Frequent Hypercyclicity of Random Entire Functions for the Differentiation Operator

Complex Anal. Oper. TheoryDOI 10.1007/s11785-013-0328-0

Complex Analysisand Operator Theory

Frequent Hypercyclicity of Random Entire Functionsfor the Differentiation Operator

Miika Nikula

Received: 5 April 2013 / Accepted: 7 September 2013© Springer Basel 2013

Abstract We study the random entire functions defined as power series f (z) =∑∞n=0(Xn/n!)zn with independent and identically distributed coefficients (Xn) and

show that, under very weak assumptions, they are frequently hypercyclic for the dif-ferentiation operator D : H(C) → H(C), f �→ D f = f ′. This gives a very simpleprobabilistic construction of D-frequently hypercyclic functions in H(C). Moreoverwe show that, under more restrictive assumptions on the distribution of the (Xn), theserandom entire functions have a growth rate that differs from the slowest growth ratepossible for D-frequently hypercyclic entire functions at most by a factor of a powerof a logarithm.

Keywords Frequently hypercyclic operator · Differentiation operator ·Rate of growth · Entire functions · Random construction

1 Introduction

1.1 Definitions and Background

Let V be a topological vector space and T : V → V a linear operator. We call theoperator T hypercyclic if there exists a f ∈ V such that {T n f : n ∈ N}, the orbit off under T , is dense in V . In this case we also say that f is hypercyclic for T .

The stronger notion of frequent hypercyclicity is defined as follows. The vector fis frequently hypercyclic for T , if for any open set U ⊂ V the sequence of iterates of

Communicated by Marek Bozejko.

M. Nikula (B)Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014 Helsinki, Finlande-mail: [email protected]

M. Nikula

f under T that belong to U has a positive lower density, or explicitly

lim infn→∞

{k ∈ {0, 1, . . . , n − 1} ∣∣ T k f ∈ U

}

n> 0.

The topological vector space we study in this note is the space of entire functions

H(C) = { f : C → C∣∣ f is holomorphic everywhere

},

equipped with the standard topology given by

fn −→n→∞ f in H(C) ⇐⇒ fn −→

n→∞ f uniformly on all compact sets K ⊂ C,

and the operator is the differentiation operator

D : H(C) → H(C), f �→ D f = f ′.

It was shown already by MacLane [11] that, in this setting, the differentiation operatoris hypercyclic. The concept of frequent hypercyclicity was defined rather recently byBayart and Grivaux (see e.g. [2,3]), and the frequent hypercyclicity of differentiationin the space of entire functions was proven soon after that by Bonilla and Grosse-Erdmann [7]. We refer the reader to [5] for more background. Bonilla and Grosse-Erdmann [8] also posed the problem which motivates the probabilistic construction ofthis note: what is the slowest growth rate possible for a D-frequently hypercyclic entirefunction? As the final answer to this question, the following theorem was establishedby Drasin and Saksman [9] by an explicit construction.

Theorem A For f ∈ H(C) and r ≥ 0, define the circle maximum

M f (r) = supθ∈[0,2π)

| f (reiθ )|

to measure the growth rate of f . Then for any c > 0 there exists a D-frequentlyhypercyclic f such that

M f (r) ≤ cr− 14 er for large r.

Conversely, any D-frequently hypercyclic f ∈ H(C) satisfies

lim supr→∞

r14 e−r M f (r) > 0.

Completing the earlier work of Blasco, Bonilla and Grosse-Erdmann [5], it wasalso proven in [9] that the optimal growth rates of the circle averages

M f,p(r) =⎛

⎝

2π∫

0

| f (reiθ )|p dθ

2π

⎞

⎠

1/p

, p ≥ 1

Frequent Hypercyclicity of Random Entire Functions

are given by

M f,p(r) ≤

⎧⎪⎪⎨

⎪⎪⎩

ϕ(r)r− 12 er for p = 1

cr− 12p er for 1 < p ≤ 2

cr− 14 er for 2 ≤ p

,

where ϕ : [0,∞) → [0,∞) is an arbitrary nondecreasing function such that ϕ(r) →∞ as r → ∞ and c > 0 is an arbitrary constant. The optimal result for p = 1 wasalready obtained by Bonet and Bonilla [6]. That these bounds are optimal meansthat the inequalities fail for D-frequently hypercyclic functions if ϕ is replaced by aconstant or if c is replaced by a nonincreasing function ψ : [0,∞) → [0,∞) suchthat ψ(r) → 0 as r → ∞.

