Elect. Comm. in Probab. 16 (2011), 507–516
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
WHITE AND COLORED GAUSSIAN NOISES AS LIMITS OF SUMS
OF RANDOM DILATIONS AND TRANSLATIONS OF A SINGLE FUNCTION
GUSTAF GRIPENBERG
Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 Aalto,
Finland
email: gustaf.gripenberg@aalto.fi
Submitted August 17, 2010, accepted in final form August 11, 2011
AMS 2000 Subject classification: 60G15; 60J65
Keywords: white noise; colored noise; Brownian motion; fractional Brownian motion; convergence; dilation; translation
Abstract
It is shown that a stochastic process obtained by taking random sums of dilations and translations
of a given function converges to Gaussian white noise if a dilation parameter grows to infinity and
that it converges to Gaussian colored noise if a scaling parameter for the translations grows to
infinity. In particular, the question of when one obtains fractional Brownian motion by integrating
this colored noise is studied.
1
Introduction
The purpose of this note is to show that if h is a given scalar function and Sn , An , Bn , and Tn are
real-valued random variables, then under fairly weak conditions the stochastic process
Xp
DX γ,t =
γSn An h γBn t − (n + Tn ) ,
(1)
n∈Z
converges to Gaussian white noise (the distribution derivative of Brownian motion) as γ → ∞,
and the process
X 1
1
DYλ,t =
(2)
p Sn An h Bn t − (n + Tn ) ,
λ
λ
n∈Z
converges to a certain kind of colored noise with spectral density depending on the function h and
the random variables An and Bn as λ → ∞.
It is not surprising that the kind of time scaling appearing in (1) gives rise to white noise and it is
not claimed here that the kind of scaling of the translations used in (2) is the only possible way
to obtain colored noise, just that it is one possibility. Thus the motivation for studying (2) is that
it is a fairly simple model which may perhaps be of help in understanding the sources of different
kinds of colored noise.
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The convergence concept considered is that the random variable R f (t)Z t dt converges in distribution to a Gaussian random variable for all test functions f , (all square integrable functions for
the case Z t = DX γ,t and a subspace of L 2 (R) in the case Z t = DYλ,t ). An important assumption is
that E(Sn ) = 0 and that the random variables Sn are independent of each other and of all other
random variables appearing in the sums.
If one takes h(t) = 1[0,1) (t), An = Bn = 1, Tn = 0, and Sn = ±1 (each with probability 12 ), then
Rt
the process t 7→ 0 DX γ,s ds is the linear interpolation of a random walk jumping up or down with
1
p
γ
step length
in each time interval of length
1
γ
so in this case it is immediately clear that the limit
when γ → ∞ is Brownian motion. It follows from Theorem 1 below that this is the case under
quite general assumptions. The convergence here means that the distributions of all finite linear
combinations of samples of the process converge to the corresponding distributions for Brownian
motion.
