
Thecreationofheuristicprobabilisticmodelsaimingatunderstandingthesequenceofprime
numbers originates in the work of Cramer [3] and has recently seen some spectacular devel-
opments with the work [13] that precises Cramer’s original idea that prime numbers behave
“at random”. The approach in this article is conceptually identical to the work of Hardy and
Littlewood (see [8] or [19]) that refines the coarse Cramer’s model to incorporate effects likely
to explain the distribution of twin primes. This refinement is done by biasing the probabilities
of the Cramer model ; an enlightening description of the biasing procedure is given in [19].
The analogy stops nevertheless here : the Cramer model consists in heuristically replacing
the sequence of primes by a random sequence and argue that they behave in a “similar way” ;
as [19] remarks, such an approach “must be taken with a liberal dose of salt”. Here, no such
heuristicreplacementiscarriedout(foracomparisonofunrelatedprobabilisticmodelinnum-
ber theory, see [20]). The problem here tackled is to construct a probabilistic approximation
of a “true” random variable, ω(U ), but the approximation has to be understood in a peculiar
N
probabilistic sense (the mod-Poisson one in place of, say, a total variation approximation),
which motivates the terminology of model such as defined previously.
Thequestion1.1ismotivatedbyasimilardevelopmentfortherandomvariableζ(1/2+itU)
where U is a random variable uniformly distributed on the interval [0,1], t > 0 and ζ is the
Riemann Zeta function. This random variable satisfies a CLT due to Selberg (see e.g. [10]) in
the same vein as the Erdös-Kac one for ω(U ), namely
n
(cid:12)   (cid:12)
sup(cid:12)(cid:12)(cid:12)Plog(cid:113)|ζ(1/2+itU)| (cid:54) x−(cid:90) x e−u2/2√du (cid:12)(cid:12)(cid:12) −−−−→ 0
x∈R(cid:12)(cid:12) 1 loglogt −∞ 2π(cid:12)(cid:12) t→+∞
2
The question of computing the limiting function for the “mod-Gaussian renormalisation”
given for all λ ∈ iR by
E(cid:0)eλlog|ζ(1/2+itU)|(cid:1)
−−−−→ Φ (λ) = Φ (λ)Φ (λ)
eλ22×12loglogt t→+∞ ζ Matrix Arithmetic
is the celebrated Keating-Snaith’s moments conjecture (see [11] for the definitions).
Inordertounderstandthislastconvergence, theauthorsof[7]constructa“hybridproduct”
model for log|ζ(1/2+itU)| that converges in the mod-* sense to the limiting function Φ .
ζ
Answering question 1.1 for the “toy model” given by ω(U ) is thus of importance for it may
n
give a hint to understand log|ζ(1/2+itU)|.
We refer to [7] for the exact details of the ζ model ; instead of describing it, let us find its
equivalent for ω(U ). Since the limiting function occuring in (5) is a product Φ = Φ Φ ,
n ω Ω C
the idea is to think of ω(U ) as being approximately an independent sum of random variables
n
4 Y.BARHOUMI-ANDRÉANI
created by means of Bernoulli random variables, i.e.
A A(cid:48)
(cid:88) (cid:88)
ω(U ) ≈ ω := B + B(cid:48) (6)
n n k k
k=1 k=1
with P(B = 1) = 1 , P(B(cid:48) = 1) = 1 and where A,A(cid:48) are choosen so that the mod-Poisson
k p k k
k
speed of convergence γ = loglogn+κ of ω(U ) matches the speed of convergence of ω . We
n n n
have set P := {p ,k (cid:62) 1}. As γ is asymptotically the mean of ω(U ), using the classical
k n n
(cid:80)
relation 1/k = logn+O(1) one finds
1(cid:54)k(cid:54)n
A A(cid:48)
(cid:88) 1 (cid:88) 1
E(ω ) = + = loglogA+log(A(cid:48))+O(1)
n
p k
k
k=1 k=1
which amounts to
A(cid:48)logA = O(logn)
the constant in the O being explicitely known.
This intuitive model, despite its degree of freedom and its artificial character, has the
advantage of being an acceptable mod-Poisson model for ω(U ) since it converges in the mod-
n
Poisson sense to Φ = Φ Φ . Nevertheless, one could ask for another reason why the limiting
ω Ω C
correction to the independence takes the form of an additive independent term, and why this
correction is again constructed by means of independent random variables. One can argue
that a natural modification of the initial model is more likely to be understood by a biasing à
la Hardy-Littlewood [8, 19] instead of a summation paradigm. This is the goal of this article.