The purpose of this note is to give a simple probabilistic construction of D-frequently hypercyclic functions in H(C) that grow almost as slowly as TheoremA permits. Our main tools are elementary—first and second moment estimates andthe Borel–Cantelli lemma—though for our growth estimate we invoke a slightly moreinvolved estimate on the suprema of random trigonometric polynomials. The construc-tion and the D-frequent hypercyclicity of the resulting functions is stated as Theorem 1and an estimate for their growth rate as Theorem 3.

Theorem 1 Let X be a complex random variable such that the support of the law ofX is whole C and such that the decay condition

for some β > 0, lim supr→∞

(log r)1+βP(|X | ≥ r) < ∞ (1)

is satisfied. Let (Xn)∞n=0 be a sequence of independent random variables with the law

of X and denote

f (z) =∞∑

n=0

Xn

n! zn .

Then for almost all realizations of the sequence (Xn)∞n=0, the power series for f

represents an entire function which is frequently hypercyclic for the differentiationoperator in H(C).

The growth estimate (3) does not directly refer to frequent hypercyclicity, and isa consequence of the following general estimate for the growth of a random entirefunction.

Proposition 2 Let X be a subnormal complex random variable, i.e. suppose the con-dition

for some C > 0, lim supt→∞

e−Ct2Eet |X | < ∞ (2)

is satisfied. Let (Xn)∞n=0 be a sequence of independent random variables with the law

of X and denote

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f (z) =∞∑

n=0

Xn

n! zn .

There exists a deterministic constant C > 0 such that almost surely the function fsatisfies

sup|z|=r

| f (z)| ≤ C√

log rer

r1/4 (3)

for all sufficiently large r .

Remark. Subnormality is a much stronger assumption on the tail of |X | than thedecay condition (1) assumed in Theorem 1. By applying Markov’s inequality to theexponential moment Eet |X | and optimizing over the parameter t one sees that any sub-normal complex random variable X satisfies P(|X | ≥ r) = O

(exp(−r2/4C)

)as r

→ ∞.Combining Theorem 1 with Proposition 2 gives the main result of this note.

Theorem 3 Let X be a complex random variable such that the support of the law ofX is whole C and that the decay condition (2) is satisfied. Then the random powerseries

f (z) =∞∑

n=0

Xn

n! zn

almost surely represents a D-frequently hypercyclic entire function that satisfies thegrowth estimate (3).

To finish this introductory section we sketch the main idea of the argument for whythe random power series in Theorem 1 should represent a D-frequently hypercyclicentire function. The details of the proof and the study of the growth rate are presentedin the following two sections.

1.2 Sketch of the Main Argument

Let X be a complex random variable such that the support of its law is the wholeplane.1 Consider an infinite sequence (Xn)

∞n=0 of independent copies of X and define

the random function f : C → C by

f (z) =∞∑

n=0

Xnzn

n! . (4)

1 This requirement is clearly necessary for the power series (4) to represent a D-frequently hypercyclicentire function.


Under the condition

lim supn→∞

(Xn

n!) 1

n = 0 almost surely,

which is a rather weak assumption on the tail of |X |, the power series (4) definesa random entire function. Under the additional but still weak assumption (1) on thetail of |X | in Theorem 1, this random entire function can be shown to be frequentlyhypercyclic for the differentiation operator.

The complex plane can be covered by a countable number of disks centered atthe origin, and in any such disk the set of polynomials with rational coefficients is acountable dense set in the space of bounded analytic functions of the disk. To showthe frequent hypercyclicity of our f , it is thus sufficient to show that for any disk Br

of radius r > 0 centered at the origin, any polynomial p : C → C and any ε > 0,almost surely we have

lim infn→∞

{k ∈ {0, 1, . . . , n − 1} ∣∣ supz∈Br

| f (k)(z)− p(z)| < ε}

n> 0.

To understand why this should be true, first note that as the event {|X − a| < ε} has apositive probability for any a ∈ C and ε > 0 we have

P

(

supz∈Br

| f (z)− p(z)| < ε

)

> 0.

But by differentiation we have

f (k)(z) =∞∑

n=0

Xn+kzn

n! for all k ∈ N,

which implies the equality in distribution

f (k)d= f.

Thus also

P

(

supz∈Br

| f (k)(z)− p(z)| < ε

)

> 0 for all k ∈ N.

Moreover the events

{

supz∈Br

| f (k)(z)− p(z)| < ε

}

and

{

supz∈Br

| f ( j)(z)− p(z)| < ε

}

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can be expected to be, in an appropriate sense, almost independent if the difference ofk and j is large, since up to a small perturbation the values of an analytic function ina disk of radius r are determined by some finite number of power series coefficients.To conclude, it remains to observe that in an infinite sequence of independent trialsan event will occur with frequency equal to the probability of the event.

Thus for the proof of Theorem 1 the main challenge is to make quantitative thesense in which the events referred to above are “almost” independent. It is in thisquantitative analysis that the assumptions on the tail of |X | enter the consideration.

