Local universality for real roots of random trigonometric polynomials

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by  Alexander Iksanov

LOCAL UNIVERSALITY FOR REAL ROOTS OF RANDOM
TRIGONOMETRIC POLYNOMIALS
6
1
0 ALEXANDERIKSANOV,ZAKHARKABLUCHKO,ANDALEXANDERMARYNYCH
2

y
Abstract. Consider arandom trigonometric polynomial Xn :R→R of the
a
form
M n
Xn(t)=X(ξksin(kt)+ηkcos(kt)),
4 k=1
1 where(ξ1,η1),(ξ2,η2),...areindependentidenticallydistributedbivariatereal
random vectors with zero mean and unit covariance matrix. Let (sn)n∈N be
any sequence of real numbers. We prove that as n → ∞, the number of
]
R real zeros of Xn in the interval [sn +a/n,sn +b/n] converges in distribu-
tion to the number of zeros in the interval [a,b] of a stationary, zero-mean
P
Gaussian process with correlation function (sint)/t. We also establish simi-
.
h lar local universality results for the centered random vectors (ξk,ηk) having
t an arbitrary covariance matrix or belonging to the domain of attraction of a
a
two-dimensionalα-stablelaw.
m
[

2
v 1. Introduction
0
We are interestedin randomtrigonometricpolynomialsX :R R of the form
4 n →
7 n
5 (1) Xn(t)= (ξksin(kt)+ηkcos(kt)),
0
k=1
X
.
1 where the coefficients ξ ,η ,ξ ,η ,... are real random variables. In a recent pa-
1 1 2 2
0 per, Aza¨ıs et al. [3] conjectured that if ξ ,η ,ξ ,η ,... are independent identically
1 1 2 2
6
distributed (i.i.d.) with zero mean and finite variance, then the number of real
1
zeros of X in the interval [a/n,b/n]convergesin distribution (without normaliza-
: n
v tion) to the number of zeros in the interval [a,b] of a stationary Gaussian process
i
X Z :=(Z(t))t∈R with zero mean and
r Cov(Z(t),Z(s))=sinc(t s), t,s R,
a − ∈
where
(sint)/t, if t=0,
sinct= 6
(1, if t=0.
The limit distribution does not depend on the distribution of ξ , a phenomenon
1
referred to as local universality. Aza¨ıs et al. [3] proved their conjecture assuming
that ξ has an infinitely smooth density that satisfies certain integrability condi-
1
tions. However, as they remarked, even the case of the Rademacher distribution
P[ξ = 1] = 1/2 remained open. Our aim is to prove the conjecture of [3] in
1
±
2010 Mathematics Subject Classification. Primary: 26C10;Secondary: 30C15,42A05, 60F17,
60G55.
Keywordsandphrases. Randomtrigonometricpolynomials,realzeros,localuniversality,sta-
tionaryprocesses,randomanalyticfunctions,functionallimittheorem, stableprocesses.
1
2 ALEXANDERIKSANOV,ZAKHARKABLUCHKO,ANDALEXANDERMARYNYCH
full generality (Theorem 2.1 below). The method of proof proposed in the present
paper is very different from the one used in [3]. Let us briefly sketch our approach
assuming that (ξ ,η ),(ξ ,η ),... are i.i.d. randomvectorssuch that ξ and η are
1 1 2 2 1 1
centered uncorrelated random variables with unit variance. We start by proving a
functional limit theorem (Theorem 3.1 below) stating that
1
(2) X s + · Z()
n n
√n n n−→→∞ ·
(cid:16) (cid:17)
weakly on some suitable space of analytic functions. Then, we use the continuous
mapping theorem to deduce the convergence of the real zeros. The basic fact
underlying this part of the proof is the Hurwitz theorem stating that the complex
zerosofananalyticfunctiondonotchange“toomuch”underaslightperturbation
of the function. Essentially, Hurwitz’s theorem tells us that the functional which
maps an analytic function to the point process of its complex zeros is continuous.
Since we are interested in real zeros, we have to ensure that real zeros remain real
after a small perturbation. If we restrict ourselves to analytic functions which are
realonR, then non-realzeroscome in complex conjugatedpairs,and a simple real
zero cannot become complex under a small perturbation of the function. These
considerations,see Lemmas 4.1 and 4.2, justify the use of the continuous mapping
theorem.