In many cases, however, it has turned out that Brownian motion is not a satisfactory process to
use as a model and that one should rather use, e.g., fractional Brownian motion, see [5] and
in particular [12] in the case of ethernet traffic. Fractional Brownian motion B tH with Hurst
parameter H ∈ (0, 1) is a centered Gaussian process with covariance function E(BsH B tH ) = 12 (|s|2H +
|t|2H − |t − s|2H ). Another way to express this is that
‚Z
E
f
(t)d B tH
Œ
Z
= (2π)1−2H Γ(2H + 1) sin(πH)
g(t)dB tH
|ξ|1−2H fˆ(ξ)ĝ(ξ) dξ,
R
R
R
Z
Rt
R
where B tH = 0 d BsH and fˆ(ξ) = R e−i2πξt f (t) dt, see e.g. [8]. In this note it is shown that for
suitable (real) functions f and g one has
‚Z
lim E
λ→∞
f (t)DYλ,t dt
R
Œ
Z
g(t)DYλ,t dt
R
=
‚
Z
E
R
2 Œ
A2n ξ ĥ
fˆ(ξ)ĝ(ξ) dξ.
|Bn |
Bn Thus we see that it is possible to obtain fractional Brownian motion as the limit of the process
Rt
DYλ,s ds but the point is that in order to get this process exactly rather special assumptions on
0
the function h and/or on the distribution of (An , Bn ) are needed whereas the assumptions that
Rt
suffice for the convergence of 0 DX γ,s ds to Brownian motion as γ → ∞ are much more general.
But this note does not study the question to what extent fractional Brownian motion may be a
Rt
reasonable approximation to the limit of 0 DYλ,s , ds.
Fractional Brownian motion can be approximated in many ways, see e.g. [1], [2], [3], [6], [9],
[10], and [11], depending partly on whether one wants to get an efficient simulation tool or
whether one wants, as in this note, to see to what extent it is a consequence of some ”universal
principles” and therefore expected to appear in many connections. One can also get a fractional
Brownian motion with Hurst parameter H = 14 as a limit from one-dimensional nearest-neighbor
symmetric simple exclusion processes, see e.g. [7].
2
Statement of results
Theorem 1. Assume that
(i) h ∈ L 1 (R; R), ĥ(0) = 1, and supξ∈R |ĥ(ξ)||ξ|α/2 < ∞ for some number α > 1.
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509
(ii) The real valued random variables Sn and the R3 -valued random variables (An , Bn , Tn ), where
n ∈ Z, are independent and the distributions of Sn and (An , Bn , Tn ) do not depend on n. In
addition P(Bn = 0) = 0, E(Sn ) = 0, and E(Sn2 ) = 1 for all n ∈ Z.
‚
(iii) E
A2n
Œ
Bn
= 1, n ∈ Z.
If f ∈ L 2 (R; R), then
R
R
f (t)DX γ,t dt converges in distribution to an N (0, k f k2L 2 (R) )-distributed
random variable as γ → ∞ and if g ∈ L 2 (R; R) as well, then
‚Z
lim E
γ→∞
f (t)DX γ,t dt
Œ
Z
R
g(t)DX γ,t dt
=
R
Z
fˆ(ξ)ĝ(ξ) dξ.
R
Note that assumption (i) implies that h ∈ L 2 (R; R).
Theorem 2. Assume that
R
(i) The function h : R → R is such that R |h(t)|(1 + |t|m )−1 dt < ∞ for some number m ≥ 0
and the (distribution) Fourier transform of h is induced by a measurable function ĥ such that
R
|ĥ(ξ)|(1 + |ξ|m̂ )−1 dξ < ∞ for some number m̂ ≥ 0.
R
(ii) The real valued random variables Sn and the R3 -valued random variables (An , Bn , Tn ), where
n ∈ Z, are independent and the distributions of Sn and (An , Bn , Tn ) do not depend on n. In
