c 2004 Society for Industrial and Applied Mathematics
SIAM J. MATH. ANAL. Vol. 36, No. 1, pp. 323–346
ASYMPTOTIC NORMALITY OF SCALING FUNCTIONS∗ LOUIS H. Y. CHEN† , TIM N. T. GOODMAN‡ , AND S. L. LEE† 2
Abstract. The Gaussian function G(x) = √1 e−x /2 , which has been a classical choice for 2π multiscale representation, is the solution of the scaling equation αG(αx − y)dg(y), x ∈ R, G(x) = R
with scale α > 1 and absolutely continuous measure dg(y) = √
1 2π(α2 − 1)
e−y
2
/2(α2 −1)
dy.
It is known that the sequence of normalized B-splines (Bn ), where Bn is the solution of the scaling equation φ(x) =
n j=0
1 2n−1
n φ(2x − j), j
x ∈ R,
converges uniformly to G. The classical results on normal approximation of binomial distributions and the uniform B-splines are studied in the broader context of normal approximation of probability measures mn , n = 1, 2, . . . , and the corresponding solutions φn of the scaling equations αφn (αx − y)dmn (y), x ∈ R. φn (x) = R
Various forms of convergence are considered and orders of convergence obtained. A class of probability densities are constructed that converge to the Gaussian function faster than the uniform B-splines. Key words. normal approximation, probability measures, scaling functions, uniform B-splines, asymptotic normality AMS subject classifications. 41A15, 41A25, 41A39, 42C40, 65T60 DOI. 10.1137/S0036141002406229
1. Introduction. The Gaussian function, G(x) = √12π e−x /2 , and its derivatives have been widely used in scale-space representation (see [1], [11], [18]). The uniform B-spline, Bn , which is the solution of the scaling equation n n 1 φ(x) = (1.1) φ(2x − j), x ∈ R, n−1 j 2 j=0 2
associated with the binomial distribution 21n nj , j = 0, 1, . . . , n, approximates the Gaussian and provides fast computational algorithms for practical implementation of Gaussian scale-space representation (see [15], [16]). The B-spline, Bn , is the probability density function of the sum of n copies of independent identically distributed ∗ Received by the editors April 23, 2002; accepted for publication (in revised form) October 3, 2003; published electronically July 14, 2004. This research was supported by the Wavelets Strategic Research Programme, National University of Singapore, under a grant from the National Science and Technology Board and the Ministry of Education, Singapore. http://www.siam.org/journals/sima/36-1/40622.html † Department of Mathematics, University of Singapore, 10 Kent Ridge Road, Singapore 119260 (
[email protected],
[email protected]). ‡ Department of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland, UK (
[email protected]).
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uniform random variables on the interval [0, 1). It is well known that the binomial distributions converge to the normal distribution in the sense that x [xn ] 2 1 1 n √ lim = e−t /2 dt, n n→∞ 2 k 2π −∞ k=0
(1.2) where xn =
√
nx/2 + n/2, and it is also known that
(1.3)
lim
n→∞
xn
1 Bn (t)dt = √ 2π −∞
x
e−t
2
/2
dt,
−∞
√ √ where xn := nx/2 3 + n/2. Further, the normalized B-splines converge uniformly on R to the Gaussian function (see [5] and [13]). In fact, Curry and Schoenberg [5] considered the more general class of Polya frequency functions as limits of nonuniform B-splines with arbitrary knots. The Gaussian function satisfies the integral scaling equation αG(αx − y)dg(y), x ∈ R, G(x) = R
where α > 1 is a scaling constant and g is the absolutely continuous measure given by dg(y) = √
2 2 1 e−y /2(α −1) dy. 2 2π(α − 1)