Our method is quite general and allows us to establish the corresponding local
universalityresultinthecasewhen(ξ ,η )hasanon-zerocorrelation(Theorem2.3)
1 1
or even does not have finite second moments but is in the domain of attraction of
some stable two-dimensional law (Theorem 2.5).
Closing the introduction, we mention that the scope of our approach is not re-
strictedto trigonometricpolynomials. The same method canbe applied to various
ensembles of random algebraic polynomials. A similar method was used in [20],
[15]forcomplexzerosofrandomTaylorseriesnearthecircleofconvergence,in[19]
for Dirichlet series with random coefficients and some other sums of analytic func-
tions with random coefficients, and in [12, 13] for complex zeros of the partition
function of the (Generalized) Random Energy Model. Unlike in these works, we
investigate real zeros. Let us also mention that the asymptotics of EN [a,b] (that
n
is, the expected number of real zeros in an interval whose length does not go to
0) was studied in the recent works [1] and [8], where more references on random
trigonometric polynomials can be found.
The structure of the paper is as follows. The main results are stated in Sec-
tion2. FunctionallimittheoremsforX andtheirproofsaregiveninSection3. In
n
Section 4 the proofs of the main theorems are presented. Some auxiliary technical
lemmas are collected in the Appendix.
d
As usual, denotes convergence in distribution of random variables and vec-
−→ w
tors. The notation is used to denote weak convergence of random elements
−→ v
withvalues in a metric space,while denotes vagueconvergenceoflocally finite
−→
measures.
2. Main results
2.1. Coefficients with finite second moments. For a real analytic function f
whichdoesnotvanishidenticallydenotebyN [a,b]thenumberofzerosoff inthe
f
REAL ROOTS OF RANDOM TRIGONOMETRIC POLYNOMIALS 3
interval [a,b]. It will become clear from our proofs that the results hold indepen-
dently of whether the zeros are counted with multiplicities or not. Theorem 2.1,
which is our first main result, proves the conjecture of [3], weakens the original
assumptionsof[3]onthe distributionofthe coefficientsandallowsforanarbitrary
sequence (sn)n∈N as the location of the scaling window.
Theorem 2.1. Let (ξ ,η ),(ξ ,η ),... bei.i.d. random vectors with zeromean and
1 1 2 2
unit covariance matrix, that is,
Eξ =Eη =0, E[ξ2]=E[η2]=1, E[ξ η ]=0.
1 1 1 1 1 1
Let (sn)n∈N be any sequence of real numbers and [a,b] R a finite interval. Then,
⊂
a b
d
N s + ,s + N [a,b],
Xn n n n n n−→→∞ Z
(cid:20) (cid:21)
where (Z(t))t∈R is the stationary Gaussian process defined in Section 1.
We can also prove the weak convergence of point processes of zeros. Given a
locally compact metric space X, denote by M (X) the space of locally finite point
p
measuresonXendowedwiththevaguetopology. Arandomelementwithvaluesin
M (X) is called a point processon X. We refer to [17] for the informationonpoint
p
processes and their weak convergence. For a real analytic function f which does
not vanish identically denote by ZerosR(f) the locally finite point measure on R
counting the real zeros of f with multiplicities. The next theorem is stronger than
Theorem 2.1 in view of the continuous mapping theorem and Lemma 4.2 below.
Theorem 2.2. Under the same assumptions as in Theorem 2.1 we have
w
ZerosR Xn sn+ · ZerosR(Z())
n n−→→∞ ·
(cid:16) (cid:16) (cid:17)(cid:17)
on M (R).
p
In the next theorem we consider i.i.d. random vectors with arbitrary covariance
matrix. Inparticular,thistheoremcoversrandomtrigonometricpolynomialsofthe
form n ξ sin(kt) and n η cos(kt) involving sin or cos terms only.
k=1 k k=1 k
TheoPrem 2.3. Let (ξ1,η1P),(ξ2,η2),... be i.i.d. random vectors with
Eξ =Eη =0, E[ξ2]=σ2 < , E[η2]=σ2 < , E[ξ η ]=ρ,
k k k 1 ∞ k 2 ∞ k k
where 0<σ2+σ2 < . Then, for every fixed s R,
1 2 ∞ ∈
ZerosR Xn s+ · w ZerosR(G())
n n−→→∞ ·
(cid:16) (cid:16) (cid:17)(cid:17)
on Mp(R), where (G(t))t∈R is a centered Gaussian process with covariance
(3) E[G(t )G(t )]
1 2
σ12+σ22 sinc(t t ), s / πZ,
= 2 1− 2 ∈
(σ12+2σ22 sinc(t1−t2)− σ12−2σ22 sinc(t1+t2)+ρ1−cto1s+(tt12+t2), s∈πZ,
with the convention that x (1 cosx)/x equals 0 at x=0.