More precisely, we will answer question 1.1 by conditioning a random proportion of primes
to be divisors with probability one in a slight modification of Ω , i.e. by modifying the proba-
n
bilities of the Bernoulli sum in the same way Hardy and Littlewood modify the probabilities
of the Cramer Bernoulli variables, namely
Theorem. Set γ := loglogn. For θ > 0, let B (θ) be the Bernoulli random variable given
n k
by
θ
P(B (θ) = 1) = = 1−P(B (θ) = 0)
k k
θ+k−1
Thereexistarealsequence(v ) , arandomintegerC(cid:48) andasequence(I ) ofi.i.d. random
n n n (cid:96) (cid:96)
integers independent of (B (v )) and C(cid:48) (all quantities explicitely defined in theorem 3.9)
k n k n
such that
(cid:18) (cid:12) (cid:19)
Ω(cid:48)n(cid:48) :=L (cid:88)Bpk(vn)(cid:12)(cid:12)(cid:12)BpI1(vn) = ··· = BpICn(cid:48) (vn) = 1
k
satisfies
(cid:16) (cid:17)
E xΩ(cid:48)n(cid:48)
−−−−→ Φ (x)
E(xPγn) n→+∞ ω
The explicit description of all involved parameters is the content of theorem 3.9. It will
be proven using a probabilistic interpretation of mod-Poisson convergence developed in [2],
which interprets it as a change of probability. In the context of discrete random variables,
A MODEL REPRODUCING THE MOD-POISSON FLUCTUATIONS IN THE SELBERG-SATHÉ THEOREM 5
conditioning and biasing can be understood in the same framework, which deepends the
analogy with the Hardy-Littlewood approach.
Notations
We gather here some notations used throughout the paper.
The set {1,2,...,n} will be denoted by 1,n . The set of prime numbers will be denoted
by P := {p ,k (cid:62) 1}. (cid:74) (cid:75)
k
The distribution of a real random variable X in the probability space endowed with the
measure P will be denoted by P : if A is a measurable set, P (A) := P(X ∈ A).
X X
If X and Y are two random variables having the same distribution, that is P = P , we
X Y
will note X =L Y or equivalently X ∼ P . We will denote by P(γ) the Poisson distribution of
Y
parameter γ > 0, by N (0,1) the standard Gaussian distribution and by U(A) the uniform
distribution on the set A.
For f ∈ L1(P ), f (cid:62) 0, the penalisation or bias of P by f is the probability measure P
X X Y
denoted by
f(X)
P := •P
Y E(f(X)) X
This definition is equivalent to the following : for all g ∈ L∞(P ),
X
E(f(X)g(X))
E(g(Y)) =
E(f(X))
Note that in a discrete setting, conditioning amounts to take f = 1 for a suitable set A.
A
A partition of an integer N is a sequence of integers λ = (λ ,...,λ ) where λ (cid:62) λ (cid:62) ...
1 k 1 2
and (cid:80)k λ = N. We define the length of such a partition by (cid:96)(λ) := k. The paintbox
i=1 i
process (see [14]) is a random partition λ = (λ ,...,λ ) constructed in the following way : let
1 k
(I ) be i.i.d. integer valued random variables and define the (random) equivalence relation by
i i
k ∼ r ⇐⇒ I = I (7)
k r
The equivalence classes of this relation define a random partition λ. In the case where
I ∼ U( 1,N ), this random partition is equal in law to the cycle structure of a random
uniform (cid:74)perm(cid:75)utation σ ∈ S and in particular, the total number of cycles of σ satisfies
N
C(σ) = (cid:96)(λ).
2. Reminder on mod-Poisson convergence
2.1. Definition and examples. Let P ∼ P(γ) with γ > 0. Recall that
γ
γk
P(P = k) = e−γ
γ
k!
which is a statement equivalent to E(cid:0)eiuPγ(cid:1) = exp(cid:0)γ(eiu−1)(cid:1) for all u ∈ R. We define the
mod-Poisson convergence in the Laplace-Fourier setting by the following
Definition 2.1. Let(Z ) beasequenceofpositiverandomvariablesand(γ ) beasequence
n n n n
of strictly positive real numbers. (Z ) is said to converge in the mod-Poisson sense at speed
n n
(γ ) if for all z ∈ C, the following convergence holds locally uniformly in z ∈ C
n n
E(cid:0)zZn(cid:1)
−−−−→ Φ(z)
E(zPγn) n→+∞
6 Y.BARHOUMI-ANDRÉANI
where Φ : C → C is a continuous function satisfying Φ(1) = 1, Φ(z) = Φ(z) and with
P ∼ P(γ ).