The Gaussian function and its derivatives and the modulated Gaussian have been used extensively in many applications such as scale-space analysis and computer vision (see [1], [11], [18]). The normal approximation of the binomial distributions and the uniform B-splines enables the binomial coefficients and B-splines to replace the Gaussian function in the Gaussian scale-space representation and vice versa (see [11], [15], [16]). The Gaussian function is optimal in time-frequency localization, amenable to statistical analysis, and provides an accurate model of human vision (see [18]). While inheriting approximately many of the rich properties of the Gaussian, the binomial distributions and B-splines have the added advantage of providing fast algorithms for practical computations. We shall consider a sequence of scaling equations (1.4) αφn (αx − y)dmn (y), x ∈ R, n = 1, 2, . . . , φn (x) = R
where α > 1 and (mn ) is a sequence of probability measures with finite first and second moments. It will be shown in the next section that for each n, (1.4) has a unique solution, which is also a probability measure. We shall call φn the mn -scaling function and mn its filter. If mn is a discrete measure concentrated on the integers Z with mass hn (j) at j ∈ Z, then (1.4) becomes the discrete scaling equation φn (x) = (1.5) αhn (j)φn (αx − j), x ∈ R. j∈Z
In particular, if mn is the discrete measure concentrated on the set {0, 1, . . . , n} with mass 21n nj at j = 0, 1, . . . , n and scale α = 2, then (1.5) reduces to (1.1). The
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object of this paper is to investigate the approximation of the Gaussian function by probability measures and the corresponding scaling functions in the same way as the normal approximation by binomial and B-spline distributions and to construct sequences of distributions that converge to the Gaussian faster than the binomial and B-spline distributions. Suppose that (mn ) is a sequence of probability measures on R with mean µ(mn ) = µn and standard deviation σ(mn ) = σn , and define m n (S) = mn (σn S + µn ) for measurable S ⊂ R, or, equivalently, (1.6)
m n (u) = eiuµn /σn m n (u/σn ),
u ∈ R.
We say that (mn ) is asymptotically normal if for all x ∈ R, x x lim (1.7) G(t)dt. dm n (t) = n→∞
−∞
−∞
If mn is absolutely continuous, then by the Radon–Nikodym theorem, dmn (t) = fn (t)dt for a probability density function fn , and then dm n (t) = fn (t)dt, where fn (t) = σn fn (σn t + µn ). The central limit theorem tells us that if mn is the probability distribution for the sum of n independent, identically distributed random variables, then (mn ) is asymptotically normal. In the case that each such random variable is uniformly distributed on the interval [0, 1), mn has density function Bn , and the asymptotic normality is also implied by the convergence of the normalized B-splines discussed earlier. Now it is well known that asymptotic normality can be stated in terms of convergence of characteristic functions, i.e., Fourier transforms of the probability density functions. To be precise, (1.7) is equivalent to (1.8)
2 m n (u) → e−u /2 locally uniformly on R,
where local uniform convergence means convergence that is uniform on compact subsets. This result is given in [7, p. 249], and more modern expositions are given in [10] and [17]. In section 2, we show that if m is a probability measure on R with finite first moment, then the solution of the scaling equation φ(x) = (1.9) αφ(αx − y)dm(y), x ∈ R, R
is also a probability measure. In (1.9), and throughout the paper, α is a number larger than 1, which we call the scale. We remark that if the solution is absolutely continuous, then its probability density satisfies (1.9). If the solution φ is not absolutely continuous, then it satisfies (1.9) in the weak sense, i.e., (1.10)
φ(u) = m(u/α) φ(u/α),
u ∈ R.
The following result puts in perspective the asymptotic normality exhibited by the binomial coefficients and the uniform B-splines.