7→ −
Remark 2.4. In Theorem 2.3 we can replace the fixed s by a general sequence
(sn)n∈N as in Theorem 2.1, but then we have to replace the condition s / πZ with
∈
lim ndist(s ,πZ) = + and s πZ with lim ndist(s ,πZ) = 0. Here,
n→∞ n n→∞ n
∞ ∈
we used the notation dist(s ,πZ)=min s πk : k Z .
n n
{| − | ∈ }
4 ALEXANDERIKSANOV,ZAKHARKABLUCHKO,ANDALEXANDERMARYNYCH
2.2. Coefficients from a stable domain of attraction. Let(ξ ,η ),(ξ ,η ),...
1 1 2 2
be i.i.d. random vectors from the strict domain of attraction of a two-dimensional
α-stable distribution, 0<α<2. This means that there exist numbers b >0 such
n
that
n n
1
(4) ξ , η d S ,
k k α,ν
bn !n−→→∞
k=1 k=1
X X
where S is a non-degeneratetwo-dimensionalα-stable randomvectorwith L´evy
α,ν
measure ν and shift parameter 0. The adjective “strict” is used to highlight that
convergence (4) holds without centering, in particular, it is assumed that Eξ =
1
Eη = 0 if α > 1. We refer to [18] for details on multivariate stable distributions
1
and stable processes. Note that ν is a locally finite measure on R2 0 which has
\{ }
the homogeneity property
ν(λB)=λ−αν(B)
for all λ>0 andall Borelsets B R2 0 . In whatfollows we identify R2 andC
⊂ \{ }
via the canonical isomorphism and consider R2-valued processes as C-valued and
vice versa.
Theorem 2.5. Assume that (4) holds and let s R be fixed. Then
∈
w
ZerosR Xn s+ · ZerosR(Zν())
n n−→→∞ ·
(cid:16) (cid:16) (cid:17)(cid:17)
on Mp(R), where (Zν(t))t∈R is a stochastic process given by
1 1 1
(5) Z (t)=Im eitudL(u)= sin(tu)dReL(u)+ cos(tu)dImL(u),
ν
Z0 Z0 Z0
for t R, and (L(u)) is a C-valued α-stable L´evy process with zero drift, no
u∈[0,1]
∈
Gaussian component, and the L´evy measure ν˜ defined by
1ν(e2πiyB)dy, if s / πQ,
(6) ν˜(B):= 0 ∈
(Rq1 qk=1ν(e2πik/qB), if s=2πp/q, with p∈Z,q ∈N coprime,
for all Borel sets BP C 0 , with ν being the L´evy measure of Sα,ν in (4).
⊂ \{ }
Remark 2.6. The integral in (5) (which need not exist in the Lebesgue–Stieltjes
sense because L has finite variation a.s. in the case α (0,1) only) is defined via
∈
integration by parts:
1 1
(7) eitudL(u)d=ef L(1)eit it L(u)eitudu.
−
Z0 Z0
See, e.g., [10, 18] for the properties of such stochastic integrals.
Remark 2.7. An interesting feature of Theorem 2.5 is that the behavior of the
zeros near s depends on whether s˜:= s/(2π) is rational or not. To see why such
arithmeticeffectsshowup,assumeforamomentthatξ andη areindependentand
k k
symmetricα-stable. Then,X (s)isalsosymmetricα-stablewithscalingparameter
n
σ , where
n
n n n n
σα = cos(ks)α+ sin(ks)α = cos(2π ks˜ )α+ sin(2π ks˜ )α
n | | | | | { } | | { } |
k=1 k=1 k=1 k=1
X X X X
and denotes the fractional part. If s˜is irrational, then the sequence ( ks˜ )k∈N
{·} { }
is uniformly distributed on the interval [0,1] by Weyl’s equidistribution theorem,
REAL ROOTS OF RANDOM TRIGONOMETRIC POLYNOMIALS 5
see,forexampleTheorem2.1andExample2.1in[14],whereasforrationals˜=p/q
it is uniformly distributed on the finite set 0,1,...,q−1 , whence
{ q q }
1 1( cos(2πu)α+ sin(2πu)α)du, if s / πQ,
lim σα = 0 | | | | ∈
n→∞n n (Rq1 qk=1(|cos(2πk/q)|α+|sin(2πk/q)|α), if s=2πp/q.