2 Frequent Hypercyclicity of Random Entire Functions

Before embarking on the proof of Theorem 1 we present a simple lemma that showsthat the assumed decay condition (1) cannot be weakened much without losing notonly the D-frequent hypercyclicity but even the infinite radius of convergence of thepower series.

Lemma 4 Let X be a complex random variable such that

lim infr→∞ (log r)P(|X | ≥ r) > 0 (5)

and let (Xn)∞n=0 be a sequence of independent copies of X. Then almost surely the

radius of convergence of the power series

∞∑

n=0

Xn

n! zn (6)

is zero.

Remark The condition (5) cannot be weakened much. For example, an easy argumentsimilar to the proof of the lemma shows that if

lim supr→∞

(log r)(log log r)αP(|X | ≥ r) < ∞ for some α > 0

the lemma fails and the power series (6) almost surely represents an entire function.

Proof Let M > 0. From (5) we get, for some c > 0 and large enough n,

P(|Xn| ≥ (Mn)n

) ≥ c

n log(Mn).

The series∑∞

n=21

n log n is divergent and the random variables (Xn)∞n=1 are indepen-

dent, so by the Borel–Cantelli lemma, almost surely there exists an infinite increasingsequence (nk)

∞k=1 of indices such that

for all k, |Xnk | ≥ (Mnk)nk .


The radius of convergence R of the power series (6) is given by

1

R= lim sup

n→∞

( |Xn|n!) 1

n

.

By using Stirling’s approximation for n!, we get

( |Xnk |nk !

) 1nk ∼ e

nk|Xnk |

1nk (2πnk)

− 12nk ≥ eM(2πnk)

− 12nk

for the sequence (nk)∞k=1. It follows that

1

R≥ eM a.s.

that is, almost surely the radius of convergence of (6) is less than 1eM .

Since M > 0 was arbitrary, it follows that the radius of convergence of (6) is almostsurely zero. �

By the remark after the statement of Lemma 4 we now know that the condition (1)ensures that the random power series

∑∞n=0(Xn/n!)zn almost surely defines an entire

function. Next we prove its frequent hypercyclicity.

Proof of Theorem 1 Let r > 0 and ε > 0 be fixed and let

g(z) =N∑

n=0

anzn

n!

be a given polynomial of degree N . As indicated in the introduction, to prove thetheorem it is sufficient to show that almost surely,

supz∈Br

∣∣∣ f (k)(z)− g(z)

∣∣∣ < ε

for a set of indices k with a positive lower density. Defining the events

Ak ={

supz∈Br

∣∣∣ f (k)(z)− g(z)

∣∣∣ < ε

}

and the random variable

S′n =

n−1∑

k=0

1Ak ,

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this is equivalent to showing that

lim infn→∞

S′n

n> 0.

A direct approach to this kind of a problem is to compute the expectation and varianceof S′

n . If the expectation grows linearly but variance slower than quadratically, we aredone.

Computation of the variance of S′n involves the conditional probabilities P(A j |Ak)

for j, k ∈ N0, which are rather difficult to get to directly. To get more manageablequantities we define the events Bk by setting, for all k ∈ N0,

Bk ={

N∑

n=0

|Xk+n − an|rn

n! <ε

2and |Xk+n| < ρn for all n > N

}

where (ρn)∞n=N+1 is a nondecreasing sequence of positive reals tending to infinity

such that

∞∑

n=N+1

ρnrn

n! <ε

2(7)

and that for some γ > 0

∞∑

n=d

P(|X | ≥ ρn) = O(d−γ ) as d → ∞. (8)

We postpone the choice of the sequence (ρn)∞n=N+1 until the end of the proof and first

show how its existence implies the frequent hypercyclicity of f .By (7) and the definition of the events Bk it is clear that Bk ⊂ Ak for all k ∈ N0.

Thus

Sn :=n−1∑

k=0

1Bk ≤n−1∑

k=0

1Ak = S′n .

We will show that almost surely

lim infn→∞

Sn

n> 0, (9)

which implies the theorem.Denote

p = P

(N∑

k=0

|Xk − ak |rk

k! <ε

2

)


and

Qd =∞∏

k=d

P(|X | < ρ j ) =∞∏

k=d

(1 − P(|X | ≥ ρ j )

)for d = N + 1, N + 2, . . . .