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Theorem 1.1. Let (mn ) be a sequence of probability measures on R with finite first and second moments and (m n ) be uniformly bounded in a neighborhood of the origin. Then (mn ) is asymptotically normal if and only if the corresponding sequence of mn -scaling functions is asymptotically normal. In order to study the asymptotic normality of scaling functions, we need only to study the asymptotic normality of their filters, because of Theorem 1.1. The binomial coefficients, which are the filters for the uniform B-splines, define a sequence of discrete probability measures that is asymptotically normal. It follows from Theorem 1.1 that the coefficients bn,k in the expansion (1.11)
1 + z + · · · + z α−1 α
n
n(α−1)
=
bn,k z k ,
k=0
where the scale α is here an integer, also define a sequence of probability measures that is asymptotically normal. This is because the uniform B-splines are also the solution of the scaling equations with measures mn (k) = bn,k , k = 0, 1, . . . , n(α − 1), for any integer scale α > 1. For such α, the roots of the polynomials on the left of (1.11) that generate bn,k are the complex αth roots of unity that are not equal to 1. The next theorem gives a general result that holds for a large class of polynomials including those with negative roots as well as those in (1.11). Theorem 1.2. Let γ ∈ [0, π/2), and define Dγ = {z ∈ C : satisfies (1.12)}:
z z Im ≤ tan γ Re (1.12) . (1 + z)2 (1 + z)2 For n = 1, 2, . . . , take rn,1 , . . . , rn,n in Dγ and define (1.13)
n
an,k z k =
k=0
n
(z + rn,j )/(1 + rn,j ).
j=1
We also assume that the rn,j , n = 1, 2, . . . , j = 1, . . . , n, are bounded away from −1, that the coefficients an,k , n = 1, 2, . . . , k = 0, . . . n, are real, and that (1.14)
σn2 =
n
rn,j /(1 + rn,j )2 → ∞ as n → ∞.
j=1
If mn , n = 1, 2, . . . , denote the discrete measures defined by mn ({k}) = an,k , k = 2 0, 1, . . . , n, it follows that m n (u) → e−u /2 locally uniformly as n → ∞. If, in addition, an,k ≥ 0, k = 0, 1, . . . , n, for all sufficiently large n, then (mn ) is asymptotically normal. Remark 1. We remark that the first part of Theorem 1.2 does not require mn to be a probability measure; i.e., some of the coefficients an,k could be negative. After some preliminary results in the next section, we shall prove Theorem 1.1 in section 3. A proof of Theorem 1.2 is given in section 4. We note that a special case of this result, when all rn,j > 0, was proved earlier using probabilistic techniques [3]. The completely different analytic techniques, which we employ here, give considerably more general results. These techniques also allow us to analyze, in the remainder of section 4, the order of convergence in the frequency domain for both the measures mn and the corresponding scaling functions. In particular we shall prove the following theorem.
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Theorem 1.3. We assume the conditions of Theorem 1.2 and that an,k ≥ 0, k = 0, 1, . . . , n. (a) Then φn − e−(·)2 /2 = O(σn−1 ). ∞
n
(b) If k=0 an,k z k is a reciprocal polynomial, i.e., an,0 = 0 and an,k = an,n−k , k = 0, 1, . . . , n, then φn − e−(·)2 /2 = O(σn−2 ). ∞
(c) If, in addition to the condition in (b), (1.15)
σn−1
n
2 rn,j (rn,j − 4rn,j + 1)/(1 + rn,j )4 is bounded,
j=1
then
φn − e−(·)2 /2
∞
= O(σn−3 ).
Asymptotic normality entails weak convergence in the time domain. We show in section 5 that, under mild conditions on the shape of the filters and the scaling functions, both the measures and the corresponding scaling functions converge uniformly in the time domain. The shape conditions are satisfied if rn,j are restricted to certain sectors of the complex plane, reminiscent of total positivity. It is noted that for a special case of the choice, when all rn,j > 0, Chui and Wang [4] consider convergence of the scaling functions. However, their approach is different, and they do not consider the related convergence of the measures mn . Finally, in the same section, we consider the order of convergence in the time domain and prove the following results. Theorem 1.4. We assume the conditions of Theorem 1.2 and that all rn,j lie in the sector | arg z| ≤ π3 . Then as n → ∞, k − µn − 12 max σn an,k − G = O(σn ), k=0,...,n σn
n and if k=0 an,k z k is reciprocal, k − µn −2 = O(σn 3 ). max σn an,k − G k=0,...,n σn We remark that in [3], this problem is considered, using probabilistic techniques, for the special case when an,0 , . . . , an,n are the Eulerian numbers. In this case σn = 1 −2 π(n + 1)/6. Thus our result gives order of convergence O(σn 3 ) = O(n− 3 ), while 1 [3] shows only convergence O(n− 4 ). Theorem 1.5. We assume the conditions of Theorem 1.2, that rn,j include 1 and all Re(rn,j ) ≥ 0. For n = 1, 2, . . . , let φn denote the scaling function corresponding to the measure mn ({k}) = an,k , k = 0, 1, . . . , n, with scale 2, and define φn (x) = σ(φn )φn (σ(φn )x + µ(φn )),
x ∈ R.