Note that for α = 2P(which corresponds to the finite variance case studied in
Section2.1),thereisnodifferencebetweentherationalandirrationalcasesbecause
sin2t+cos2t=1.
Remark 2.8. In Theorem 2.5 it is possible to replace the fixed s by a sequence
(sn)n∈R assuming that s := limn→∞sn exists and either s / πQ (the first case
∈
in (6)) or s πQ and s s =o(1/n) as n (the second case in (6)).
n
∈ | − | →∞
3. Convergence of random trigonometric polynomials as random
analytic functions
3.1. Spaces of analytic functions and analytic continuations of the pro-
cesses Z, G and Z . Let be the space of functions which are analytic on the
ν
H
entire complex plane. We endow with the topology of uniform convergence on
H
compact sets. This topology is generated by the complete separable metric
1 f g D¯
d(f,g)= k − k k ,
2k1+ f g D¯
k≥1 k − k k
X
where D¯ = z r is the closed disk or radius r > 0 around the origin, and
r
{| | ≤ }
f = sup f(z) is the sup-norm of f on a compact set K C; see [6,
k kK z∈K| | ⊂
pp. 151–152]. A random analytic function is a random element taking values in
the space endowed with the Borel σ-algebra. We refer to [9] and [19] for more
H
information on random analytic functions.
Let R be a closed subspace of consisting of all functions f which take
H H ∈ H
real values on R. Note that for every f R we have f(z¯) = f(z) for all z C.
∈ H ∈
Indeed, the functions f(z) and f(z¯) are analytic and coincide on R. Hence, they
must coincide everywhere on C by the uniqueness theorem for analytic functions.
The space R is endowed with the induced topology and metric.
H
Following the approachoutlined in the introduction, we shall show that conver-
gence (2) and its counterparts in the case of correlated (ξ ,η ) and in the stable
1 1
case hold weakly on the space R. But, first of all, we have to construct ana-
H
lytic continuations of the limit processes Z, G and Z appearing in Theorems 2.1,
ν
Theorems 2.3 and 2.5, respectively.
3.1.1. TheprocessZ. ThestationaryGaussianprocess(Z(t))t∈R appearinginThe-
orem2.1canbeextendedanalyticallytothecomplexplaneusingtherepresentation
(8) Z(t)= sinc(t πk)N , t C,
k
− ∈
k∈Z
X
where (Nk)k∈Z are i.i.d. real standard Gaussian random variables. The series
in (8) converges uniformly on compact subsets of C because so does the series
k∈Z|sinc(t−πk)|2; see [9, Lemma 2.2.3]. It follows that (Z(t))t∈C is an analytic
function on C with probability 1.
P
6 ALEXANDERIKSANOV,ZAKHARKABLUCHKO,ANDALEXANDERMARYNYCH
The R2-valued process ((ReZ(t),ImZ(t)))t∈C is jointly real Gaussian in the
sense that for all t ,...,t C, the 2d-dimensional random vector
1 d
∈
(ReZ(t ),ImZ(t ),...,ReZ(t ),ImZ(t ))
1 1 d d
is real Gaussian. Clearly, EZ(t) = 0 for all t C. The covariance structure of
∈
(Z(t))t∈C is given by
(9) E[Z(t)Z(s)]=sinc(t s), t,s C,
− ∈
(10) E[Z(t)Z(s)]=sinc(t s¯), t,s C.
− ∈
For instance, in the case when t,s / πZ, we have
∈
sin(t πk) sin(s πk)
E[Z(t)Z(s)]= − −
t πk s πk
k∈Z − (cid:18) − (cid:19)
X
1
=(sint)(sins¯)
(t πk)(s¯ πk)
k∈Z − −
X
(sint)(sins¯) 1 1
=
s¯ t t πk − s¯ πk
− k∈Z(cid:18) − − (cid:19)
X
(sint)(sins¯)
= (cott cots¯)
s¯ t −
−
sin(t s¯)
= − ,
t s¯
−
where we used the partial fraction expansion of the cotangent. In the case when
t=πj for some j Z, we have
∈
E[Z(t)Z(s)]= sinc(t πk)sinc(s πk)=sinc(s πj)=sinc(t s¯)
− − − −
k∈Z
X
because sinc(t πk)=1 for k =j and 0 for k=j. The proof of (9) is similar.
