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Electron. J. Probab. 18 (2013), no. 69, 1–21.
ISSN: 1083-6489 DOI: 10.1214/EJP.v18-2234
On convergence of general wavelet decompositions
of nonstationary stochastic processes
Yuriy Kozachenko∗
Andriy Olenko†
Olga Polosmak‡
Abstract
The paper investigates uniform convergence of wavelet expansions of Gaussian random processes. The convergence is obtained under simple general conditions on
processes and wavelets which can be easily verified. Applications of the developed
technique are shown for several classes of stochastic processes. In particular, the
main theorem is adjusted to the fractional Brownian motion case. New results on
the rate of convergence of the wavelet expansions in the space C([0, T ]) are also presented.
Keywords: Convergence in probability ; uniform convergence ; convergence rate ; Gaussian
process ; fractional Brownian motion ; wavelets.
AMS MSC 2010: 60G10 ; 60G15 ; 42C40.
Submitted to EJP on August 11, 2012, final version accepted on July 13, 2013.
1
Introduction
In the book [11] wavelet expansions of non-random functions bounded on R were
studied in different spaces. However, developed deterministic methods may not be
appropriate to investigate wavelet expansions of stochastic processes. For example, in
the majority of cases, which are interesting from theoretical and practical application
points of view, stochastic processes have almost surely unbounded sample paths on R.
It indicates the necessity of elaborating special stochastic techniques.
Recently, a considerable attention was given to the properties of the wavelet orthonormal series representation of random processes. More information on convergence of wavelet expansions of random processes in various spaces, references and
numerous applications can be found in [3, 7, 14, 15, 16, 17, 18, 21, 20, 24]. Most
known stochastic results concern the mean-square or almost sure convergence, but for
various practical applications one needs to require uniform convergence. To give an
∗ Taras Shevchenko Kyiv National University, Ukraine.
E-mail: ykoz@ukr.net
† La Trobe University, Australia.
E-mail: a.olenko@latrobe.edu.au, https://sites.google.com/site/olenkoandriy/
‡ Taras Shevchenko Kyiv National University, Ukraine.
E-mail: DidenkoOlga@yandex.ru
Uniform convergence of wavelet decompositions
0.0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.0
−0.5
−1.0
reconstruction of W(t)
0.0
1.0
−1.5
−0.5
−1.5
−1.0
reconstruction of W(t)
−0.5
−1.5
−1.0
W(t)
0.0
0.0
adequate description of the performance of wavelet approximations in both cases, for
points where the processes are relatively smooth and points where spikes occur, we
can use the uniform distance instead of global integral Lp metrics. A more in depth
discussion, further references and various applications in econometrics, simulations of
stochastic processes and functional data analysis can be found in [6, 9, 10, 20, 23].
In his 2010 Szekeres Medal inauguration speech, an eminent leader in the field, Prof.
P. Hall stated the development of uniform stochastic approximation methods as one of
frontiers in modern functional data analysis.
Figures 1 and 2 illustrate some features of wavelet expansions of stochastic processes. Figure 1 presents a simulated realization of the Wiener process and its wavelet
reconstructions by two sums with different numbers of terms. The figure has been
generated by the R package wmtsa [22]. Besides providing a realization of the Wiener
process and its wavelet reconstructions, we also plot corresponding reconstruction errors. Figure 2 shows maximum absolute reconstruction errors for 100 simulated realizations. To reconstruct each realization of the Wiener process two approximation
sums (as in Figure 1) were used. We clearly see that empirical probabilities of obtaining large reconstruction errors become smaller if the number of terms in the wavelet
expansions increases. Although this effect is expected, it has to be established theoretically in a stringent way for different classes of stochastic processes and wavelet bases.
It is also important to obtain theoretical estimations of the rate of convergence for such
stochastic wavelet expansions.
1.0
0.0
0.2
0.4
t
t
0.6
0.8
1.0
t
0
20
40
60
80
100
3e−12
1e−12
max abs reconstruction errors
0.09
0.07
max abs reconstruction errors
Figure 1: Plots of the Wiener process and its wavelet reconstructions.
0
20
40
60
80
100
Figure 2: Plots of reconstruction errors for 100 simulated realizations.
In this paper we make an attempt to derive general results on stochastic uniform
convergence which are valid for wavelet expansions of wide classes of stochastic processes. The paper deals with the most general class of such wavelet expansions in
EJP 18 (2013), paper 69.
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Uniform convergence of wavelet decompositions
comparison with particular cases considered by different authors, see, for example,
[1, 3, 8, 13, 20]. Applications of the main theorem to special cases of practical importance (stationary processes, fractional Brownian motion, etc.) are demonstrated. We
also prove the exponential rate of convergence of the wavelet expansions.
Throughout the paper, we impose minimal assumptions on the wavelet bases. The
results are obtained under simple conditions which can be easily verified. The conditions are weaker than those in the former literature.
These are novel results on stochastic uniform convergence of general finite wavelet
expansions of nonstationary random processes. The specifications of established results
are also new (for example, for the case of stationary stochastic processes, compare
[15, 16]).
Finally, it should be mentioned that the analysis of the rate of convergence gives a
constructive algorithm for determining the number of terms in the wavelet expansions
to ensure the uniform approximation of stochastic processes with given accuracy. It
provides a practical way to obtain explicit bounds on the sharpness of finite wavelet
series approximations.
The organization of the article is the following. In the second section we introduce
the necessary background from wavelet theory and certain sufficient conditions for
mean-square convergence of wavelet expansions in the space L2 (Ω). In §3 we formulate
and discuss the main theorem on uniform convergence in probability of the wavelet
expansions of Gaussian random processes. The next section contains the proof of the
main theorem. Two applications of the developed technique are shown in section 4.
In §5 the main theorem is adjusted to the fractional Brownian motion case. Lastly, we
obtain the rate of convergence of the wavelet expansions in the space C([0, T ]).
In what follows we use the symbol C to denote constants which are not important
for our discussion. Moreover, the same symbol C may be used for different constants
appearing in the same proof.
2
Wavelet representation of random processes
b
b
Let φ(x), x ∈ R, be a function from the space L2 (R) such that φ(0)
6= 0 and φ(y)
is
continuous at 0, where
Z
e−iyx φ(x) dx
b
φ(y)
=
R
is the Fourier transform of φ.
Suppose that the following assumption holds true:
X
b + 2πk)|2 = 1 (a.e.)
|φ(y
k∈Z
There exists a function m0 (x) ∈ L2 ([0, 2π]), such that m0 (x) has the period 2π and
b
φ(y)
= m0 (y/2) φb (y/2) (a.e.)
In this case the function φ(x) is called the f -wavelet.
Let ψ(x) be the inverse Fourier transform of the function
y
n yo y
b
ψ(y)
= m0
+ π · exp −i
· φb
.
2
2
2
Then the function
1
ψ(x) =
2π
Z
b dy
eiyx ψ(y)
R
is called the m-wavelet.
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Uniform convergence of wavelet decompositions
Let
φjk (x) = 2j/2 φ(2j x − k),
ψjk (x) = 2j/2 ψ(2j x − k),
j, k ∈ Z ,
(2.1)
where φ(x) and ψ(x) are defined as above.
b
b
The conditions φ(0)
6= 0 and φ(y)
is continuous at 0 guarantee the completeness of
system (2.1), see [12]. Then the family of functions {φ0k ; ψjk , j ∈ N0 , k ∈ Z}, N0 :=
{0, 1, 2, ...}, is an orthonormal basis in L2 (R) (see, for example, [11]. A necessary and
sufficient condition on m0 is given in [4, 5]).
Definition 2.1. Let δ(x), x ∈ R, be a function from the space L2 (R). If the system of
functions {δ(x − k), k ∈ Z} is an orthonormal system, then the function δ(x) is called a
scaling function.
Remark 2.2. f -wavelets and m-wavelets are scaling functions.
Definition 2.3. Let δ(x), x ∈ R, be a scaling function. If there exists a bounded function
Φ(x),
x ≥ 0, such that Φ(x) is a decreasing function, |δ(x)| ≤ Φ(|x|) a.e. on R, and
R
γ
(Φ(|x|))
dx < ∞ for some γ > 0, then δ(x) satisfies assumption S 0 (γ).
R
Remark 2.4. If assumption S 0 (γ) is satisfied for some γ > 0 then the assumption is also
true for all γ1 > γ.
Lemma 2.5. If for some function Φφ (·) the f -wavelet φ satisfies assumption S 0 (γ), then
there exist a function Φψ (·) such that for the corresponding m-wavelet ψ assumption
S 0 (γ) holds true and Φψ (x) ≥ Φφ (x), for all x ∈ R.
Proof. The wavelets φ and ψ admit the following representations, see [11],
φ(x) =
√ X
2
hk φ (2x − k) ,
k∈Z
√ X
ψ(x) = 2
λk φ (2x − k) ,
(2.2)
k∈Z
where hk =
√ R
P
2
2 R φ (u) φ (2u − k)du,
|hk | < ∞, and λk = (−1)1−k h1−k .
k∈Z
If k ≥ 0, then
|hk | ≤
k
3
√ Z
2
Φφ (|u|) Φφ (|2u − k|) du +
≤
Φφ (u) Φφ (|2u − k|) du
k
3
−∞
√
∞
√ Z
2
Z
k
3
2Φφ (k/3)
Φφ (|u|) du +
√
Z
√
Z
2Φ (k/3)
Φφ (|2u − k|) du
k
3
−∞
≤
∞
2Φφ (k/3)
Φφ (|2u − k|) du = CΦφ (k/3) ,
Z
Φφ (|u|) du +
R
R
where
Z
3
C := √
Φφ (|u|) du.
2 R
Similarly, for k ≤ 0 we get |hk | ≤ CΦφ (|k|/3) .
Thus, for all k ∈ Z,
|hk | ≤ C Φφ (|k|/3) .
(2.3)
Note that the series in the right-hand side of (2.2) converges in the L2 (R)-norm.
Therefore, there exists a subsequence of partial sums which converges to ψ (x) a.e. on
R. Thus, by (2.2) and (2.3) we obtain
|ψ (x)| ≤
√
2C
X
Φφ (|2x − k|) · Φφ
k∈Z
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Uniform convergence of wavelet decompositions
=
√
2C
X
Φφ (|2x − 1 − k|) · Φφ (|k|/3) =: I (2x − 1)
(2.4)
k∈Z
a.e. on R.
If u := 2x − 1, then for u > 0
I(u) ≤
√