(10)

Since the Xn are independent and the support of their law is the whole plane, wehave p > 0. The assumption (8) in turn implies that all the products Qd are alsopositive. Using these notations, the invariance of the law of f under differentiationgives

ESn = E

n−1∑

k=0

1Bk =n−1∑

k=0

P(Bk) = nP(B0)

= n P

(N∑

n=0

|Xn − an|rn

n! <ε

2

) ∞∏

n=N+1

P (|Xn| < ρn)

= npQN+1. (11)

Similarly, the variance of Sn is given by

E(Sn − ESn)2 = E

(n−1∑

k=0

1Bk

)2

− (ESn)2

=n−1∑

k=0

n−1∑

j=0

P(Bk ∩ B j )− n2 p2 Q2N+1

= npQN+1 + 2n−1∑

d=1

(n − d)P(B0 ∩ Bd)− n2 p2 Q2N+1. (12)

The probability P(B0 ∩ Bd) can be explicitly computed for large d. Since the sequence(ρk)

∞k=N+1 is nondecreasing and tends to infinity, there exists a constant M ∈ N, M >

N , the choice of which depends on g, r and (ρk)∞k=N+1, such that for all d ≥ M ,

|ak | + ε

2

k!rk

≤ ρk+d for all k = 0, 1, . . . , N . (13)

This condition ensures that

{N∑

k=0

|Xd+k − ak |rk

k! <ε

2

}

⊂ {|Xk+d | ≤ ρk+d for k = 0, 1, . . . , N }

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for all d ≥ M . Thus by the definition of the events B j , for all d ≥ M we have

P(B0 ∩ Bd) = P

(N∑

k=0

|Xk − ak |rk

k! <ε

2,

N∑

k=0

|Xd+k − ak |rk

k! <ε

2,

|Xk | < ρk for k > N and |Xd+k | < ρk for k > N

)

= P

(N∑

k=0

|Xk − ak |rk

k! <ε

2,

N∑

k=0

|Xd+k − ak |rk

k! <ε

2,

|Xk | < ρk for N < k < d and |Xd+k | < ρk for k > N

)

= p2d−1∏

k=N+1

P(|X | < ρk)

∞∏

k=N+1

P(|X | < ρk) = p2 Q2N+1

Qd.

Continuing from (12),

E(Sn − ESn)2 = npQN+1 + 2

M−1∑

d=1

(n − d)P(B0 ∩ Bd)

+ 2n−1∑

d=M

(n − d)p2 Q2N+1

Qd− n2 p2 Q2

N+1.

for all n > M . We write the last term as

n2 p2 Q2N+1 = p2 Q2

N+1

(

2n−1∑

d=1

(n − d)+ n

)

= np2 Q2N+1 + 2

(M−1∑

d=1

+n−1∑

d=M

)

(n − d)p2 Q2N+1

and regroup the terms to get

E(Sn −ESn)2 =n

(pQN+1− p2 Q2

N+1

)+ 2

M−1∑

d=1

(n − d)(P(B0 ∩ Bd)− p2 Q2

N+1

)

+2n−1∑

d=M

(n − d)

(

p2 Q2N+1

Qd− p2 Q2

N+1

)

. (14)

Obviously the first two terms grow as O(n) as n tends to infinity. To bound theasymptotic growth of the final sum we use the elementary inequalities


1

1 − x≤ ec1x and ec1x − 1 ≤ c1c2x

that hold for some constants c1, c2 > 0 in some neighbourhood of the origin. Bytaking the constants large enough, these inequalities can be assumed to hold on theinterval [0, 1/2]. Recalling (10), we may estimate

1

Qd− 1 =

∞∏

k=d

(1 − P(|X | ≥ ρk)

)−1 − 1 ≤∞∏

k=d

ec1P(|X |≥ρk ) − 1

= ec1∑∞

k=d P(|X |≥ρk ) − 1 ≤ c1c2

∞∑

k=d

P(|X | ≥ ρk)

for d large enough that∑∞

k=d P(|X | ≥ ρk) ≤ 1/2. Without loss of generality, we maysuppose that M has been taken large enough that this estimate holds for all d ≥ M .We use this estimate in (14) to get

E(Sn − ESn)2 ≤ O(n)+ 2p2 Q2

N+1c1c2

n−1∑

d=M

(n − d)∞∑

k=d

P(|X | ≥ ρk),

and by the property (8) of the sequence (ρk)∞k=N+1 this further implies

E(Sn − ESn)2 ≤ O(n)+ O

(n−1∑

d=M

(n − d) d−γ)

for some γ > 0.

For 0 < γ < 1 we determine the asymptotics of the sum by writing

n−1∑

d=M

(n − d) d−γ = n2−γn−1∑

d=M

(

1 − d

n

)(d

n

)−γ 1

n

and noting that the last sum tends to the integral∫ 1

0 (1 − x)x−γ dx < ∞ as n → ∞.For 0 < γ < 1 we thus have

E(Sn − ESn)2 = O(n2−γ ).