addition P(Bn = 0) = 0, E(Sn ) = 0, and E(Sn2 ) = 1 for all n ∈ Z.
(iii) There is a number α > 1 and a finite set K ⊂ R such that the function
‚
ξ 7→ E
A2n
α Œ 2 Œ
ξ ĥ
,
B ξ
1 + B
‚
Bn
n
n
is bounded on compact subsets of R \ K.
(iv) f and g ∈ L 2 (R; R) are such that
‚
Z
E
R
A2n
ξ
1 + B
‚
|Bn |
n
α Œ 2 Œ
ξ ĥ
| fˆ(ξ)|2 + |ĝ(ξ)|2 dξ < ∞.
B
n
R
Then R f (t)DYλ,t dt converges in distribution to an N (0, σ2 )-distributed random variable as λ → ∞
where
‚ 2 2 Œ
Z
An ξ 2
ĥ
| fˆ(ξ)|2 dξ,
σ =
E
|Bn |
Bn R
and
‚Z
lim E
λ→∞
R
f (t)DYλ,t dt
Œ
Z
g(t)DYλ,t dt
R
=
‚
Z
E
R
2 Œ
A2n ξ ĥ
fˆ(ξ)ĝ(ξ) dξ.
|B |
B n
n
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(In this case the distribution Fourier transform of h is defined by the requirement that R ĥ(ξ)ϕ(ξ) dξ =
R
R
h(t)ϕ̂(t) dt where ϕ̂(t) = R e−i2πtξ ϕ(ξ) dξ.)
R
If the probability density of the random
variable Bn is b with b(x) > 0 for all x > 0 and b(x) = 0
p
−H+1/2
for all x ≤ 0, and An = Bn
/ b(Bn ), (that is, there is a very specific relation between the
parts of the model) then
‚ 2 2 Œ
Z∞
An ξ 1−2H
= |ξ|
ĥ
s2H−2 |ĥ(s)|2 ds,
E
|Bn | Bn 0
and provided the integral is finite we see that the limit of DYλ,t is the distribution derivative of a
multiple of fractional Brownian motion with Hurst parameter H.
If, on the other hand h(t) = sign (t)|t|H−3/2 , then
‚
E
Œ2 2 1
‚
2 Œ
sin( π4 (1 − 2H))
π Γ( 2 + H)2
A2n ξ 2
2H−2
ĥ
= E(A |B |
|ξ|1−2H ,
)
π
n n
2H−1
(1
−
2H)
|Bn | Bn (2π)
4
and if E(A2n |Bn |2H−2 ) < ∞, then we again get the distribution derivative of a multiple of fractional
Brownian motion. But the function h is locally integrable in the case H ∈ ( 12 , 1) only. If H ∈ (0, 12 ]
R
we can, however, interpret the integral R p1λ Sn An h(Bn t − λ1 (n + Tn )) f (t) dt as a Cauchy principal
value, or in other words, consider h as a tempered distribution. Then the conclusion still holds
but in these cases hypothesis (iv) may be much more restrictive and exclude indicator functions
of intervals.
€
Š
By considering h Bn t − λ1 (n + Tn ) as dilation and translation of a tempered distribution one can
see that the proof of Theorem 2 extends to a proof of the following claim.
Corollary 3. The conclusion of Theorem 2 remains true provided hypothesis (i) is replaced by the
assumption
(i’) h is a real valued tempered distribution such
R that the distribution Fourier transform of h is
induced by a measurable function ĥ such that R |ĥ(ξ)|(1 + |ξ|m̂ )−1 dξ < ∞ for some number
m̂ ≥ 0,
and integrals involving h are interpreted as the value of a tempered distribution at a test function.
3
Proofs
First we derive some results that are common for the processes DX γ,t and DYλ,t and throughout we
assume that all random variables Sn , An , Bn , and Tn are defined on a probability space (Ω, F , P).
Assume that f and g ∈ L 2 (R) in the case of Theorem 1 and that hypothesis (iv) of Theorem 2
holds otherwise. Let
Z Ç
γ
1
U(n, γ, λ, f , ω) =
An (ω)
h γBn (ω)t − (n + Tn (ω)) f (t) dt.
λ
λ
R
(In many cases below we leave out the argument ω of the random variables.) In the case of Theorem 1 we have h ∈ L 2 (R) and it is immediately clear that this is a well defined random variable. In