X
2C 
Φφ (|u − k|) Φφ (|k|/3)
|k|≤ 34 u

X
+
Φφ (|u − k|) · Φφ (|k|/3) =:
√
2C (A1 (u) + A2 (u)) .
|k|≥ 43 u
Notice also that
X
A1 (u) ≤ Φφ (u/4) ·
X
Φφ (|k|/3) ≤ Φφ (u/4) ·
|k|≤ 34 u
A2 (u) ≤ Φφ (u/4) ·
X
Φφ (|k|/3) ,
k∈Z
Φφ (|u − k|) ≤ Φ (u/4) ·
|k|≥ 34 u
X
Φφ (|u − k|) .
k∈Z
Therefore, for u > 0
I(u) ≤
√
!
X
2CΦφ (u/4)
Φφ (|u − k|) +
k∈Z
We are to prove that,
P
Φφ (|k|/3) .
(2.5)
k∈Z
Φφ (|u − k|) +
k∈Z
X
P
Φφ (|k|/3) is bounded.
k∈Z
Note that
X
Φφ (k − u) ≤
k≥u+1
X Z
k
0
and, similarly,
∞
Z
X
Φφ (u − k) ≤
Φφ (v) dv.
0
k≤u−1
Thus,
Z
Φφ (|u − k|) ≤ 2 Φφ (0) +
X
∞
X
Φφ (k/3) ≤
k=1
it follows that
∞
Φφ (v) dv
=: D1 .
0
k∈Z
Since
Φφ (v) dv < ∞,
k−1
k≥u+1
∞
Z
Φφ (v − u) dv ≤
∞ Z
X
k=1
∞
X
k
Z
∞
Φφ (v/3) dv = 3
k−1
Φφ (u)du,
0
Z
Φφ (|k|/3) ≤ Φφ (0) + 6
∞
Φφ (u) du =: D2 .
0
k=1
Thus, we conclude that for u > 0
I(u) ≤
√
2C (D1 + D2 ) Φφ (u/4) .
(2.6)
Since for u < 0
Φφ (|u + k|) = Φφ (|−u − k|) = Φφ (||u| − k|) ,
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Uniform convergence of wavelet decompositions
it follows that
I(u) ≤
√
2C (D1 + D2 ) Φφ (|u|/4)
(2.7)
for every u ∈ R.
By (2.4), (2.5), (2.6), and (2.7),
2x − 1 a.e. on R.
|ψ(x)| ≤ C̃ Φφ 4 (2.8)
The desired result follows from (2.8) if we chose
(
Φψ (x) :=
max(1, C̃),
x ∈ (0, 1/2),
, x∈
max(1, C̃) · max Φφ (x), Φφ 2x−1
6 (0, 1/2).
4
Motivated by Lemma 2.5, we will use the following assumption instead of two separate assumptions S 0 (γ) for the f -wavelet φ and the m-wavelet ψ.
Assumption S(γ). For some function Φ(·) and γ > 0 both the father and mother
wavelets satisfy assumption S 0 (γ).
Let {Ω, B, P} be a standard probability space. Let X(t), t ∈ R, be a random process
such that EX(t) = 0, E|X(t)|2 < ∞.
If sample trajectories of this process are in the space L2 (R) with probability one,
then it is possible to obtain the representation (wavelet representation)
X(t) =
X
ξ0k φ0k (t) +
Z
ξ0k =
ηjk ψjk (t) ,
(2.9)
j=0 k∈Z
k∈Z
where
∞ X
X
Z
X(t)φ0k (t) dt,
ηjk =
R
X(t)ψjk (t) dt .
(2.10)
R
The majority of random processes does not possess the required property. For example, sample paths of stationary processes are not in the space L2 (R) (a.s.). However,
in many cases it is possible to construct a representation of type (2.9) for X(t).
Consider the approximants of X(t) defined by
Xn,kn (t) =
X
|k|≤k00
ξ0k φ0k (t) +
n−1
X
X
ηjk ψjk (t) ,
(2.11)
j=0 |k|≤kj
where kn := (k00 , k0 , ..., kn−1 ).
Theorem 2.6 below guarantees the mean-square convergence of Xn,kn (t) to X(t) if
k00 → ∞, kj → ∞, j ∈ N0 , and n → ∞. The latter means that we increase the number
n of multiresolution analysis subspaces which are used to approximate X(t). For each
multiresolution analysis subspace j = 00 , 0, 1, 2... the number kj of its basis vectors,
which are used in the approximation, increases too, as n tends to infinity. Thus, for
each fixed k and j there is n0 ∈ N0 that the terms ξ0k φ0k (t) and ηjk ψjk (t) are included in
all Xn,kn (t) for n ≥ n0 (i.e., each ξ0k φ0k (t) and ηjk ψjk (t) can be absent only in the finite
number of Xn,kn (t)).
Theorem 2.6. [18] Let X(t), t ∈ R, be a random process such that EX(t) = 0, E|X(t)|2 <
∞ for all t ∈ R, and its covariance function R(t, s) := EX(t)X(s) is continuous. Let the
f -wavelet φ and the m-wavelet ψ be continuous functions which satisfy assumption
S(1). Let c(x), x ∈ R, denote a non decreasing on [0, ∞) even function with c(0) > 0.
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Uniform convergence of wavelet decompositions
Suppose that there exists a function A : (0, ∞) → (0, ∞), and a0 , x0 ∈ (0, ∞) such that
c(ax) ≤ c(x) · A(a), for all a ≥ a0 , x ≥ x0 .
If
Z
c(x)Φ(|x|) dx < ∞ and |R(t, t)|1/2 ≤ c(t),
R
then
1. Xn,kn (t) ∈ L2 (Ω) ;
2. Xn,kn (t) → X(t) in mean square when n → ∞, k00 → ∞, and kj → ∞ for all j ∈ N0 .
3
Uniform convergence of wavelet expansions for Gaussian random processes
In this section we show that, under suitable conditions, the sequence Xn,kn (t) converges in probability in Banach space C([0, T ]), T > 0, i.e.
P
sup |X(t) − Xn,kn (t)| > ε → 0,
0≤t≤T
when n → ∞, k00 → ∞, and kj → ∞ for all j ∈ N0 := {0, 1, ...} . More details on the
general theory of random processes in the space C([0, T ]) can be found in [2].
Theorem 3.1. [19] Let Xn (t), t ∈ [0, T ], be a sequence of Gaussian stochastic processes
such that all Xn (t) are separable in ([0, T ], ρ), ρ(t, s) = |t − s|, and
E|Xn (t) − Xn (s)|2
sup sup
1/2
≤ σ(h) ,
n≥1 |t−s|≤h
where σ(h) is a function, which is monotone increasing in a neighborhood of the origin
and σ(h) → 0 when h → 0 .
Assume that for some ε > 0
Z
ε
q
− ln σ (−1) (u) du < ∞ ,
(3.1)
0
where σ (−1) (u) is the inverse function of σ(u). If the random variables Xn (t) converge
in probability to the random variable X(t) for all t ∈ [0, T ], then Xn (t) converges to X(t)
in the space C([0, T ]).
Remark 3.2. For example, it is easy to check that assumption (3.1) holds true for
σ(h) =
β
ln
C
eα + h1
and
σ(h) = Chκ ,
when C > 0, β > 1/2, α > 0, and κ > 0.