For γ ≥ 1 the asymptotic growth of the variance is even slower, since we may estimate

n−1∑

d=M

(n − d) d−γ ≤ nn−1∑

d=M

(1

d− 1

n

)

= O(n log n).

To summarize (11) and the preceding estimates, we have deduced that under theassumptions (7) and (8), the expectation and variance of Sn satisfy

ESn = npQN+1 and E(Sn − ESn)2 = O(n2−α) for some 0 < α < 1.

M. Nikula

These facts imply that (9) holds almost surely. To see this we start by using Chebychev’sinequality to get

P

(|Sn − ESn| ≥ n1−α/4) ≤ n−2(1−α/4)

E(Sn − ESn)2 = O

(n−α/2) .

Choose a δ > 1 so that −δα/2 < −1 and consider a sequence (n j )∞j=1 of indices such

that n j ∼ jδ . For this sequence the series

∞∑

j=1

P

(∣∣Sn j − ESn j

∣∣ ≥ n1−α/4

j

)

is summable, so by the Borel–Cantelli lemma, almost surely

n1−α/4j > |Sn j −ESn j | �⇒ pQN+1 − n−α/4

j <Sn j

n jfor all but finitely many j.

Since N is fixed, this already implies that lim inf j→∞Sn jn j

> 0 almost surely. Toextend this from the subsequence (n j ) to the full result, let n ≥ n1 be arbitrary andtake an index j so that n j ≤ n ≤ n j+1. Since Sn is nondecreasing we have

Sn

n>

Sn j

n j+1= n j

n j+1

Sn j

n j>

n j

n j+1

(pQN+1 − n−α/4

j

).

Noting that the asymptotic growth n j ∼ jδ implies lim j→∞ n j/n j+1 = 1, the fullclaim (9) follows.

It remains to show that under the assumption (1) it is possible to choose a sequence(ρk)

∞k=N+1 satisfying (7) and (8). Our choice is ρn = aen , where a > 0 is taken so

that

a∞∑

n=N+1

(er)n

n! <ε

2.

This choice ensures that (7) holds. To see that (8) holds as well, observe that by (1)there exists a constant c > 0 such that

P(|X | ≥ aen) ≤ c

(n + log a)1+β

for all n ∈ N such that aen ≥ 1. Thus for large d we have

∞∑

n=d

P(|X | ≥ aen) ≤∞∑

n=d

c

(n + log a)1+β ≤∞∫

d−1+log a

c

x1+β dx = O(d−β),

which means that (8) holds with γ = β. The proof is complete. �


Remark Bayart and Matheron [4] give an eigenvalue characterization for an operatorto be weakly mixing with respect to a Gaussian measure. If the distribution of X inour construction is Gaussian, the measure we obtain on H(C) is clearly an explicitexample of a measure of the type considered by Bayart and Matheron.

Remark 2 It is not absolutely necessary to assume that the random variables (Xn) areindependent: with more involved analysis our methods would extend to cases in whichthe mutual dependence, measured in some appropriate sense, between Xn and Xm

decays sufficiently fast as |n − m| → ∞.

3 Growth of Random Analytic Functions

In this section we show that the growth rate of our random frequently hypercyclicentire functions is close to the minimum possible for frequently hypercyclic entirefunctions.

Let us first consider a random entire function f defined as in Theorem 1 for a stan-dard complex Gaussian random variable X , i.e. we suppose that that Re X and Im Xare independent real Gaussians with mean 0 and variance 1

2 . Since sums of indepen-dent Gaussian variables are Gaussian, the values { f (z)}z∈C are also Gaussian randomvariables. Moreover, the variance of the sum of independent random variables is thesum of the variances, so we have

E| f (z)|2 = E

∣∣∣∣∣

∞∑

n=0

Xn

n! zn

∣∣∣∣∣

2

=∞∑

n=0

|z|2n

(n!)2 . (15)

Let us denote

I (r) =∞∑

n=0

r2n

(n!)2 for r ≥ 0.

The growth rates of our random analytic functions are determined by the asymptoticsof the function I (r). Our function I (r) is, up to a scaling of the argument, the modifiedBessel function of the second kind, usually denoted by I0; namely we have I (r) =I0(2r) for all r ≥ 0. By the asymptotics of I0 we have2 (see e.g. formula 9.7.1 in[1])

I (r) ∼ 1

2√π

e2r

√r

as r → ∞.

The formula E|X |p = �(p2 + 1)(E|X |2)p/2 for the moments of a complex Gaussian

random variable now gives, for all p > 0,

2 For two positive functions f and g, we denote f ∼ g if limx→∞ f (x)/g(x) = 1 and f � g if for someC > 0 we have C−1 ≤ f (x)/g(x) ≤ C for all large x .