− 6
Representation (8) appeared, for example, in [2]. Note that the analytically
continued process (Z(t))t∈C is stationary with respect to shifts along the real axis,
but it is not stationary with respect to shifts along the imaginary axis.
3.1.2. The process G. In the case s / πZ we can simply take
∈
σ2+σ2
G(t):= 1 2Z(t), t C,
2 ∈
r
wheretheprocess(Z(t))t∈C isthesameasinSection3.1.1. Inthecases πZtake
∈
a centered C-valued Brownian motion (W(u)) with covariance structure
u∈[0,1]
E[(ReW(1))2]=σ2, E[(ImW(1))2]=σ2, E[(ImW(1))(ReW(1))]=ρ,
1 2
and put
1
U(t)= eitudW(u), t C,
∈
Z0
REAL ROOTS OF RANDOM TRIGONOMETRIC POLYNOMIALS 7
where the integralis defined via the formalintegrationby parts,as in (7). Clearly,
this defines U as a random analytic function on C. Now put1
U(t) U(t) 1 1
G(t)= − = sin(tu)dReW(u)+ cos(tu)dImW(u), t C.
2i ∈
Z0 Z0
Integrating by parts and using the identities
1 1 1
sin(t u)sin(t u)du= sinc(t t ) sinc(t +t ),
1 2 1 2 1 2
2 − − 2
Z0
1 1 1
cos(t u)cos(t u)du= sinc(t t )+ sinc(t +t ),
1 2 1 2 1 2
2 − 2
Z0
1 1 cos(t +t ) 1 cos(t t )
1 2 1 2
sin(t u)cos(t u)du= − + − − ,
1 2
2(t +t ) 2(t t )
Z0 1 2 1− 2
it is easy to check that the covariance function of (G(t))t∈R is given by the second
line in (3).
3.1.3. TheprocessZ . Let(L(u)) beaC-valuedα-stableL´evyprocessdefined
ν u∈[0,1]
in Theorem 2.5. As in the construction of G above, put
1
(11) U (t)= eitudL(u), t C,
ν
∈
Z0
where the integral is understood as in (7). Obviously, U is a random analytic
ν
function on C and we can take
U (t) U (t) 1 1
Z (t)= ν − ν = sin(tu)dReL(u)+ cos(tu)dImL(u), t C.
ν
2i ∈
Z0 Z0
3.2. Functional limit theorems for random trigonometric polynomials.
Now we are ready to prove convergence (2) and its counterparts corresponding to
Theorems 2.3 and 2.5.
Theorem 3.1. Let (ξ ,η ),(ξ ,η ),... bei.i.d. random vectors with zeromean and
1 1 2 2
unit covariance matrix. Fix any sequence of real numbers (sn)n∈N and consider a
random process (Yn(t))t∈C defined by
1 t
(12) Y (t):= X s +
n n n
√n n
(cid:18) (cid:19)
n
1 t t
= ξ sin k s + +η cos k s + .
k n k n
√n n n
k=1(cid:18) (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19)(cid:19)
X
w
Then Yn Z on R.
n−→→∞ H
Proof. The proof consists of two steps.
Convergence of finite-dimensional distributions. Take any t ,...,t C. We have
1 d
∈
the representationY (t)=V (t)+...+V (t), where
n n,1 n,n
1 t t
V (t):= ξ sin k s + +η cos k s + .
n,k k n k n
√n n n
(cid:18) (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19)(cid:19)
1Here the following observation is used: if f is an analytic function, so is g(z) := (f(z)−
f(z))/(2i). Moreover,g∈HR andfort∈Rwehaveg(t)=Imf(t). Inparticular,G(t)=ImU(t)
fort∈R,but,generallyspeaking,thisrelationfailsfort∈C\R.