The following theorem is the main result of the paper.
Theorem 3.3. Let a Gaussian process X(t), t ∈ R, its covariance function, the f wavelet φ, and the corresponding m-wavelet ψ satisfy the assumptions of Theorem 2.6.
Suppose that
(i) assumption S(γ), γ ∈ (0, 1), holds true for φ and ψ;
(ii) the integrals
R
R
α > 1/2(1 − γ);
b
lnα (1 + |u|)|ψ(u)|
du and
R
R
b
lnα (1 + |u|)|φ(u)|
du converge for some
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Uniform convergence of wavelet decompositions
(iii) there exist constants b0 and cj , j ∈ N0 , such that for all integer k E|ξ0k |2 ≤ b0 ,
E|ηjk |2 ≤ cj , and
∞
X
√
j
cj 2 2 j α(1−γ) < ∞.
(3.2)
j=0
Then Xn,kn (t) → X(t) uniformly in probability on each interval [0, T ] when n → ∞,
k00 → ∞, and kj → ∞ for all j ∈ N0 .
Remark 3.4. If both wavelets φ and ψ have compact supports, then some assumptions of Theorem 3.3 are superfluous. In the following theorem we give an example by
considering approximants of the form
Xn (t) :=
X
ξ0k φ0k (t) +
n−1
X
X
ηjk ψjk (t) .
j=0 k∈Z
k∈Z
Theorem 3.5. Let X(t), t ∈ R, be a separable centered Gaussian random process such
that its covariance function R(t, s) is continuous. Let the f -wavelet φ and the correm-wavelet ψ be continuous
functions with compact supports and the integrals
Rsponding
R
α
α
b
b
ln
(1
+
|u|)|
ψ(u)|
du
and
ln
(1
+
|u|)|φ(u)|
du converge for some α > 1/2(1 − γ),
R
R
γ ∈ (0, 1). If there exist constants cj , j ∈ N0 , such that E|ηjk |2 ≤ cj for all k ∈ Z, and assumption (3.2) is satisfied, then Xn (t) → X(t) uniformly in probability on each interval
[0, T ] when n → ∞.
Proof. The assumptions of Theorem 2.6 and S(γ), 0 < γ < 1, are satisfied because φ and
ψ have compact supports. Therefore, the desired result follows from Theorem 3.3.
Remark 3.6. For example, Daubechies wavelets satisfy the assumptions of Theorem 3.5.
4
Proof of the main theorem
To prove Theorem 3.3 we need some auxiliary results.
Lemma 4.1. If δ(x) is a scaling function satisfying assumption S 0 (γ), then
sup Sγ (x) ≤ 3Φγ (0) + 4
x∈R
Z
∞
Φγ (t) dt,
(4.1)
1/2
where
Sγ (x) =
X
|δ(x − k)|γ .
k∈Z
Proof. The lemma is a simple generalization of a result from [11].
Since Sγ (x) is a periodic function with period 1, it is sufficient to prove (4.1) for
x ∈ [0, 1].
Notice, that for x ∈ [0, 1] and integer |k| ≥ 2 the inequality |x − k| ≥ |k|/2 holds true.
Hence, Φ(|x − k|) ≤ Φ (|k|/2) and
Sγ (x) ≤ Φγ (|x|) + Φγ (|x + 1|) + Φγ (|x − 1|) +
X
Φγ (|k|/2)
|k|≥2
γ
≤ 3Φ (0) + 2
∞ Z
X
k=2
k
γ
γ
Z
∞
Φγ (t) dt.
Φ (t/2) dt = 3Φ (0) + 4
1
2
k−1
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Uniform convergence of wavelet decompositions
Lemma 4.2. Let δ̂(x) denote the Fourier transform of the scaling function δ(x), δjk (x) :=
j
2 2 δ(2j x − k), j ∈ N0 , k ∈ Z. If for some α > 0
Z
b
lnα (1 + |u|)|δ(u)|
du < ∞,
R
then for all x, y ∈ R and k ∈ Z
1
,
|x − y|
j
1
|δjk (x) − δjk (y)| ≤ 2 2 j α R1α · ln−α eα +
, j ∈ N0 ,
|x − y|
R
R
b
b
|δ(u)|
du and R1α := π1 R lnα (eα + |u| + 1) |δ(u)|
du.
:= π1 R lnα eα + |u|
2
|δ0k (x) − δ0k (y)| ≤ R0α · ln−α eα +
where R0α
1
2π
Proof. Since δ(x) =
j
22
|δjk (x) − δjk (y)| ≤
2π
R
R
eixu δ̂(u) du, we have
j Z
Z 22
(x − y)u2j ixu2j
iyu2j −e
sin
e
|δ̂ (u) | du =
|δ̂ (u) | du.(4.2)
π R
2
R
Note that for v 6= 0 the following inequality holds:
s |ln (eα + |s|)|α
sin
≤
α .
v
|ln (eα + |v|)|
(4.3)
By (4.2) and (4.3) we obtain
j
|δjk (x) − δjk (y)| ≤
|u|2j
α
ln e +
|δ̂ (u) | du.
2
R
Z
22
π lnα eα +
1
|x−y|
α
The assertion of the lemma follows from this inequality.
Lemma 4.3. If a scaling function δ(x) satisfies the assumptions of Lemmata 4.1 and
4.2, then for γ ∈ (0, 1) and α > 0 :
X
|δjk (x) − δjk (y)| ≤ 2
j
2 +1
j
α(1−γ)
R11−γ (α)
Z
γ
!
∞
3Φ (0) + 4
γ
Φ (t) dt
1/2
k∈Z
× ln eα +
X
|δ0k (x) − δ0k (y)| ≤
2R01−γ (α)
1
|x − y|
γ
−α(1−γ)
, j 6= 0,
Z
(4.4)
!
∞
γ
3Φ (0) + 4
Φ (t) dt
1/2
k∈Z
× ln eα +
1
|x − y|
−α(1−γ)
.
(4.5)
Proof. By lemma 4.2 for j ≥ 1 we obtain
X
|δjk (x) − δjk (y)| =
k∈Z
≤ ln eα +
X
|δjk (x) − δjk (y)|1−γ · |δjk (x) − δjk (y)|γ
k∈Z
1
|x − y|
−α(1−γ) j
2 2 j α R1 (α)
1−γ X
·
|δjk (x) − δjk (y)|γ .
k∈Z
EJP 18 (2013), paper 69.
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Uniform convergence of wavelet decompositions
We now make use of the inequality |a + b|α ≤ qα (|a|α + |b|α ) , where
(
qα =
α ≤ 1,
1,
(4.6)
2α−1 , α > 1.
By lemma 4.1 we get
X
|δjk (x) − δjk (y)|γ
≤
2 sup
x∈R
k∈Z
≤
2
jγ
2
X
|δjk (x)|γ = 2
jγ
2
+1
x∈R
k∈Z
+1
sup
Z
γ
X
|δ(2j x − k)|γ
k∈Z
!
∞
γ
Φ (t) dt .
3Φ (0) + 4
1/2
Inequality (4.4) follows from this estimate. The proof of inequality (4.5) is similar.
Now we are ready to prove Theorem 3.3.
Proof. In virtue of Theorem 3.1 and Remark 3.2 it is sufficient to prove that Xn,kn (t) →
X(t) in mean square and
2
E |Xn,kn (t) − Xn,kn (s)|
12
C
≤
ln
β
eα
+
1
|t−s|
for some C > 0, β > 1/2, and α > 0.
Since γ < 1, assumption S(1) holds true for φ(x) and ψ(x). Hence, by Theorem 2.6
we get Xn,kn (t) → X(t) in mean square.
Note that (2.11) implies
E |Xn,kn (t) − Xn,kn (s)|
2
21
 2 1/2
X