M. Nikula

E

2π∫

0

| f (reiθ )|p dθ

2π=

2π∫

0

E| f (reiθ )|p dθ

2π= E| f (r)|p

= �( p

2+ 1)

I (r)p2

�(

er

r14

)p

as r → ∞;

we remark that the implied constants on the last line depend on p and tend to infinityas p → ∞. Recalling Theorem A, this calculation indicates that the growth of theintegral means of the functions considered in Theorem 1 is nearly minimal amongamong all D-frequently hypercyclic functions, at least in the Gaussian case and whenaveraged over the possible realizations of the randomness. In Proposition 2 we considerthe individual realizations of the construction in a slightly more general case.

The proof of Proposition 2 relies on the following estimate, given by Kahane [10,p. 69], for the probability of a random trigonometric polynomial taking large values.

Lemma 5 Let ξ be a real random variable that satisfies the decay condition (2). Let(ξn)

Kn=1 be a finite sequence of independent copies of ξ, (an)

Kn=1 a finite sequence

of complex constants and (qn)Kn=1 a finite sequence of trigonometric polynomials of

degree less than or equal to N. Then there exists a constant c > 0 that depends onlyon the distribution of ξ such that the tail estimate

P

⎛

⎜⎝ supθ∈[0,2π ]

∣∣∣∣∣

K∑

n=1

anξnqn(θ)

∣∣∣∣∣> c√

log N

(K∑

n=1

|an|2) 1

2

⎞

⎟⎠ ≤ 1

N 2 (16)

holds.

Proof of Proposition 2 Write

f (z) =∞∑

n=0

Xnzn

n! =∞∑

n=0

(Re Xn)zn

n! + i∞∑

n=0

(Im Xn)zn

n! =:g(z)+ h(z).

It is clearly enough to prove the estimate (3) for g.Let R ∈ N and further decompose g by writing

g(z) =∞∑

n=0

(Re Xn)zn

n! =3R∑

n=0

(Re Xn)zn

n! +∞∑

j=1

3 j+1 R∑

n=3 j R+1

(Re Xn)zn

n!

= : gR,0(z)+∞∑

j=1

gR, j (z).

We will show that almost surely for all but finitely many R ∈ N, around |z| = R thecontribution of the series

∑∞j=1 gR, j is negligible and that gR,0 satisfies (3).


We start with the analysis of gR,0. As

sup|z|=R

|gR,0(z)| = supθ∈[0,2π ]

∣∣∣∣∣

3R∑

n=0

XnRn

n! einθ

∣∣∣∣∣

and

3R∑

n=0

(Rn

n!)2

≤∞∑

n=0

R2n

(n!)2 = I (R),

by Lemma 5 there exists a constant c1 depending only on the distribution of Re X suchthat

P

(

sup|z|=R

|gR,0(z)| > c1√

log 3R√

I (R)

)

≤ 1

(3R)2.

The series∑∞

R=1 R−2 is summable, so by the Borel–Cantelli lemma we have

sup|z|=R

|gR,0(z)| ≤ c1√

log 3R√

I (R) (17)

almost surely for all but finitely many R ∈ N. Since

√I (R) � eR

R1/4

as R → ∞, we see that indeed gR,0(z) has the claimed order of magnitude close to|z| = R.

For treating the functions gR, j we need an estimate for the tail of the series of I (R).By bounding from above by a geometric series we see that for any integer A ≥ 3Rwe have

∞∑

n=A+1

R2n

(n!)2 <R2A

(A!)2∞∑

n=1

R2n

A2n= R2A

(A!)2(R/A)2

1 − (R/A)2≤ 1

8

R2A

(A!)2 .

Applying this and Lemma 5 to the functions gR, j gives

P

(

sup|z|=R

|gR, j (z)| > c1

√log 3 j+1 R

1

8

R2·3 j R

((3 j R)!)2)

≤ 1

(3 j+1 R)2.

The double series∑∞

j=1∑∞

R=11

(3 j+1 R)2is summable. The Borel–Cantelli lemma thus

implies that almost surely

M. Nikula

sup|z|=R

|gR, j (z)| ≤ c1

√log 3 j+1 R

1

8

R2·3 j R

((3 j R)!)2

for all but finitely many pairs (R, j) ∈ N2. We use Stirling’s approximation to write,

for large R,

c1

√log 3 j+1 R

1

8

R2·3 j R

((3 j R)!)2 ∼ c1

8

√( j + 1) log 3 + log R

R2·3 j R

(3 j R/e

)2·3 j R 2π 3 j R

= c1

16π

√( j + 1) log 3 + log R

( e

3 j

)2·3 j R 1

3 j R.