the other case we first make the additional assumption that f and g ∈ S (R), the Schwartz space
of rapidly decreasing smooth functions, and then combine the argument below with a limiting
Sums of dilations and translations
511
procedure, and here hypothesis (iii) of Theorem 2 is used,R to show that we indeed
get a random
P
variable under hypothesis (iv) of Theorem 2. Note that R f (t)DX γ,t dt = n∈Z Sn U(n, γ, 1, f )
R
P
and R f (t)DYλ,t dt = n∈Z Sn U(n, 1, λ, f ) so it follows from hypothesis (ii) that
‚Z
f (t)DX γ,t dt
E
Œ
Z
=
g(t)DX γ,t dt
R
R
X
E U(n, γ, 1, f )U(n, γ, 1, g) ,
(3)
E U(n, 1, λ, f )U(n, 1, λ, g) .
(4)
n∈Z
and
‚Z
f (t)DYλ,t dt
E
Œ
Z
g(t)DYλ,t dt
R
R
=
X
n∈Z
If ĥ is the Fourier-transform of h then the Fourier-transform of the function t 7→ h(γBn (ω)t − λ1 (n+
Tn )ω))) (or the tempered distribution induced by this function) is the function
ξ 7→
1
γ|Bn (ω)|
e−i2πξn/γλBn (ω) e−i2πξTn (ω)/γλBn (ω) ĥ
ξ
γBn (ω)
(or the tempered distribution induced by this function) and we deduce by Plancherel’s theorem
(or the definition of the distribution Fourier transform) that
U(n, γ, λ, f , ω) = p
An (ω)
γλ|Bn (ω)|
In order to get an estimate for
U(n, γ, λ, f , ω) =
p
Z
P
γλAn (ω)
ei2πξn/γλBn (ω) ei2πξTn (ω)/γλBn (ω) ĥ
R
n∈Z E(U(n, γ, λ,
Z
ξ
γBn (ω)
fˆ(ξ) dξ.
(5)
f )2 ) we rewrite U(n, γ, λ, f , ω) as
ei2πξn ei2πTn (ω)ξ ĥ (λξ) fˆ γλBn (ω)ξ dξ
R
=
p
γλAn (ω)
Z
1
2
− 21
ei2πξn
X
ei2πTn (ω)(ξ+k) ĥ λ(ξ + k) fˆ γλBn (ω)(ξ + k) dξ.
(6)
k∈Z
Let A, B, and T be random variables on a probability space (Ω∗ , F∗ , P∗ ) with the same distribution
as An , Bn , and Tn . It is crucial for the proof that when taking expectations one can use random
variables that P
do not depend on n. In the argument below one should first take g = f and show
that the sum n∈Z E(U(n, γ, λ, f )2 is finite and then go through the argument again, using the
fact that one may now invoke Fubini’s theorem and other results needed, in the case where f and
g may be different as well. By (6) we see that when one calculates expectations of the random
variables U(n, γ, λ, f ) one gets Fourier-coefficients of a periodic function and hence it follows from
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Electronic Communications in Probability
Parseval’s theorem that
X
E(U(n, γ, λ, f )U(n, γ, λ, g)) =
n∈Z
X
Z
n∈Z
=
Z
X
2
γλA(ω)
1
2
Z
ei2πξn
×
− 12
=
Z
X
X
ei2πT (ω)(ξ+ j) ĥ λ(ξ + j) fˆ γλB(ω)(ξ + j) dξ
j∈Z
k∈Z
‚
γλA(ω)
2
1
2
Z
ei2πξn
X
− 12
1
2
ei2πξn
×
− 12
=
X
ei2πT (ω)(ξ+k) ĥ (λ(ξ + k))ĝ γλB(ω)(ξ + k) dξ P∗ (dω)
Ω∗ n∈Z
Z
ei2πξn
− 12
Ω∗
n∈Z
1
2
Z
U(n, γ, λ, f , ω)U(n, γ, λ, g, ω) P(dω)
Ω
X
ei2πT (ω)(ξ+ j) ĥ λ(ξ + j) fˆ γλB(ω)(ξ + j) dξ
j∈Z
Œ
ei2πT (ω)(ξ+k) ĥ (λ(ξ + k))ĝ γλB(ω)(ξ + k) dξ P∗ (dω)
k∈Z
Z ‚
2
γλA(ω)
X
ei2πT (ω)(ξ+ j) ĥ λ(ξ + j) fˆ γλB(ω)(ξ + j)
− 12 j∈Z
Ω∗
×
1
2
Z
X
−i2πT (ω)(ξ+k)
e
Œ
ĥ (λ(ξ + k)) ĝ γλB(ω)(ξ + k) dξ P∗ (dω)
k∈Z
‚
2
= E γλA
Z
1
2
X
ei2πT (ξ+ j) ĥ λ(ξ + j) fˆ γλB(ξ + j)
− 12 j∈Z
×
X
−i2πT (ξ+k)
e
Œ
ĥ (λ(ξ + k)) ĝ γλB(ξ + k) dξ .
(7)
k∈Z
Let Ẑ be either Z or Z \ {0}. By the Cauchy-Schwarz inequality we have
Z
1
2
− 21
2
X