γn n
When such a convergence holds, we write it as
mod-P
(Z ,γ ) −−−−→ Φ
n n
n→+∞
Remark 2.2. The limiting function Φ is not unique, since it is defined up to multiplication by
an exponential (see [9]).
Remark 2.3. The original definition used in [12] is in the Fourier setting, i.e. for |z| = 1.
The advantage of this definition is that the Fourier transform of a random variable always
exists. Other restrictions of the domain of convergence are possible. For instance, [6] uses
{−c < Re < c} for a certain c > 0. In the Laplace case, one has, locally uniformly in x ∈ R
+
E(cid:0)xZn(cid:1)
−−−−→ Φ(x)
E(xPγn) n→+∞
and in particular, Φ(x) (cid:62) 0 for all x ∈ R .
+
From now on, we restrict ourselves to the Laplace setting. Mod-Poisson convergence
will always mean “in the Laplace setting” unless specified. In particular, the limiting mod-
Poisson function Φ will be defined on R and a quantity such as ||Φ|| will be understood as
+ ∞
sup |Φ(x)|.
x∈R+
The following example is fundamental to understand mod-Poisson convergence :
Example 2.4. Let (B ) be a sequence of Bernoulli random variables such that
k k
p := P(B = 1) = 1−P(B = 0)
k k k
where (p ) is a sequence of real numbers satisfying the conditions
k k
(cid:88)
(i) p = +∞
k
k(cid:62)1
(cid:88)
(ii) p2 < +∞
k
k(cid:62)1
Let Z := (cid:80)n B and γ := (cid:80)n p . Then,
n k=1 k n k=1 k
mod-P
(Z ,γ ) −−−−→ Φ
n n
n→+∞
where
(cid:89)
Φ(x) = (1+p (x−1))e−pk(x−1) (8)
k
k(cid:62)1
Indeed, setting P ∼ P(γ ) one has, locally uniformly in x
γn n
E(cid:0)xZn(cid:1) (cid:81)n E(cid:0)xBk(cid:1) (cid:89)n
= k=1 = (1+p (x−1))e−pk(x−1)
E(xPγn) eγn(x−1) k
k=1
(cid:89)
−−−−→ (1+p (x−1))e−pk(x−1)
k
n→+∞
k(cid:62)1
since (cid:80)kp2k < ∞ and (1+pk(x−1))e−pk(x−1) = exp(cid:0)−p2k(x−1)2/2+o(p2k)(cid:1).
A MODEL REPRODUCING THE MOD-POISSON FLUCTUATIONS IN THE SELBERG-SATHÉ THEOREM 7
One can see that equation (5) is a particular case of this last theorem with p = 1/p
k k
where P := {p ,k (cid:62) 1}. As pointed out in the introduction, this is also the case of equation
k
(3) using the “hybrid sum” ω of (6) that incorporates the corrective term
n
(cid:18) (cid:19)
ΦC(x) := (cid:89) 1+ x−1 e−x−k1
k
k(cid:62)1
This term is the limiting mod-Poisson function of the random variable C := (cid:80)n B
n k=1 k
where (B ) is the last sequence of Bernoulli random variables with p = 1/k, and with speed
k k k
n
(cid:88) 1
H := = logn+γ +o(1)
n
k
k=1
where γ is the Euler-Mascheroni constant.
Remark 2.5. Sucharandomvariablehasalsothedistributionofthetotalnumberofcyclesofa
randompermutationselectedaccordingtotheuniformdistributionP definedbyP (σ) = 1/n!
n n
for all σ ∈ S (see e.g. [1]). Using the formula (see e.g. [23] 12.11)
n
(cid:18) (cid:19)
1 = e(z−1)γ (cid:89) 1+ z−1 e−z−k1
Γ(z) k
k(cid:62)1
Φ (z) can be replaced by 1/Γ(z) when the speed H is replaced by H −γ.
C n n
2.2. A probabilistic interpretation of mod-Poisson convergence. Werecallthefollow-
ing theorem from [2] :
Theorem 2.6. Let Φ be a bounded function on R and γ > 0. Define the distribution
+
Q(Φ,γ) by
(cid:16) (cid:17)
Φ Pγ
γ
Q ∼ Q(Φ,γ) ⇐⇒ P := •P
γ Qγ E(cid:16)Φ(cid:16)Pγ(cid:17)(cid:17) Pγ
γ
where P ∼ P(γ).
γ
Then, if Q (Φ) ∼ Q(Φ,γ ), we have
γn n
mod-P
(Q (Φ),γ ) −−−−→ Φ
γn n
n→+∞
For the reader’s convenience, we remind the proof of this result.