From this form of the estimate for sup|z|=R |gR, j (z)| it is easy to see that we haveactually shown that almost surely

∞∑

j=1

sup|z|=R

|gR, j (z)| → 0 as R → ∞. (18)

The estimates (17) and (18) also show that almost surely, for all but finitely manyR ∈ N we have

sup|z|=R

|g(z)| ≤ sup|z|=R

|gR,0(z)| +∞∑

j=1

sup|z|=R

|gR, j (z)| ≤ C√

log ReR

R14

for some deterministic constant C > 0 that depends only on the distribution of Re X .The extension from integer to real radii is immediate, as the ratio

C√

log(R + 1) eR+1

(R+1)14

C√

log R eR

R14

= e

√log(R + 1)

log R

(R

R + 1

) 14

is bounded as R → ∞ and by the maximum principle, the function R �→sup|z|=R |g(z)| is increasing. �

While our construction of a random D-frequently hypercyclic entire function issimple it cannot reach the optimal asymptotics for D-frequently hypercyclic entirefunctions, essentially since it is impossible to use independent power series coefficientsand get the required amount of cancellations. We substantiate this assertion by thefollowing proposition in which we show that in the case of a Gaussian X the boundin Proposition 2 in fact cannot be improved at all.

Proposition 6 Let (Xn)∞n=0 be an independent sequence of standard complex

Gaussian random variables. There exists a constant c > 0 such that for almostevery realization of the sequence (Xn) there exists a sequence (zk)k∈N of complex


numbers such that |zk | → ∞ and the entire function defined by the power seriesf (z) =∑∞

n=0(Xn/n!)zn satisfies

| f (zk)| ≥ c√

log |zk | e|zk |

|zk |1/4 for all k.

Proof In (15) we already noted that in the case of a standard complex Gaussian X thevalues { f (z)}z∈C are a centered (jointly) Gaussian family with variances given by

E| f (z)|2 = E

∣∣∣∣∣

∞∑

n=0

Xn

n! zn

∣∣∣∣∣

2

=∞∑

n=0

|z|2n

(n!)2 = I (|z|) � e2|z|

|z|1/2 .

Let α ∈ (1, 2) be fixed. For a given k ∈ N, we decompose the series defining f as

f (z) =k2−�kα�∑

n=0

Xnzn

n! +k2+�kα�−1∑

n=k2−�kα�+1

Xnzn

n! +∞∑

n=k2+�kα�Xn

zn

n!= : f1,k(z)+ f2,k(z)+ f3,k(z). (19)

It is clear that there exists a K ∈ N (depending only on α) so that the family { f2,k}k≥K

is independent. We will first show that

E| f1,k(k2)|2E| f (k2)|2 = O

(e−kα−1

)and

E| f3,k(k2)|2E| f (k2)|2 = O

(e−kα−1

)(20)

as k → ∞. This implies that in (19) the only nonnegligible term for z = k2 is f2,k(z),and it is easy to analyze the sequence ( f2,k(k2)) of independent Gaussian randomvariables and show that the claimed lower bound holds for infinitely many k in thesequence (zk)k∈N = (k2)k∈N.

We start our estimation by noting that the maximal term in the series

E| f (k2)|2 = I (k2) =∞∑

n=0

((k2)n

n!)2

is the one with n = k2. We denote the maximal term by Mk := (k2)2k2

(k2!)2 . The ratio of a

general term and the maximal term is given, for n < k2, by

((k2)n

n!)2

/

((k2)k

2

k2!

)2

=(

k2

k2 · k2 − 1

k2 · . . . · n + 1

k2

)2

=(

1 ·(

1 − 1

k2

)

· . . . ·(

1 − k2 − (n + 1)

k2

))2

(21)

M. Nikula

and for n > k2 by

((k2)n

n!)2

/

((k2)k

2

k2!

)2

=(

k2

k2 + 1· k2

k2 + 2· . . . · k2

n

)2

=((

1+ 1

k2

)−1

·(

1 + 2

k2

)−1

· . . . ·(

1+ n − k2

k2

)−1)2

.

(22)

For n < k2 we use the inequality 1 − x ≤ e−x for x ≥ 0 in (21) to get the estimate

((k2)n

n!)2

/

((k2)k

2

k2!

)2

≤ exp

⎛

⎝−2k2−(n+1)∑

j=0

j

k2

⎞

⎠ = exp

(

− (k2 − n − 1)(k2 − n)

k2

)

= exp

(

−k2 + 2n + 1 − n(n + 1)

k2

)

.

By elementary calculus, the last estimate is increasing in n for n < k2, so for 0 ≤ n ≤k2 − �kα� we have

((k2)n

n!)2

/

((k2)k

2

k2!