ĥ λ(ξ + j) fˆ γλB(ω)(ξ + j)

 dξ

j∈Ẑ
Z
≤
1
2
X
− 12 j∈Z
1
1 + |λ(ξ + j)|α
X
2 2
1 + |λ(ξ + j)|α ĥ λ(ξ + j) fˆ γλB(ω)(ξ + j) dξ.
(8)
j∈Ẑ
Let j0 = 0 if Ẑ = Z and j0 = 1 if Ẑ = Z \ {0}. Changing variables and adding intervals of
Sums of dilations and translations
513
integration we get,
Z
1
2
2 2
1 + |λ(ξ + j)|α ĥ λ(ξ + j) fˆ γλB(ω)(ξ + j) dξ
X
− 12 j∈Ẑ
=
Z
2 2
(1 + |λξ|α ) ĥ (λξ) fˆ γλB(ω)ξ dξ
|ξ|≥ j0 /2
=
γλ|B(ω)|
Π2
ξ α ξ
ĥ
fˆ (ξ)2 dξ.
1 + γB(ω)
γB(ω)
‚
Z
1
|ξ|≥γλ|B(ω)| j0 /2
Thus we conclude from (7), (8), and (9) (with Ẑ = Z) that
‚ 2 ‚
Z
α Œ 2 Œ
X
ξ A
ξ 2
| fˆ(ξ)|2 dξ,
E(U(n, γ, λ, f ) ) ≤ cα
E
1 + ĥ
|B|
γB
γB
R
n∈Z
(provided λ ≥ 1) where cα = supξ∈R
P
(9)
(10)
1
j∈Z 1+|(ξ+ j)|α .
By (7) we have (with Ẑ = Z \ {0})
X
E(U(n, γ, λ, f )U(n, γ, λ, g))
n∈Z
‚
2
= E γλA
1
2
Z
i2πT (ξ+0)
e
ĥ (λ(ξ + 0)) fˆ γλB(ξ + j) e−i2πT (ξ+0) ĥ (λ(ξ + k)) ĝ γλB(ξ + 0) dξ
Œ
− 21
‚
Z
2
+ E γλA
1
2
ei2πT (ξ+0) ĥ (λ(ξ + 0)) fˆ γλB(ξ + 0)
− 21
×
X
−i2πT (ξ+k)
e
Œ
ĥ (λ(ξ + k)) ĝ γλB(ξ + k) dξ
k∈Ẑ
Z
+ E γλA2
1
2
X
ei2πT (ξ+ j) ĥ λ(ξ + j) fˆ γλB(ξ + j)
− 12 j∈Ẑ
×
X
−i2πT (ξ+k)
e
ĥ (λ(ξ + k)) ĝ γλB(ξ + k) dξ .
(11)
k∈Z
By the same change of variable as used above we see that
‚
2
E γλA
Z
1
2
ei2πT (ξ+0) ĥ (λ(ξ + 0)) fˆ γλB(ξ + j)
− 12
−i2πT (ξ+0)
×e
Œ
ĥ (λ(ξ + k)) ĝ γλB(ξ + 0) dξ

2
A
= E
|B|
Z
γλ|B|
2
−
γλ|B|
2

2
ξ
ĥ
fˆ(ξ)ĝ(ξ) dξ

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Electronic Communications in Probability
and for the remaining terms in (7) we use the Cauchy-Schwarz inequality together with (8) and
(9) in order to conclude that

Z γλ|B|
X
2
2
A
E(U(n, γ, 1, f )U(n, γ, 1, g)) − E 
n∈Z
|B| − γλ|B|
2
 
≤ cα E 
A2
‚ ‚
×
E
|ξ|≥
A2
+ cα  E 
A2
γλ|B|
2
ŒŒ 12
α Œ 2
2
ξ ξ ĝ (ξ) dξ
1 + ĥ
γB
γB Z ‚
|B|
 
 1
α Œ 2
2
ξ 2
ξ fˆ (ξ) dξ
1 + ĥ
γB
γB ‚
Z
|B|

2
ξ
fˆ(ξ)ĝ(ξ) dξ
ĥ

γB R
|B|
|ξ|≥
 1
α Œ 2
2
ξ 2
ξ ĝ (ξ) dξ
1 + ĥ
γB
γB ‚
Z
γλ|B|
2
A2
‚ ‚
×
E
ŒŒ 12
α Œ 2
ξ 2
ξ fˆ (ξ) dξ
1 + ĥ
.
γB
γB Z ‚
|B|
R
It follows from the dominated convergence theorem and the assumptions (for the cases γ → ∞
and λ → ∞, respectively) that the right-hand side of this inequality tends to 0 when γ → ∞ or
λ → ∞. Thus we conclude under the assumptions of Theorem 1 that
Z
X
lim
E(U(n, γ, 1, f )U(n, γ, 1, g)) =
fˆ(ξ)ĝ(ξ) dξ,
(12)
γ→∞
R
n∈Z
and under the assumptions of Theorem 2 that
lim
λ→∞
X
E(U(n, 1, λ, f )U(n, 1, λ, g)) =
E
R
n∈Z
‚
Z
2 Œ
A2 ξ ĥ
f (ξ)ĝ(ξ) dξ,
|B|
B (13)
By the Cauchy-Schwarz inequality and (5) we have in the case of Theorem 1 when λ = 1,
U(n, γ, 1, f , ω)
p
|ξ|≤ γ
!1
2
2
ξ
ĥ
dξ
γB (ω) p
|ξ|> γ
!1
2
2
ξ
ĥ
dξ
γB (ω) |An (ω)|
≤p
γ|Bn (ω)|
Z
|An (ω)|
+p
γ|Bn (ω)|
Z
Z
p
|ξ|≤ γ
n
Z
p
|ξ|> γ
n
| fˆ(ξ)|2 dξ
| fˆ(ξ)| dξ
!1
2
!1
2
2
1
2