 ξ0k (φ0k (t) − φ0k (s)) 
≤ E |k|≤k0
0
 2 1/2
n−1
n−1
X X
Xp
√

=: S +
ηjk (ψjk (t) − ψjk (s)) 
+
Sj .
E |k|≤kj
j=0
j=0
We will only show how to handle Sj . A similar approach can be used to deal with the
remaining term S.
Sj can be bounded as follows
!2
Sj =
X X
X
Eηjk ηjl (ψjk (t) − ψjk (s))(ψjl (t) − ψjl (s)) ≤ cj
|k|≤kj |l|≤kj
|ψjk (t) − ψjk (s)|
.
k∈Z
By lemma 4.3
Sj ≤ 2j j 2α(1−γ) cj L2 ln eα +
1
|t − s|
−2α(1−γ)
where
L :=
2R11−γ (α)
3Φγ (0) + 4
Z
,
!
∞
Φγ (t) dt .
1/2
Similarly,
S ≤ b0 L21 ln eα +
1
|t − s|
−2α(1−γ)
EJP 18 (2013), paper 69.
,
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Uniform convergence of wavelet decompositions
where
2R01−γ (α)
L1 :=
Z
γ
!
∞
3Φ (0) + 4
γ
Φ (t) dt .
1/2
Therefore, we conclude that
where
2
E |Xn,kn (t) − Xn,kn (s)|
21
≤ ln eα +
C
1
|t−s|
α(1−γ) ,
∞
X
p
√ 2j α(1−γ)
b0 L1 + L
cj 2 j
< ∞ and β := α(1 − γ) > 1/2.
C :=
j=0
5
Examples
In this section we consider some examples of wavelets and stochastic processes
which satisfy assumption (3.2) of Theorem 3.3.
b be a Lipschitz function of order κ > 0 (κ = 1 for the case
Example 5.1. Let ψ
0
b
supu∈R |ψ (u)| ≤ C ), i.e.
b
b
|ψ(u)
− ψ(v)|
≤ C|u − v|κ .
Assume that for the covariance function R(t, s)
Z Z
|R(t, s)| dt ds < ∞
R
and
R
Z Z b
κ
κ
R2 (z, w) · |z| · |w| dz dw < ∞,
R
R
where
Z Z
R(u, v) e−izu e−iwv du dv.
b2 (z, w) =
R
R
R
Now we show that E|ηjk |2 ≤ cj for all k ∈ Z and find suitable upper bounds for cj .
By Parseval’s theorem,
Z Z
Eηjk ηjl =
EX(u)X(v) ψjk (u)ψjl (v) dudv
R
1
=
(2π)2
R
Z Z
b2 (z, w) ψbjk (z) ψbjl (w) dz dw.
R
R
R
Since
k
z
e−i 2j z
ψbjk (z) = j/2 · ψb j ,
2
2
it follows that
|Eηjk ηjl | ≤
1
j
2 (2π)2
Z Z z w b
R2 (z, w) · ψb j · ψb j dz dw.
2
2
R R
b
By properties of the m-wavelet ψ we have ψ(0)
= 0. Therefore, using the Lipschitz
conditions, we obtain
|Eηjk ηjl | ≤
This means that
√
C2
2
(2π) 2j(1+2κ)
j/2(1+2κ)
cj ≤ C/2
Z Z b
κ
κ
R2 (z, w) · |z| · |w| dz dw.
R
R
and assumption (3.2) holds.
EJP 18 (2013), paper 69.
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Uniform convergence of wavelet decompositions
In the following example we consider the case of stationary stochastic processes.
This case was studied in detail by us in [15]. Note that assumptions in the example are
much simpler than those used in [15].
Example 5.2. Let X(t) be a centered short-memory stationary stochastic process and
ψb be a Lipschitz function of order κ > 0. Assume that the covariance function R(t−s) :=
EX(t)X(s) satisfies the following condition
Z b 2κ
R(z) · |z| dz < ∞.
R
By Parseval’s theorem we deduce
Z Z
Z Z −ivz
e
b
b
|Eηjk ηjl | = R(u − v)ψjk (u) du ψjl (v) dv = R(z)ψjk (z) dzψjl (v) dv R R 2π
R R
Z Z 1
1
b b
b b z b z ≤
R(z) ψjk (z) ψbjl (z) dz = j+1
R(z) · ψ j · ψ j dz.
2π R
2 π R
2
2
Thus, by the Lipschitz conditions, for all k, l ∈ Z :
Z C
b 2κ
|Eηjk ηjl | ≤
R(z)
· |z| dz.
π 21+j(1+2κ) R
√
This means that cj ≤ C/2j/2(1+2κ) and assumption (3.2) is satisfied.
6
Application to fractional Brownian motion
In this section we show how to adjust the main theorem to the fractional Brownian
motion case.
Let Wα (t), t ∈ R, be a separable centered Gaussian random process such that
Wα (−t) = Wα (t) and its covariance function is
1
(|t|α + |s|α − ||t| − |s||α ) , 0 < α < 2.
2
R(t, s) = EWα (t)Wα (s) =
(6.1)
Lemma 6.1. If assumption S(γ), 0 < γ < 1, holds true and for some α > 0
Z
cψ :=
|u|α |ψ(u)| du < ∞,
R
then for the coefficients of the process Wα (t), defined by (2.10),
|Eηjk ηjl | ≤
C
2j(1+α)
for all k, l ∈ Z.
Proof. Since |Eηjk ηjl | ≤ E|ηjk |2
By (2.1) and (6.1) we obtain
2
E|ηjk |
12
E|ηjl |2
12
, it is sufficient to estimate E|ηjk |2 .
Z Z
=
=
≤
+
+
Z Z
u v
1
R(t, s) ψjk (t)ψjk (s) dtds = j
R j , j ψ(u − k)ψ(v − k) du dv
2 R R
2 2
R R
Z Z
1
1 u α v α u v α + j − j − j ψ(u − k)ψ(v − k) du dv
2j R R 2 2j
2
2
2
Z Z
1
α
|u| ψ(u − k)ψ(v − k) du dv
21+j(1+α)
R R
Z Z
α
|v| ψ(u − k)ψ(v − k) du dv
R R
Z Z
z1 + z2 + z3