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Then −1
If
n n=0
φn − G ∞ = O(σn 2 ). an,k z k is reciprocal for large enough n, then φn − G ∞ = O(σn−1 ).
If, in addition, (1.15) is satisfied, then −3
φn − G ∞ = O(σn 2 ). It is noted that certain sequences of scaling functions give a faster rate of convergence to the Gaussian than the uniform B-splines. Also on considering Theorems 1.3 and 1.5, it might be expected that the second part of Theorem 1.4 should give order −2/3 of convergence O(σn−1 ) instead of O(σn ) and that under the additional condition −3/2 (1.15) we should obtain order O(σn ). We have been unable to prove orders bet−2/3 ter than O(σn ) due to a technical restriction in Lemma 5.4, and we do not know whether this restriction can be removed. 2. Probability measures and scaling equations. Consider the scaling equation (1.9) where m is a probability measure and, as before, α is a number (not necessarily an integer) satisfying α > 1. We shall show that (1.9) has a unique solution, which is a probability measure. Further, if m has finite first and second moments, then the solution of (1.9) also has finite first and second moments. Equation (1.10) suggests that, when φ(0) = 1, φ(u) is given by the infinite product (2.1) below but with n replaced by ∞. We remark that products of the form (2.1) occur in the study of groups of transformations in Hilbert space (see, for example, [6, section 38]). For the case when φ is the B-spline Bn and α = 2, this reduces to the classical formula of Vi`ete: sin x/x =
∞
cos(x/2j ).
j=1
So as a preliminary result we need to consider the convergence of (2.1) in Lemma 2.1 below. Lemma 2.1. Suppose that m is a probability measure with finite first moment. Then the products (2.1)
n
j m(u/α ),
u ∈ R,
j=1
converge locally uniformly as n → ∞. Proof. Since m is a probability measure, |m(u)| ≤ 1 for all u ∈ R. Then for every nonnegative integer n and all u, n j m(u/α ) ≤ 1 for all u ∈ R. j=1 Also, since m has finite first moment, m is bounded, and so j m(u/α ) − 1 ≤ C|u|/αj , j = 1, 2, . . . ,
ASYMPOTIC NORMALITY OF SCALING FUNCTIONS
329
for a constant C > 0. Thus for integers n > , n− n j j +j ) m(u/α )| m(u/α ) − |1 − m(u/α ≤ j=1 j=1 j=1 ≤ C|u|(α− − α−n )/(α − 1), which tends on compact subsets of R as , n → ∞. Therefore, the n to zero uniformly j product j=1 m(u/α ) converges uniformly on compact sets as n → ∞. Proposition 2.2. If m is a probability measure with finite first and second moments, then the scaling equation (1.9) has a unique solution φ, which is also a probability measure with finite first and second moments. Further, (2.2)
µ(φ) = (α − 1)−1 µ(m)
σ(φ)2 = (α2 − 1)−1 σ(m)2 .
and
Proof. Choose a nonnegative initial function f0 ∈ C(R) with compact support and f0 (0) = 1, and for n = 1, 2, . . . define (2.3) αfn−1 (αx − y)dm(y), x ∈ R. fn (x) = R
Then (2.4)
fn (u) = fn−1 (u/α)m(u/α) =
n
j m(u/α )f0 (u/αn ),
u ∈ R.