8 ALEXANDERIKSANOV,ZAKHARKABLUCHKO,ANDALEXANDERMARYNYCH
The d-dimensional complex random vector Y := (Y (t ),...,Y (t )) can be rep-
n n 1 n d
resentedasasumofindependent,zeromeanrandomvectors(V (t ),...,V (t ))
n,k 1 n,k d
overk =1,...,n. ToshowthatY convergesindistributiontoY:=(Z(t ),...,Z(t )),
n 1 d
we shall use the Lindeberg central limit theorem. First we need to check that for
all i,j =1,...,d,
lim E[Y (t )Y (t )]=E[Z(t )Z(t )]=sinc(t t ),
n i n j i j i j
n→∞ −
lim E[Y (t )Y (t )]=E[Z(t )Z(t )]=sinc(t t¯).
n i n j i j i j
n→∞ −
It follows from (12) that
1 n k(t t ) 1
E[Y (t )Y (t )]= cos i− j cos(u(t t ))du=sinc(t t )
n i n j i j i j
n n n−→→∞ − −
k=1 Z0
X
and, similarly,
1 n k(t t¯) 1
E[Y (t )Y (t )]= cos i− j cos(u(t t¯))du=sinc(t t¯).
n i n j i j i j
nk=1 n n−→→∞Z0 − −
X
It remains to verify the Lindeberg condition. Take some t C. For every ε>0 we
∈
need to show that
n
lim E V (t)2 =0.
n→∞ | n,k | 1{|Vn,k(t)|≥ε}
k=1
X (cid:2) (cid:3)
Usingthe inequalities z +z 2 2z 2+2z 2 and sinz cosh(Imz), cosz
1 2 1 2
| | ≤ | | | | | |≤ | |≤
cosh(Imz), we obtain that for all k =1,...,n,
2
V (t)2 (cosh2(Imt)) (ξ2+η2).
| n,k | ≤ n · k k
With C =2cosh2(Imt) we get
n n
C
E V (t)2 E (ξ2+η2)
| n,k | 1{|Vn,k(t)|≥ε} ≤ n k k 1{C(ξk2+ηk2)≥nε2}
kX=1 (cid:2) (cid:3) Xk=1 h i
=CE (ξ2+η2)
1 1 1{ξ12+η12≥nε2/C}
h i
which converges to 0 as n because E[ξ2+η2]< .
→∞ 1 1 ∞
Tightness. In order to prove that the sequence (Yn)n∈N is tight on , it suffices to
H
show that for every R>0,
(13) sup sup EY (t)2 < ,
n
n∈N|t|≤R | | ∞
see [12, Lemma 4.2] or the remark after Lemma 2.6 in [19]. For all t R and
| | ≤
n N we have
∈
1 n k(t t¯) 1 n 2k(Imt)
EY (t)2 = cos − = cosh cosh(2R)<
n
| | n n n n ≤ ∞
k=1 k=1
X X
because R k Imt R for all k =1,...,n.
− ≤ n ≤
It follows that Yn convergesto Z weakly on , as n . Since R is a closed
H →∞ H
subset of and all processes under consideration have their sample paths in R,
the converHgence holds weakly on R, as well. H(cid:3)
H
REAL ROOTS OF RANDOM TRIGONOMETRIC POLYNOMIALS 9
The next theorem provides convergence of random trigonometric polynomials
under the assumptions of Theorem 2.3.
Theorem 3.2. Let s R be fixed and define a random process (Yn(t))t∈C by
∈
1 t
Y (t):= X s+ .
n n
√n n
(cid:18) (cid:19)
w
Under assumptions of Theorem 2.3 we have Yn G on R.
n−→→∞ H
Proof. We use the same idea as in the proof Theorem 3.1. We have Y (t) =
n
n V (t) where
k=1 n,k
P 1 t t
V (t):= ξ sin k s+ +η cos k s+ .
n,k k k
√n n n
(cid:18) (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19)(cid:19)
We shall need the standard trigonometric identities
n 1 sin (n+ 1)θ n 1 θ cos (n+ 1)θ
(14) cos(kθ)= + 2 , sin(kθ)= cot 2 .
−2 2sinθ 2 2− 2sinθ
k=1 (cid:0) 2 (cid:1) k=1 (cid:0) 2 (cid:1)
X X
wherethecaseθ 2πZisunderstoodbycontinuity. AsintheproofofTheorem3.1
∈
the subsequent argument is divided into two steps.