α
||u| − |v|| ψ(u − k)ψ(v − k) du dv =: 1+j(1+α) .
(6.2)
2
R R
EJP 18 (2013), paper 69.
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Uniform convergence of wavelet decompositions
It follows from assumption S(γ) that
Z
|ψ(u)| du < ∞.
N :=
R
Hence, we get the estimate
Z Z
z1
Z
α
≤
|u| |ψ(u − k)||ψ(v − k)| du dv =
ZR
×
R
|ψ(v)| dv
R
α
|u + k| |ψ(u)| du ≤ qα N (N |k|α + cψ ) < ∞,
(6.3)
R
where qα is defined by (4.6).
R
Using Fubini’s theorem and R ψ(u) du = 0 we obtain
Z
Z
α
|u| ψ(u − k) du
z1 =
ψ(v − k) dv = 0.
R
R
Similarly, z2 = 0.
Finally, we estimate z3 as follows
Z Z
z3 ≤
α
||u + k| − |v + k|| |ψ(u)||ψ(v)| du dv.
R
R
By the reverse triangle inequality and (4.6) we obtain
α
α
α
||u + k| − |v + k|| ≤ qα (|u| + |v| ) .
Hence,
|z3 |
≤ qα 2α+1
Z
α
Z
|u| |ψ(u)| du
R
|ψ(v)| dv = qα cψ 2α+1 N,
(6.4)
R
which completes the proof of the lemma.
In some case, for example, for the fractional Brownian motion the assumption b0 ≥
|Eξ0k ξ0l | of Theorem 3.3 doesn’t hold true. The following theorem gives the uniform
convergence of wavelet expansions without this assumption.
Theorem 6.2. Let a random process X(t), t ∈ R, the f -wavelet φ, and the corresponding m-wavelet ψ satisfy the assumptions of Theorem 2.6 and assumptions (i) and (ii) of
Theorem 3.3.
Suppose that there exist
(iii’) constants cj , j ∈ N0 , such that E|ηjk |2 ≤ cj for all k ∈ Z and (3.2) holds true;
(iv) some ε > 0 such that
2
X
C
S := E ξ0k (φ0k (t) − φ0k (s)) ≤ 2α(1−γ) , α > 1/2(1 − γ),
1
|k|<|k0 |
ln eα + |t−s|
0
if |t − s| < ε.
Then Xn,kn (t) → X(t) uniformly in probability on each interval [0, T ] when n → ∞,
k00 → ∞, and kj → ∞ for all j ∈ N0 .
Proof. The assertion of the theorem follows from the proof of Theorem 3.3.
EJP 18 (2013), paper 69.
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Uniform convergence of wavelet decompositions
Now, under some mild additional conditions on the f -wavelet φ, we show that estimate (iv) holds true in the fractional Brownian motion case.
Lemma 6.3. Let the f -wavelet φ satisfy the assumptions of Theorem 2.6,
b
• φ(z)
→ 0 and φbR0 (z) → 0, when
±∞;
R z→
R
b0 (u)| du and
• the integrals R |u|α |φ(u)|, R |φ
lnα (1 + |u|)|φb(i) (u)| du, i = 0, 1, 2,
R
converge for some α > 1/2(1 − γ).
Then assumption (iv) of Theorem 6.2 holds true for Wα (t).
Proof. S can be bounded as follows
X X
S≤
|Eξ0k ξ0l ||φ0k (t) − φ0k (s)||φ0l (t) − φ0l (s)| .
|k|≤k0 |l|≤k0
1
1
Since Eξ0k ξ0l ≤ E|ξ0k |2 2 E|ξ0l |2 2 , it is sufficient to estimate E|ξ0k |2 .
Similarly to (6.2) we get
2
E|ξ0k |
Z Z
Z Z
α
|u| φ(u − k)φ(v − k) du dv +
≤
R
Z Z
α
|v| φ(u − k)φ(v − k) du dv
R
R
R
α
||u| − |v|| φ(u − k)φ(v − k) du dv.
+
R
R
Analogously to (6.3) and (6.4) we obtain
Z Z
R
Z Z
R
α
|u| |φ(u − k)φ(v − k)| du dv ≤ qα M (M |k|α + cφ ) ,
R
α
||u| − |v|| |φ(u − k)φ(v − k)| du dv ≤ qα cφ M 2α+1 ,
R
where
Z
Z
|φ(u)| du < ∞, cφ :=
M :=
R
|u|α |φ(u)| du < ∞.
R
Then
Eξ0k ξ0l 1
1
≤ 2qα M (M |k|α + cφ (2α + 1)) 2 (M |l|α + cφ (2α + 1)) 2
q
q
α
α
≤ 2qα M M |k| 2 + cφ 2α+1
M |l| 2 + cφ 2α+1 .
To estimate |φ0k (t) − φ0k (s)|, we use the representations
φb0k (z) = e−ikz · φb (z) ,
Z
1
eitz e−ikz φb (z) dz.
φ0k (t) =
2π R
Repeatedly using integration by parts and the assumptions of the lemma, we obtain that
for k 6= 0 :
Z
1
|φ0k (t) − φ0k (s)| = eitz − eisz φb (z) d(e−ikz )
2ikπ R
Z
i
1 h
itz
isz b
itz
isz b0
−ikz =
i
te
−
se
φ(z)
+
e
−
e
φ
(z)
d(e
)
2
2πk
ZR h
1 b + 2i teitz − seisz φb0 (z)
=
− t2 eitz − s2 eisz φ(z)
2
2πk
R
Z
i
2 itz
1
itz
isz b00
−ikz
b
t e − s2 eisz |φ(z)|
+ e −e
φ (z) e
dz ≤
dz
2
2πk
R
Z
Z
itz
itz
te − seisz |φb0 (z)| dz +
e − eisz |φb00 (z)| dz .
+2
R
(6.5)
R
EJP 18 (2013), paper 69.
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Uniform convergence of wavelet decompositions
By inequalities (8) and (12) given in [15] we get
|teitz − seisz |
≤
|t − s| + t|eitz − eisz |
≤
α
|z|
α
ln
e
+
2
c
α,T
+ 2T   ,
α
1
1
ln eα + |t−s|
ln eα + |t−s|