j=1
Further, fn is nonnegative, and fn (0) = 1 for n = 0, 1, . . . . Therefore, fn defines a sequence of probability measures µn ∈ C0 (R)∗ , where dµn (x) = fn (x)dx and C0 (R)∗ is the dual of the space C0 (R) of continuous functions that vanish at infinity. Therefore, µ n = fn , n = 0, 1, . . . . Since the unit ball in C0 (R)∗ is weak* compact, there exist a subsequence µn and a probability measure φ on R such that µn → φ as → ∞ in the weak* topology. It follows (see [7, p. 249]) that µ n converges locally uniformly to φ as n → ∞. By Lemma 2.1 and (2.4), φ(u) =
∞
j m(u/α ),
u ∈ R,
j=1
which satisfies (1.10). Define Πn (u) :=
(2.5)
n
j m(u/α ),
u ∈ R.
j=1
Then (2.6)
Πn (u) → φ(u) locally uniformly on R,
where φ is the solution of (1.9). We shall show that Πn converges uniformly in a neighborhood of the origin. Since m(0) = 1, there exists a closed disc D centered at the origin such that m(u) = 0 for all u ∈ D. Differentiating (2.5) gives (2.7)
n 1 m (u/αj ) Πn (u) = m(u/α ) , j) αj m(u/α j=1 j=1
n
j
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LOUIS H. Y. CHEN, TIM N. T. GOODMAN, AND S. L. LEE
which shows that Πn is uniformly convergent on D. It follows that φ exists and Πn converges uniformly to φ on D. Hence φ is continuous on D, and (0). φ (0) = (α − 1)−1 m
(2.8) Differentiating (2.7) gives
(2.9)
⎞2 ⎛ n j 1 m (u/α ) j ⎝ ⎠ m(u/α ) Πn (u) = j m(u/α j) α j=1 j=1 n
+
n
j m(u/α )
j=1
n j )−m (u/αj )2 1 m (u/αj )m(u/α , j )2 α2j m(u/α j=1
which shows that Πn is uniformly convergent on D. Thus φ exists and is continuous on D. A straightforward computation using (2.9) leads to (2.10)
φ (0) =
1 (α2 − 1)
m (0) +
2m (0)2 (α − 1)
.
It follows that φ has finite first and second moments, and the relationships (2.2) follow from (2.8) and (2.10). 3. Proof of Theorem 1.1. We shall prove a slightly stronger result than that of Theorem 1.1. This result is contained in Theorem 3.1. Theorem 3.1. Let (mn ) be a sequence of probability measures on R with finite first and second moments, and (m n ) is uniformly bounded in a neighborhood of 0. Then the following are equivalent: 2 (a) m n (u) → e−u /2 locally uniformly on R as n → ∞. 2 (b) φn (u) → e−u /2 locally uniformly on R as n → ∞. (c) (mn ) is asymptotically normal. (d) (φn ) is asymptotically normal. Further, if (a) holds locally uniformly on R, then (b) holds uniformly on R. Proof. By Proposition 2.2, for each n = 0, 1, . . . , (1.4) has a unique solution φn , which is also a probability measure with finite first and second order moments, and (3.1)
µ(mn ) = (α − 1)µ(φn ) and σ(mn )2 = (α2 − 1)σ(φn )2 .
By (1.6), (1.10), and (3.1), (3.2)
n (α−1 φn (u) = m
α2 − 1 u)φn (α−1 u),
u ∈ R.
Iterating (3.2) leads to (3.3)
∞ m n (α−j α2 − 1 u), φn (u) =
u ∈ R,
j=1
where the infinite product on the right converges locally uniformly on R and uniformly in n, since (m n ) is uniformly bounded in a neighborhood of 0.