)2

≤ exp

(

−k2 + 2(

k2 − �kα�)

+ 1 −(k2 − �kα�)2

k2

)

= exp

(

−�kα�2

k2

)

.

Since α ∈ (1, 2), the bound on the right decays faster than any polynomial. This finallyallows us to estimate

E| f1,k(k2)|2 =

k2−�kα�∑

n=0

(k2)2n

(n!)2 = Mk

k2−�kα�∑

n=0

((k2)n

n!)2

/

((k2)k

2

k2!

)2

≤ Mk

k2−�kα�∑

n=0

exp

(

−�kα�2

k2

)

< Mkk2 exp

(

−�kα�2

k2

)

.

Since trivially Mk ≤ E| f (k2)|2, this implies the first claim in (20). The second claimfollows from a similar estimation starting from the estimate (22). We omit the details.

For σ > 0, the probability of the scalar multiple σ X of the standard complexGaussian X taking a large value can be computed exactly as

P (|σ X | ≥ |σ |a) = 1

π

∞∫

a

e−|z|2 dA(z) = e−a2,


where dA is the area measure on the complex plane. It follows that for any ε > 0,

P

(

| f j,k(k2)| ≥ ε

√

E| f (k)|2)

= exp

(

−ε2 E| f (k)|2E| f j,k(k2)|2

)

for j = 1, 3

whence from (20) we get

∞∑

k=1

P

(

| f j,k(k2)| ≥ ε

√

E| f (k)|2)

< ∞ for j = 1, 3.

Thus from the Borel–Cantelli lemma we deduce that for any given ε > 0, almostsurely for all but finitely many k we have

| f j,k(k2)| < ε

√

E| f (k2)|2 ≤ Cεek2

(k2)1/4for j = 1, 3, (23)

where Cε > 0 is a constant depending only on ε. It remains to estimate the contribution

of the term f2,k(k2). From (20) we get that E| f2,k (k2)|2E| f (k2)|2 → 1 as k → ∞. Thus there

exists a c2 < 1 such that

P

(

| f2,k(k2)| ≥ c

√log k

√

E| f (k)|2)

= exp

(

−(

c√

log k)2 E| f (k)|2

E| f2,k(k2)|2)

> exp

(

−(

c√

log k)2

c−2)

= 1

k,

which gives

∞∑

k=1

P

(

| f2,k(k2)| ≥ c

√log k

√

E| f (k)|2)

= ∞.

Since the variables { f2,k(k2)} are independent, the Borel–Cantelli lemma implies thatalmost surely for all but finitely many k,

| f2,k(k2)| ≥ c

√log k

√

E| f (k)|2 ≥ c′√

log k2 ek2

(k2)1/4.

Combined with (23), this gives the claim. �Acknowledgments I thank David Drasin and my advisor Eero Saksman for helpful comments and fordirecting me to this problem. I also thank the anonymous referee for a close reading of the manuscript andfor numerous helpful comments.

M. Nikula

References

1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Math-ematical Tables. Dover Publications, NY (1972) (10th printing)

2. Bayart, F., Grivaux, S.: Hypercyclicité: le rôle du spectre ponctuel unimodulaire. C. R. Math. Acad.Sci. Paris 338(9), 703–708 (2004)

3. Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Am. Math. Soc. 358(11), 5083–5117(2006)

4. Bayart, F., Matheron, É.: Mixing operators and small subsets of the circle. arXiv:1112.1289 (2011)5. Blasco, O., Bonilla, A., Grosse-Erdmann, K.-G.: Rate of growth of frequently hypercyclic functions.

Proc. Edinb. Math. Soc. (2) 53(1), 39–59 (2010)6. Bonet, J., Bonilla, A.: Chaos of the differentiation operator on weighted Banach spaces of entire

functions. Complex Anal. Oper. Theory (to appear)7. Bonilla, A., Grosse-Erdmann, K.-G.: On a theorem of Godefroy and Shapiro. Integral Equ. Oper.

Theory 56(2), 151–162 (2006)8. Bonilla, A., Grosse-Erdmann, K.-G.: A problem concerning the permissible rates of growth of fre-

quently hypercyclic entire functions. In: Topics in Complex Analysis and Operator Theory, pp. 155–158. Univ. Málaga, Málaga (2007)

9. Drasin, D., Saksman, E.: Optimal growth of frequently hypercyclic entire functions. J. Funct. Anal.263(11), 3675–3688 (2012)

10. Kahane, J.P.: Some Random Series of Functions, 2nd edn. Cambridge University Press, Cambridge(1985)

11. MacLane, G.: Sequences of derivatives and normal families. J. Anal. Math. 2, 72–87 (1952)

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Frequent Hypercyclicity of Random Entire Functions for the Differentiation Operator