|t2 eitz − s2 eisz |
|t2 − s2 | + t2 |eitz − eisz |
α

|z|
α
ln
e
+
2
c̃
α,T
+ 2T 2   ,
α
1
1
ln eα + |t−s|
ln eα + |t−s|
≤
≤
where cα,T and c̃α,T are constants which do not depend on t, s and z.
Applying these inequalities to (6.5) we obtain
|φ0k (t) − φ0k (s)| ≤
k2
Bφ,α,T
ln eα +
α
1
|t−s|
,
k 6= 0,
where
Bφ,α,T
:=
+
+
Z
Z
|z| b
b
c̃α,T
|φ(z)|
dz + 2T 2
lnα eα +
|φ(z)| dz
2
R
R
Z
Z
|z| b0
|φ (z)| dz
2cα,T
|φb0 (z)| dz + 4T 2
lnα eα +
2
R
R
Z
|z| b00
2
lnα eα +
|φ (z)| dz .
2
R
1
2π
If k = 0, then
|φ00 (t) − φ00 (s)| ≤
1
2π
Z
itz
e − eisz |φb (z) | dz ≤
R
where
B0,α,T
1
:=
π
Z
α
ln
R
|z|
e +
2
α
B
,
0,α,T
1
ln eα + |t−s|
α
b
|φ(z)|
dz .
Consequently, we can estimate S as follows
S≤ ln eα +
C
1
|t−s|
2α(1−γ) , α > 1/2(1 − γ),
where
C := 2qα M
q
cφ 2α+1 B0,α,T + Bφ,α,T
X
|k| ≤ k0
k 6= 0
!2
p
α
M |k| 2 + cφ 2α+1
.
k2
Theorem 6.4. If the assumptions of Lemmata 6.1, 6.3, and assumptions (i) and (ii)
of Theorem 3.3 are satisfied, then the wavelet expansions of the fractional Brownian
motion uniformly converge to Wα (t).
EJP 18 (2013), paper 69.
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Uniform convergence of wavelet decompositions
7
Convergence rate in the space C[0, T ]
Returning now to the general case introduced in Theorem 3.3, let us investigate
what happens when the number of terms in the approximants (2.11) becomes large.
First we specify an estimate for the supremum of Gaussian processes.
Definition 7.1. [2, §3.2] A set Q ⊂ S ⊂ R is called an ε-net in the set S with respect to
the semimetric ρ if for any point x ∈ S there exists at least one point y ∈ Q such that
ρ(x, y) ≤ ε.
Definition 7.2. [2, §3.2] Let
Hρ (S, ε) :=
ln(Nρ (S, ε)), if Nρ (S, ε) < +∞;
+∞,
if Nρ (S, ε) = +∞,
where Nρ (S, ε) is the number of point in a minimal ε-net in the set S.
The function Hρ (S, ε), ε > 0, is called the metric entropy of the set S.
Lemma 7.3. [2, (4.10)] Let Y(t), t ∈ [0, T ] be a separable Gaussian random process,
ε0 := sup
E|Y(t)|2
1/2
< ∞,
0≤t≤T
1
I(ε0 ) := √
2
Zε0 p
H(ε) dε < ∞ ,
(7.1)
0
where H(ε) is the metric entropy of the space ([0, T ], ρ), ρ(t, s) = (E|Y(t) − Y(s)|2 )1/2 .
Then
 2 
p
P

 u − 8uI(ε0 )
sup |Y(t)| > u ≤ 2 exp −

2ε20
0≤t≤T



,


where u > 8I(ε0 ).
Assume that there exists a nonnegative monotone nondecreasing in some neighborhood of the origin function σ(ε), ε > 0, such that σ(ε) → 0 when ε → 0 and
E|Y(t) − Y(s)|2
sup
1/2
≤ σ(ε) .
(7.2)
|t − s| ≤ ε
t, s ∈ [0, T ]
Lemma 7.4. [16] If
σ(ε) =
β
ln
C
, β > 1/2, α > 0 ,
eα + 1ε
(7.3)
then (7.1) holds true and
γ
I(ε0 ) ≤ δ(ε0 ) := √
2
where γ := min ε0 , σ T2
.
1 !
−1 2β
p
1
C
ln(T + 1) + 1 −
,
2β
γ
Lemma 7.5. If a scaling function δ(x) satisfies assumption S 0 (γ), then
sup
|x|≤T
X
|k|≥k1
γ
Z∞
|δ(x − k)| ≤
γ
Z∞
Φ (t) dt +
k1 −T −1
Φγ (t) dt
k1 −1
for k1 ≥ T + 1.
EJP 18 (2013), paper 69.
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Uniform convergence of wavelet decompositions
Proof. Since
X
X
|δ(x − k)|γ ≤
|k|≥k1
(Φγ (|x + k|) + Φγ (|x − k|)) =: zk1 (x),
k≥k1
zk1 (x) is an even function.
Then the assertion of the theorem follows from
X
sup
|x|≤T