ASYMPOTIC NORMALITY OF SCALING FUNCTIONS
331
If (a) holds, then by (3.3) we have ∞ m n (α−j α2 − 1 u) lim φn (u) = lim n→∞
n→∞
=
∞
j=1
e−(α
2
−1)u2 /2α2j
= e−u
2
/2
u ∈ R.
,
j=1 2 Conversely, if limn→∞ φn (u) = e−u /2 , then by (3.2)
√ φn (α u/ α2 − 1) , m n (u) = √ φn (u/ α2 − 1)
u ∈ R,
for sufficiently large n. It follows that 2 e−α u /2(α −1) lim m n (u) = −u2 /2(α2 −1) = e−u /2 , n→∞ e 2
2
2
u ∈ R.
A similar argument shows that (a) holds locally uniformly on R if and only if (b) holds locally uniformly on R. Now suppose that (a) holds uniformly on compact subsets of R. Note that for any u ∈ R and n ≥ 1, −∞ −∞ −iux e dmn (x) ≤ dmn (x) = 1. |m n (u)| = −∞
−∞
So for any k ≥ 1, ∞ −j 2 − 1 u) φn (u) = m (α α n j=1 ∞
(3.4)
−j 2 n (α α − 1 u) ≤ m j=k+1 −k = φn (α u) , u ∈ R.
For any > 0, we choose A > 0 and integer N so that e−A /2 < and φn (u) − e−u2 /2 < , |u| ≤ αA, n > N. 2
Take any u with |u| > A. Then there is a nonnegative integer k such that A < α−k |u| ≤ αA, and so −k
e−(α
u)2 /2
< e−A
2
/2
< .
−k 2 Also for n > N, |φn (α−k u) − e−(α u) /2 | < , and so (α−k u) < 2. φn (u) ≤ φ n
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LOUIS H. Y. CHEN, TIM N. T. GOODMAN, AND S. L. LEE
2 2 < e−A /2 < , it follows that |φn (u) − e−u /2 | < 3. Thus for all n > N 2 and u ∈ R, |φn (u) − e−u /2 | < 3, and hence (b) holds uniformly on R. Recall that the asymptotic normality of a sequence of distribution functions is equivalent to the local uniform convergence of their characteristic functions (see, for instance, [7, p. 249]). We remark that if (mn ) is a sequence of discrete probability measures on Z with finite first and second moments, then the condition that (m n ) be uniformly bounded in a neighborhood of 0 is automatically satisfied. The following lemma gives a slightly stronger result. Lemma 3.2. If (mn ) is a sequence of discrete probability measures on Z with finite first and second moments, then (m n ) is uniformly bounded on any compact subset of R.
∞ Proof. Let mn ({k}) = bn,k ≥ 0, n = 1, 2, . . . , k ∈ Z, where k=−∞ bn,k = 1. As before, we write
Since e−u
2
/2
µn :=
∞
kbn,k and σn2 :=
∞
(k − µn )2 bn,k .
k=−∞
k=−∞
Then ∞
m n (u) =
bn,k ei(µn −k)u/σn ,
k=−∞
and so ∞ i bn,k (µn − k)ei(µn −k)u/σn m n (u) = σn k=−∞
∞ i = bn,k (µn − k)(ei(µn −k)u/σn − 1). σn k=−∞
Since |eiu − 1| ≤ 2|u| for all u ∈ R, ∞ 2|u| m ≤ (u) (k − µn )2 bn,k = 2|u|. n σn2 k=−∞
Corollary 3.3. Let (mn ) be a sequence of discrete probability measures on Z with finite first and second moments. Then (mn ) is asymptotically normal if and only if the corresponding sequence of mn -scaling functions with scale α is asymptotically normal. 4. Convergence in the frequency domain. In order to apply Theorem 1.1 to study the asymptotic normality of scaling functions, we need first to study the asymptotic normality of their filters. We begin with a proof of Theorem 1.2. Proof of Theorem 1.2. Let (4.1)
n k=0
k
an,k z =
n j=1
(pn,j z + qn,j ),
ASYMPOTIC NORMALITY OF SCALING FUNCTIONS
333
where qn,j = 1 − pn,j . Then m n (u) =
n
(pn,j e−iu + qn,j )
j=1
and m n (u) = eiuµn /σn
n
(pn,j e−iu/σn + qn,j ),
j=1
where µn = µ(mn ) =
(4.2)
n
pn,j ,
j=1
and σn2
(4.3)
n
2
= σ(mn ) =
pn,j qn,j .