Z
|An (ω)| 

≤p
|ĥ(ξ)|2  k fˆk L 2 (R)

|Bn (ω)|
|ξ|≤ pγB1 (ω)
n
|An (ω)|
+p
kĥk L 2 (R)
|Bn (ω)|
Z
p
|ξ|> γ
| fˆ(ξ)| dξ
2
!1
2
.
Sums of dilations and translations
515
It follows from this inequality that if δ > 0, |Bn (ω)| ≥ δ, and |An (ω)| ≤ δ1 then for each ε > 0
there is a number γδ,ε so that if γ > γδ,ε then |U(n, γ, 1, f , ω)| < ε. Thus it follows from (7) and
(10) when we replace An by An (1{A> 1 } + 1{|B|<δ} ) that for each δ > 0 we have
δ
‚ 2
ŒZ
X
A
2
2
lim sup
E(U(n, γ, 1, f ) 1|U(n,γ,1, f )|>ε ) ≤ c1 E
1 1 + 1{|B|<δ}
| fˆ(ξ)|2 dξ,
|B| {A> δ }
γ→∞ n∈Z
R
2
A
where c1 = cα supξ∈R (1 + |ξ|α )|ĥ(ξ)|2 . Since E( |B|
) < ∞ it follows that
X
lim
E(U(n, γ, 1, f )2 1{|U(n,γ,1, f )|>ε} ) = 0,
γ→∞
(14)
n∈Z
for each ε > 0. This isRthe well known Lindeberg condition and thus by (12) and the assumption
E(Sn ) = 0 the limit of R f (t)DX γ,t dt is normally distributed with mean 0 and variance k f k2L 2 (R) ,
see [4, Thm. 5.12]. The statement about the covariance follows from (3) and (12).
If γ = 1 and we consider the case of Theorem 2, then we can use the Cauchy-Schwarz inequality
and (5) to get
!1
Œ
2
ξ α −1
1
1+
dξ
Bn (ω) |Bn (ω)|
R
‚Z
‚
Π12
Π2
ξ α An (ω)2
ξ
ĥ
| fˆ(ξ)|2 dξ
×
1 + |Bn (ω)|
Bn (ω) Bn (ω) R
Z ‚
1
U(n, 1, λ, f , ω) ≤ p
λ
(15)
Let
Π2
ξ α ξ
| fˆ(ξ)|2 dξ,
ĥ
qn (ω) =
1+
|Bn (ω)|
Bn (ω)
Bn (ω) R
‚
Z
Π2
ξ α A(ω)2
ξ
| fˆ(ξ)|2 dξ,
q(ω) =
ĥ
1+
|B(ω)|
B(ω)
B(ω) R
Z
An (ω)2
‚
and we note that qn and q have
R the same distributions. From (15) we see (note that the first
term on the right hand side is R (1 + |ξ|α )−1 dξ < ∞) that for each ε > 0 and m ≥ 1 there is
a number λm,ε so that |U(n, 1, λ, f , ω)| ≤ 1{qn (ω)>m} |U(n, 1, λ, f , ω)| if |U(n, 1, λ, f , ω)| ≥ ε and
λ ≥ λm,ε . Thus we conclude from (10) with the aid of the same argument as above (replace An by
An 1{qn >m} ) that
X
E(U(n, 1, λ, f )2 1{|U(n,1,λ, f )|>ε} ) ≤ cα E(1{q>m} q)
lim sup
λ→∞
n∈Z
and since m was arbitrary we get
X
lim
E(U(n, 1, λ, f )2 1{|U(n,1,λ, f )|>ε} ) = 0.
λ→∞
(16)
n∈Z
for
R each ε > 0. Since this is again the Lindeberg condition we can deduce from (13) that
f (t)DYλ,t dt converges in distribution to a normally distributed random variable with mean
R
R
€ ξ Š2
A2 0 and variance R E |B|
ĥ B | fˆ(ξ)|2 dξ. The statement about the covariance follows from
(4) and (13).
Acknowledgment
I would like to thank the referee for a number of suggestions to improve the presentation.
516
Electronic Communications in Probability
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