≤
X
Zk
0≤x≤T
|k|≥k1
γ
Zk
Φ (t − T ) dt +

k≥k1
|δ(x − k)|γ ≤ sup zk1 (x) ≤
k−1
X
(Φγ (k − T ) + Φγ (k))
k≥k1
Z∞

γ
Φ (t) dt ≤
Z∞
γ
k1 −T −1
k−1
Φγ (t) dt.
Φ (t) dt +
k1 −1
Now we formulate the main result of this section.
Theorem 7.6. Let a separable Gaussian random process X(t), t ∈ [0, T ], the f -wavelet
φ, and the corresponding m-wavelet ψ satisfy the assumptions of Theorem 3.3.
Then
P
)
(
p
(u − 8uδ(εkn ))2
,
sup |X(t) − Xn,kn (t)| > u ≤ 2 exp −
2ε2kn
0≤t≤T
where u > 8δ(εkn ) and the decreasing sequence εkn is defined by (7.5) in the proof of
the theorem.
Proof. Let us verify that Y(t) := X(t) − Xn,kn (t) satisfies (7.2) with σ(ε) given by (7.3).
First, we observe that
E|Y(t) − Y(s)|
2 1/2
 X
= E ξ0k (φ0k (t) − φ0k (s))
|k|>k0
0
2 1/2
X
X

+
ηjk (ψjk (t) − ψjk (s)) +
ηjk (ψjk (t) − ψjk (s)) 
j=0 |k|>kj
j=n k∈Z
∞
X
n−1
X
  2 1/2
2 1/2
X
n−1
X X


 ≤ E ξ0k (φ0k (t) − φ0k (s))  +
ηjk (ψjk (t) − ψjk (s)) 
E |k|>k0
|k|>kj
j=0
0
+
∞
X
j=n
 2 1/2
n−1
∞ q
X
Xq
X
√
E Sj0 +
Rj0 .
ηjk (ψjk (t) − ψjk (s)) 
:= S 0 +
j=0
k∈Z
j=n
We will only show how to handle Sj0 . A similar approach can be used to deal with the
remaining terms S 0 and Rj0 .
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Uniform convergence of wavelet decompositions
By Lemmata 4.1 and 4.2 we get
Sj0
≤
X X
|Eηjk ηjl ||ψjk (t) − ψjk (s)||ψjl (t) − ψjl (s)|
|k|>kj |l|>kj
2

X
≤ cj 
|ψjk (t) − ψjk (s)|γ |ψjk (t) − ψjk (s)|1−γ 
|k|>kj
2(1−γ)
j/2 α
2

2 j R1α
|ψjk (t) − ψjk (s)|γ 
2α(1−γ) 
1
|k|>kj
ln eα + |t−s|

2
2(1−γ)
X
2j/2 j α R1α
≤ cj |ψjk (t)|γ 
2α(1−γ) 2 sup
|t|≤T
1
α
|k|>kj
ln e + |t−s|
!2
Z ∞
2(1−γ)
cj 2j+2 (j α R1α )
γ
≤ Φγ (t) dt ,
2α(1−γ) 3Φ (0) + 4
1/2
1
α
ln e + |t−s|
X
≤ cj where R1α =
Hence
1
π
R
R
b
lnα (eα + |u| + 1) |ψ(u)|
du.
n−1
Xq
Sj0 ≤ j=0
where
Bα :=
∞
X
√
(j+2)/2
cj 2
Bα
ln eα +
α
1−γ
(j R1α )
1
|t−s|
α(1−γ) ,
Z
γ
(7.4)
!
∞
γ
3Φ (0) + 4
Φ (t) dt
< ∞.
1/2
j=0
From Lemma 4.3, it follows that
∞ q
X
Rj0
j=n
P∞ √ j
L · j=n cj 2 2 j α(1−γ)
≤ α(1−γ) ,
1
ln eα + |t−s|
where
L :=
2R11−γ (α)
Z
γ
!
∞
3Φ (0) + 4
γ
Φ (t) dt
.
1/2
Similarly to (7.4) by lemma 7.4 we obtain
√
S0
1−γ
b0 R0α
≤ α(1−γ)
1
ln eα + |t−s|
where R0α =
1
π
R
R
lnα eα +
|u|
2
Z
∞
γ
Z
Φ (t) dt +
k00 −T −1
!
∞
γ
Φ (t) dt ,
k00 −1
b
|φ(u)|
du and k00 ≥ T + 1.
Thus,
E|Y(t) − Y(s)|2
1/2
≤ ln eα +
1/2
2
= E |(X(t) − Xn,kn (t)) − (X(s) − Xn,kn (s))|
C
1
|t−s|
α(1−γ) =: σ(|t − s|), α(1 − γ) > 1/2,
EJP 18 (2013), paper 69.
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Uniform convergence of wavelet decompositions
where
C := L ·
∞
X
√
j
2
cj 2 j
α(1−γ)
+ Bα +
∞
Z
1−γ
b0 R0α
Z
γ
γ
Φ (t) dt +
Φ (t) dt .
k00 −T −1
j=n
!
∞
k00 −1
Then by lemmata 7.3 and 7.4
P
)
(
p
(u − 8uI(ε̃kn ))2
,
sup |X(t) − Xn,kn (t)| > u ≤ 2 exp −
2ε̃2kn
0≤t≤T
where
ε̃kn = sup
2
E |X(t) − Xn,kn (t)|
1/2
.
0≤t≤T
The proof will be completed by investigating ε̃kn as a function of kn .
Note that
E |X(t) − Xn,kn (t)|
2
1/2
 2 1/2
X

 ξ0k φ0k (t)  +
≤ E |k|>k0
0
 2 1/2
 2 1/2
X
∞
X
X

 E +
ηjk ψjk (t)  +
ηjk ψjk (t) 
.
E j=0
j=n
k∈Z
|k|>kj
n−1
X
Let J := min n, min{j ∈ N0 : kj < 2j T + 1} . Notice that J → ∞, when n → ∞ and
kj → ∞ for all j ∈ N0 .
By Lemmata 4.2 and 7.5 for k00 ≥ T + 1
Z
ε̃kn
≤
∞
Z
γ
b0
γ
Φ (t) dt +
k00 −T −1
Z
!
∞
+
γ
Φ (t) dt
kj −1
+
k00 −1
!
∞
Φ (t) dt
+
∞
X
√
cj 2
j
2
J−1
X
√
j
Z
Z
!
∞
γ
3Φ (0) + 4
Φγ (t) dt
kj −2j T −1
j=0
γ
∞
cj 2 2
Φ (t) dt
=: εkn .
(7.5)
1/2
j=J
The choice of J and assumption (3.2) imply that εkn → 0, when n → ∞, k00 → ∞, and
kj → ∞ for all j ∈ N0 .
It is worth noticing that δ(·) is an increasing function, I(ε0 ) ≤ δ(ε0 ) for any ε0 , and
ε̃kn ≤ εkn . Hence, for u > 8δ(εkn ) we get
  2 
2 
p
p