Proof. Recall the change of probability, for x,γ > 0
xPγ
•P = P (9)
E(xPγ) Pγ Pxγ
easily seen writing, for all θ ∈ R
E(cid:0)xPγ eiθPγ(cid:1) eγ(xeiθ−1) (cid:16) (cid:17)
= = eγx(eiθ−1) = E eiθPxγ
E(xPγ) eγ(x−1)
8 Y.BARHOUMI-ANDRÉANI
Then, we have
(cid:32) (cid:33)
E(cid:0)xQγn(Φ)(cid:1) = E(cid:16)Φ(cid:16)Pγγnn(cid:17)xPγn(cid:17) = E E(cid:16)xxPPγγnn(cid:17)Φ(cid:16)Pγγnn(cid:17) = E(cid:16)Φ(cid:16)Pγxnγn(cid:17)(cid:17)
E(xPγn) E(xPγn)E(cid:16)Φ(cid:16)Pγn(cid:17)(cid:17) E(cid:16)Φ(cid:16)Pγn(cid:17)(cid:17) E(cid:16)Φ(cid:16)Pγn(cid:17)(cid:17)
γn γn γn
By continuity and boundedness of Φ, and using dominated convergence and the fact that
Pxγn −−−L−→ x
γn n→+∞
one gets, locally uniformly in x ∈ R (and in particular for x = 1)
+
(cid:18) (cid:18) (cid:19)(cid:19)
P
E Φ xγn −−−−→ Φ(x)
γn n→+∞
As Φ(1) = 1, one finally gets the result. (cid:3)
Example 2.7. Continuing example 2.4, we see that in the case of a function given by (8), i.e.
(cid:89)
Φ(x) = (1+p (x−1))e−pk(x−1)
k
k(cid:62)1
one has, for all x ∈ R
+
0 (cid:54) Φ(x) (cid:54) 1
and the last theorem applies. The positivity of Φ on R is obvious, and as 1+y (cid:54) ey for
+
all y ∈ R, setting y = pk(x−1) one has (1+pk(x−1))e−pk(x−1) (cid:54) 1 which gives the upper
bound.
A probabilistic interpretation of mod-Poisson convergence follows from this last theorem :
if (Z ) is a sequence of random variables converging in the mod-Poisson sense at speed (γ )
n n n n
to Φ, one may think of the distribution of Z as close to the distribution of Q (Φ). The
n γn
limitingfunctionΦ, once correctly scaled, would thus be aparticular correction to the Poisson
distribution that would allow a refined speed of convergence in the CLT, and mod-Poisson
convergence could thus be understood as a certain second-order convergence in distribution.
This is the case in the mod-Gaussian setting (see [2]), but also in the mod-Poisson setting
since [21, II.6 (20)] gives
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
k k
P(ω(U ) = k) = P(P = k) Φ +O
n loglogn ω loglogn (loglogn)2
uniformly in n (cid:62) 3 and k ∈ 1,(2−δ)loglogn for all δ > 0. Moreover, using 3.7, one has
||Φ(cid:48) || < ∞, which implies us(cid:74)ing the Gaussian C(cid:75)LT for P
ω ∞ γ
(cid:18)(cid:12) (cid:18) (cid:19) (cid:12)(cid:19) (cid:18)(cid:12) (cid:18) (cid:19) (cid:12)(cid:19) (cid:18)(cid:12) (cid:12)(cid:19) (cid:18) (cid:19)
E (cid:12)(cid:12)(cid:12)Φω Pγγ −1(cid:12)(cid:12)(cid:12) = E (cid:12)(cid:12)(cid:12)Φω Pγγ −Φω(1)(cid:12)(cid:12)(cid:12) (cid:54) (cid:12)(cid:12)(cid:12)(cid:12)Φ(cid:48)ω(cid:12)(cid:12)(cid:12)(cid:12)∞E (cid:12)(cid:12)(cid:12)Pγγ −1(cid:12)(cid:12)(cid:12) = O √1γ
This last result can hence be transformed into
 Φ (cid:16) k (cid:17) E(cid:16)Φ (cid:16)Ploglogn(cid:17)(cid:17) (cid:18) (cid:19)
ω loglogn ω loglogn k
P(ω(Un) = k) = P(Ploglogn = k)E(cid:16)Φ (cid:16)Ploglogn(cid:17)(cid:17) +O √loglogn +O (loglogn)2 
ω loglogn
A MODEL REPRODUCING THE MOD-POISSON FLUCTUATIONS IN THE SELBERG-SATHÉ THEOREM 9
that is
(cid:32) (cid:33)
(loglogn)k−1/2
P(ω(U ) = k) = P(Q (Φ ) = k)+O
n loglogn ω
k! logn
3. A model that converges in the mod-Poisson sense
In order to construct a probabilistic model for ω(U ) that converges in the mod-Poisson
n
sense to Φ , we remind some classical probabilistic biases.