Convergenceoffinite-dimensionaldistributions. Firstweprovethatthecovariances
of Y converge to those of G. For all t ,t C, we have
n 1 2
∈
σ2 n t t
(15) E[Y (t )Y (t )]= 1 sin k s+ 1 sin k s+ 2
n 1 n 2
n n n
k=1 (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
X
σ2 n t t
+ 2 cos k s+ 1 cos k s+ 2
n n n
k=1 (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
X
n
ρ t +t
1 2
+ sin k 2s+ .
n n
k=1 (cid:18) (cid:18) (cid:19)(cid:19)
X
Denote the three terms on the right-hand side by S (n),S (n),S (n). Using the
1 2 3
formula 2sinxsiny =cos(x y) cos(x+y) and then the first identity in (14) we
− −
obtain
σ2 n t t σ2 n t +t
S (n)= 1 cos k 1− 2 1 cos k 2s+ 1 2
1
2n n − 2n n
k=1 (cid:18) (cid:19) k=1 (cid:18) (cid:18) (cid:19)(cid:19)
X X
σ2 σ2 sin (2n+1) s+ t1+t2
= 1 sinc(t t ) 1 2n +o(1), as n .
2 1− 2 − 2n 2sin s+ t1+t2 →∞
(cid:0) (cid:0) 2n (cid:1)(cid:1)
Sending n and considering the cases(cid:0)sins = 0(cid:1)and sins = 0 separately, we
→ ∞ 6
infer
σ12 sinc(t t ), if s / πZ,
lim S (n)= 2 1− 2 ∈
n→∞ 1 (σ212 sinc(t1−t2)− σ212 sinc(t1+t2), if s∈πZ.
Similarly, using the formula 2cosxcosy = cos(x y)+cos(x+y) for the second
−
sum we arrive at
σ22 sinc(t t ), if s / πZ,
lim S (n)= 2 1− 2 ∈
n→∞ 2 (σ222 sinc(t1−t2)+ σ222 sinc(t1+t2), if s∈πZ.
10 ALEXANDERIKSANOV,ZAKHARKABLUCHKO,ANDALEXANDERMARYNYCH
Finally, in view of the second formula in (14),
ρ 1 t +t cos (2n+1) s+ t1+t2
S (n)= cot s+ 1 2 2n .
3 n 2 (cid:18) 2n (cid:19)− (cid:0)2sin s+(cid:0)t12+nt2 (cid:1)(cid:1)!
Sending n gives (cid:0) (cid:1)
→∞
0, if s / πZ,
lim S (n)= ∈
n→∞ 3 (ρ1−cos(t1+t2), if s πZ.
t1+t2 ∈
Taking everything together and recalling the definition of the process G, see (3),
we obtain
(16) lim E[Y (t )Y (t )]=E[G(t )G(t )].
n 1 n 2 1 2
n→∞
Similar computation (with t replaced by t¯), yields
2 2
(17) lim E[Y (t )Y (t )]=E[G(t )G(t )].
n 1 n 2 1 2
n→∞
Fix t ,...,t C. In view of the convergence of the covariances established
1 d
∈
in (16) and (17), to prove that (Y (t ),...,Y (t )) converges in distribution to
n 1 n d
(G(t ),...,G(t )) it is enough to verify the Lindeberg condition:
1 d
n
lim E V (t)2 =0,
n→∞ | n,k | 1{|Vn,k(t)|≥ε}
k=1
X (cid:2) (cid:3)
for every fixed t C and ε > 0. This can be done exactly in the same way as in
∈
the proof of Theorem 3.1.
Tightness. It is sufficient to check condition (13). Starting with the equality
EY (t)2 =E[Y (t)Y (t¯)] and applying (15) with t =t=t¯, we arrive at
n n n 1 2
| |
σ2 n t 2 σ2 n t 2
EY (t)2 = 1 sin k s+ + 2 cos k s+
n
| | n n n n
k=1(cid:12) (cid:18) (cid:18) (cid:19)(cid:19)(cid:12) k=1(cid:12) (cid:18) (cid:18) (cid:19)(cid:19)(cid:12)
X(cid:12) (cid:12) X(cid:12) (cid:12)
ρ n (cid:12) 2Ret(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
+ sin k 2s+ .
n n
k=1 (cid:18) (cid:18) (cid:19)(cid:19)
X
Together with the inequalities sinz cosh(Imz) and cosz cosh(Imz), this
| | ≤ | | ≤
implies condition (13). Combining pieces together, we see that Y G weakly on
n
and hence, also on R. → (cid:3)
H H
In the case of attraction to a stable law we have the following functional limit
theorem. Since its proof is more involved than in the previous cases, it is given in
the separate Section 3.3.
Theorem 3.3. Fix s R. Under the assumptions of Theorem 2.5,
∈
1 t
w
X s+ (Z (t))
(cid:18)bn n(cid:18) n(cid:19)(cid:19)t∈C n−→→∞ ν t∈C
on R.
H