 u − 8uI(ε̃kn ) 

 u − 8uδ(εkn ) 

exp −
≤ exp −
.
2
2




2ε̃kn
2εkn




Finally, an application of Lemmata 7.3 and 7.4 completes the proof.
Remark 7.7. Note, that εkn → 0, if and only if n, k00 , and all kj , j ≥ 0, approach infinity.
Remark 7.8. If εkn → 0 then δ(εkn ) → 0. Therefore
the
(
) convergence in the theorem is
exponential with the rate bounded by 2 exp
−
const
ε2kn
, where one can choose const ≈
u2 for δ(εkn ) u.
EJP 18 (2013), paper 69.
ejp.ejpecp.org
Page 19/21
Uniform convergence of wavelet decompositions
Remark 7.9. In the theorem we only require that k00 , and all kj , j ≥ 0, approach infinity. If we narrow our general class of wavelet expansions Xn,kn (t) by specifying rates
of growth of the sequences kn we can enlarge classes of wavelets bases and random
processes in the theorem and obtain explicit rates of convergence by specifying εkn .
√
For instance, consider the examples in Section 5. It was shown that cj ≤ C/2j/2(1+2κ) .
R
∞
Let the sequence kn := (k00 , k0 , ..., kn−1 ) be chosen so that k∗ Φγ (t) dt ≤ C/2Jκ for
k ∗ := min(k00 − T, k0 − T, k1 − 2T, ..., kJ−1 − 2J−1 T ) − 1. Then, by (7.5) we get εkn ≤ C/2Jκ .
Remark 7.10. Lemma 7.4 and formula (7.5) provide simple expressions to computer
εkn and δ(εkn ). It allows specifying Theorem 7.6 for various stochastic processes and
wavelets.
References
[1] A. Ayache, W. Linde, Series representations of fractional Gaussian processes by trigonometric and Haar systems, Electron. J. Probab. 14(94) (2009) 2691-2719. MR-2576756
[2] V.V. Buldygin, Yu.V. Kozachenko, Metric Characterization of Random Variables and Random
Processes, American Mathematical Society, Providence R.I., 2000. MR-1743716
[3] S. Cambanis, E. Masry, Wavelet approximation of deterministic and random signals: convergence properties and rates, IEEE Trans. Inf. Theory. 40(4) (1994) 1013-1029. MR-1301417
[4] C.K. Chui, An Introduction to Wavelets, Academic Press, New York, 1992. MR-1150048
[5] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. MR-1162107
[6] A. Delaigle, P. Hall, Methodology and theory for partial least squares applied to functional
data, Ann. Statist. 40(1) (2012) 322-352. MR-3014309
[7] G. Didier, V. Pipiras, Gaussian stationary processes: adaptive wavelet decompositions, discrete approximations and their convergence, J. Fourier Anal. and Appl. 14 (2008) 203-234.
MR-2383723
[8] K. Dzhaparidze, H. van Zanten, A series expansion of fractional Brownian motion, Probab.
Theory Related Fields. 130(1) (2004) 39-55. MR-2092872
[9] P.P.B. Eggermont, V.N. LaRiccia, Uniform error bounds for smoothing splines. High Dimensional Probability, 220-237, IMS Lecture Notes Monogr. Ser., 51, Inst. Math. Statist., Beachwood, OH, 2006. MR-2387772
[10] J. Fan, P. Hall, M. Martin, P. Patil, Adaptation to high spatial inhomogeneity using wavelet
methods, Statist. Sinica. 9(1) (1999) 85-102. MR-1678882
[11] W. Hardle, G. Kerkyacharian, D. Picard, A. Tsybakov, Wavelets, Approximation and Statistical
Applications, Springer, New York, 1998. MR-1618204
[12] E. Hernandez, G. Weiss, A First Course on Wavelets, CRC press Inc., Boca Ratan FL, 1996.
MR-1408902
[13] E. Iglói, A rate-optimal trigonometric series expansion of the fractional Brownian motion,
Electron. J. Probab. 10 (2005) 1381-1397. MR-2183006
[14] J. Istas, Wavelet coefficients of a gaussian process and applications, Ann. Inst. H. Poincaré
(B) Probab. Statist. 28(4) (1992) 537-556. MR-1193084
[15] Yu.V. Kozachenko, A. Olenko, O.V. Polosmak, Uniform convergence of wavelet expansions of
Gaussian random processes, Stoch. Anal. Appl. 29(2) (2011) 169-184. MR-2774235
[16] Yu.V. Kozachenko, A. Olenko, O.V. Polosmak, Convergence rate of wavelet expansions of
Gaussian random processes, will appear in Comm. Statist. Theory Methods. (2013)
[17] Yu.V. Kozachenko, A. Olenko, O.V. Polosmak, Convergence in Lp ([0, T ]) of wavelet expansions
of ϕ-sub-Gaussian random processes, will appear in Methodol. Comput. Appl. Probab. (2014)
arXiv:1307.1859
[18] Yu.V. Kozachenko, O.V. Polosmak, Uniform convergence in probability of wavelet expansions of random processes from L2 (Ω), Random Oper. Stoch. Equ. 16(4) (2008) 12-37. MR2494935
EJP 18 (2013), paper 69.
ejp.ejpecp.org
Page 20/21
Uniform convergence of wavelet decompositions
[19] Yu.V. Kozachenko, G.I. Slivka, Justification of the Fourier method for hyperbolic equations
with random initial conditions, Theory Probab. Math. Statist. 69 (2004) 67-83. MR-2110906
[20] O. Kurbanmuradov, K. Sabelfeld, Convergence of Fourier-wavelet models for Gaussian random processes, SIAM J. Numer. Anal. 46(6) (2008) 3084-3112. MR-2439503
[21] Y. Meyer, F. Sellan, M.S. Taqqu, Wavelets, generalized white noise and fractional integration:
the synthesis of fractional Brownian motion, J. Fourier Anal. and Appl. 5(5) (1999) 465-494.
MR-1755100
[22] wmtsa: Wavelet Methods for Time Series Analysis, available online from http://cran.rproject.org/web/packages/wmtsa/ (accessed 15 July 2013)
[23] P.C.B. Phillips, Z. Liao, Series estimation of stochastic processes: recent developments and
econometric applications, to appear in A. Ullah, J. Racine and L. Su (eds.) Handbook of
Applied Nonparametric and Semiparametric Econometrics and Statistics, Oxford University
Press, Oxford, 2013.
[24] J. Zhang, G. Waiter, A wavelet-based KL-like expansion for wide-sense stationary random
processes, IEEE Trans. Signal Proc. 42(7) (1994) 1737-1745.
Acknowledgments. The research of first two authors was partially supported by La
Trobe University Research Grant "Stochastic Approximation in Finance and Signal Processing". The authors are grateful for the referee’s comments, which helped to improve
the style of the presentation.
EJP 18 (2013), paper 69.
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