ω
3.1. Classical biases and changes of probability. Afundamentaloperationinprobability
theory is the change of probability by means of a weight on the initial probability measure.
This weight is called bias or penalisation and we will use undifferently both terminology.
Definition 3.1 (Bias/penalisation of measure). Let X be a real random variable in the
probability space endowed with the measure P and denote by P its law. For f ∈ L1(P ),
X X
f (cid:62) 0, the penalisation (or bias) of P by f is the probability measure P denoted by
X Y
f(X)
P := •P
Y E(f(X)) X
Classicalbias inprobability theoryallowto understand“pathwise transformations” induced
by such a transformation.
Example 3.2. The most classical change of probability concerns the passage from N (0,1)
to N (µ,1) =L µ+N (0,1). Indeed, if X ∼ N (0,1), one easily checks that
eµX
P = •P = eµX−µ2/2•P
X+µ E(eµX) X X
Hence, in the Gaussian setting, an exponential bias is equivalent to a translation of the
canonical evaluation. Note that the Poisson counterpart of this exponential bias was given in
equation (9).
A classical transform in probability theory is made using f : x (cid:55)→ x when the random
variable is positive.
Definition 3.3 (Size-bias transform). Let X (cid:62) 0 be a random variable with expectation
µ := E(X) < ∞. A random variable X(s) is said to be a size-bias transform of X if, for all
real functions f such that E(|Xf(X)|) < ∞
(cid:16) (cid:16) (cid:17)(cid:17)
E(Xf(X)) = µE f X(s)
An equivalent definition is thus
X
P := •P (10)
X(s) E(X) X
Example 3.4. A classical change of measure for a random walk is given by its size-bias cou-
pling, i.e. given(X ) asequenceofi.i.d. positiverandomvariablesofexpectationE(X ) := 1
k k k
defined on the same probability space, the random walk (S ) of increments (X ) is given
n n k k
by
n
(cid:88)
S := X
n k
k=1
10 Y.BARHOUMI-ANDRÉANI
(s)
The size-bias transform of S is the random variable S whose law is given by
n n
S
P := n •P
Sn(s) n Sn
A pathwise construction of such a random variable is implied by the following
Lemma 3.5 (Size-bias coupling of an independent sum). Let (Y ) be a sequence of in-
k k
dependent positive integrable random variables, independent of (X ) and having the same
k k
distribution as (X ) and let I ∈ 1,n be a random index independent of (X ) and (Y ) of
k k k k k k
law given by (cid:74) (cid:75)
E(X )
P(I = k) = k
(cid:80)n E(X )
(cid:96)=1 (cid:96)
Then,
S(s) =L S −X +Y(s)
n n I I
and in particular, if (Y ) is defined on the same probability space as (X ) , one has a natural
k k k k
(cid:16) (cid:17)
(s)
coupling S ,S .
n n
For the reader’s convenience, we recall the proof of this lemma (see also e.g. [1, pp 78-79]).
(cid:104)−k(cid:105) (cid:80)
Proof. Let f be a bounded measurable function and S := X . Then, by indepen-
n (cid:96)(cid:54)=k (cid:96)
dence,
n
(cid:16) (cid:16) (cid:17)(cid:17) 1 1 (cid:88)
E f S(s) := E(S f(S )) = E(X f(S ))
n E(S ) n n E(S ) k n
n n
k=1
n
1 (cid:88) (cid:16) (cid:16) (cid:17)(cid:17)
= E X f S(cid:104)−k(cid:105)+X
E(S ) k n k
n
k=1
n
= 1 (cid:88)E(X )E(cid:16)f(cid:16)S(cid:104)−k(cid:105)+Y(s)(cid:17)(cid:17)
E(S ) k n k
n
k=1
(cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
= E f S(cid:104)−I(cid:105)+Y(s) = E f S −X +Y(s)
n I n I I
(cid:3)
A last type of useful bias concerns the exponential bias of a sum of independent terms, and
in particular Bernoulli random variables.
Lemma3.6(Exponentialbiasofanindependentsum). Let(Y ) beasequenceofindependent
k k
random variables. Define, for a certain x > 0,
n
(cid:88)
S := Y
n k
k=1
xSn
P = •P
Sn(x) E(xSn) Sn