j=1
Therefore, iuµn + F log m n (u) = σn j=1 n
(4.4)
pn,j ,
−iu σn
,
where F (p, t) = log(pet + q),
q = 1 − p.
By induction, for n = 2, 3, . . . , (4.5)
F (n) (p, t) :=
n−2 ∂n t −n F (p, t) = (pe + q) pq (−1)j cn (j)pj q n−2−j e(j+1)t , n ∂t j=0
where c2 (j) = δ0 (j), j ∈ Z, and for n = 2, 3, . . . , cn satisfies the recursive relation cn+1 (j) = (j + 1)cn (j) + (n − j)cn (j − 1), j ∈ Z.
∞
∞
∞ From (4.6) we have j=−∞ cn+1 (j) = n j=−∞ cn (j), and since j=−∞ c2 (j) = 1, we have (4.6)
(4.7)
∞
cn (j) = (n − 1)!,
n = 2, 3. . . . .
j=−∞
By (4.5) the Taylor series of F (p, t) is given by (4.8)
F (p, t) =
∞
aν (p)tν ,
ν=0
where (4.9)
a0 (p) = 0, a1 (p) = p, a2 (p) =
1 pq, 2
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LOUIS H. Y. CHEN, TIM N. T. GOODMAN, AND S. L. LEE
and for ν = 3, 4, . . . , aν (p) =
(4.10)
ν−2 pq (−1)k cν (k)pk q ν−2−k . ν! k=0
By (4.4) and (4.8), ∞
iuµn log m n (u) = + aν (pn,j )σn−ν (−iu)ν . σn j=1 ν=0 n
(4.11)
By (4.2), (4.3), and (4.9), n
a1 (pn,j )σn−1 (−iu) = −
j=1 n
iuµn , σn
a2 (pn,j )σn−2 (−iu)2 = −
j=1
u2 , 2
so that (4.11) becomes (4.12)
log m n (u) = −
n ∞ u2 −ν + σn (−iu)ν aν (pn,j ). 2 ν=3 j=1
Now rn,j ∈ Dγ if and only if
rn,j rn,j ≤ tanγ Re Im (1 + rn,j )2 (1 + rn,j )2 or |Im (pn,j qn,j )| ≤ tanγ Re (pn,j qn,j ) . Therefore, (4.13)
|pn,j qn,j | ≤ secγ Re (pn,j qn,j ) .
On the other hand, rn,j being bounded away from −1 is equivalent to (4.14)
|pn,j | ≤ A − 1,
n = 1, 2, . . . ,
j = 1, 2, . . . , n,
for some constant A. By (4.10), (4.13), and (4.14), |pn,j qn,j | cν (k)|pn,j |k |qn,j |ν−2−k ν! ν−2
|aν (pn,j )| ≤
k=0
(4.15)
≤ sec γ Re(pn,j qn,j )Aν−2 /ν.
By (4.12) and (4.15), ∞ n σn−ν |u|ν u2 log m ≤ sec γ (u) + Re(pn,j qn,j )Aν−2 n 2 ν ν=3 j=1 ν−2 ∞ ν A |u| ≤ sec γ ν σ n ν=3 −1 3 A|u| A|u| 1− (4.16) ≤ sec γ σn σn
ASYMPOTIC NORMALITY OF SCALING FUNCTIONS
335
whenever A|u| < σn . Since σn → ∞ as n → ∞, taking the limits as n → ∞, (4.16) 2 gives limn→∞ m n (u) = e−u /2 locally uniformly. Recall that the region Dγ in Theorem 1.2 comprises all z ∈ C satisfying
z z Im ≤ tan γ Re . (1 + z)2 (1 + z)2 It can be seen that Dγ contains the sector | arg z| ≤ γ, and for z = ±reiθ , r > 0, γ ≤ θ ≤ π, (1.12) is equivalent to sin( θ−γ 2 ) sin( θ+γ 2 )
≤r≤
sin( θ+γ 2 ) sin( θ−γ 2 )
.
In particular Dγ contains the unit circle r = 1. For the special case of Theorem 1.2, when all rn,j > 0, the result was proved using probabilistic methods in [3] and [12]. Our analytic techniques allow us not only to prove asymptotic normality for a much larger class of measures but also, in the next result, to give information on the order of convergence in the frequency domain. Proposition 4.1. We assume the conditions of Theorem 1.2 (except that we do not require an,k ≥ 0, k = 0, 1, . . . , n). As before, σn2
=
n j=1
rn,j . (1 + rn,j )2
Then there is a constant K > 0 so that for Sn := {u : |u| ≤ Kσn } the following hold. (a) There is a constant B such that −u2 /2 −1 m (4.17) (u) − e n ≤ Bσn , u ∈ Sn , n = 1, 2, . . . .
n (b) If k=0 an,k z k is a reciprocal polynomial, then there is a constant C such that −u2 /2 −2 m (4.18) (u) − e n ≤ Cσn , u ∈ Sn , n = 1, 2, . . . . (c) Finally, if in addition to the condition in (b), (1.15) is satisfied, then there is a constant D such that −u2 /2 −3 m (4.19) (u) − e n ≤ Dσn , u ∈ Sn , n = 1, 2, . . . . Proof. (a) From (4.16) we see that for |u| ≤ 14 A−1 cos γ σn , log m n (u) +
4A|u|3 u2 1 ≤ sec γ ≤ u2 , 2 3σn 3
and so log m n (u) ≤ − 16 u2 . By the mean value theorem, u2 −u2 /2 −u2 /6 m n (u) + ≤e log m n (u) − e 2 3 2 4A|u| ≤ sec γ e−u /6 3σn ≤ Bσn−1
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LOUIS H. Y. CHEN, TIM N. T. GOODMAN, AND S. L. LEE
for a constant B, which gives (4.17). (b) We note from (4.6) and (4.10) that (4.20)
a3 (p) =
a4 (p) =
(4.21)
pq (q − p), 3!
pq 2 (q − 4pq + p2 ). 4!
n Suppose that Pn (z) = k=0 an,k z k is a reciprocal polynomial. Then Pn (z) = 0 if and −1 , then pn,j = qn,k and qn,j = pn,k , it only if Pn (z −1 ) = 0. Noting that if rn,j = rn,k follows that n
(4.22)
a3 (pn,j ) = 0.
j=1
So from (4.12) and (4.15), −1 A2 |u|4 A|u| u2 log m 1− n (u) + ≤ sec γ 2 σn2 σn whenever A|u| < σn . Then (4.18) follows in a similar manner as before. (c) Finally, we assume (1.15). Then (4.12), (4.21), (4.22), and (4.15) give (4.19). √ We note that r2 − 4r + 1 = 0 when r = 2 ± 3, and so (1.15) √ requires that in n some sense the roots of Pn (z) := k=0 an,k z k are close to −2 ± 3. In particular, (1.15) will be satisfied if Pn (z) = Qn (z)(z 2 + 4z + 1)kn , where Qn is a reciprocal polynomial of degree n = n − 2kn and n−1/2 n is bounded over n. In this case (4.19) takes the form 2 n (u) − e−u /2 ≤ Cn−3/2 , u ∈ Sn , n = 1, 2, . . . . m We now consider the order of convergence of the normalized mn -scaling functions φn as in Theorem 1.1, again in the frequency domain. From (3.3) it follows as in (4.4) that ∞ n iuµn iu log φn (u) = + F pn,k , − j σn α σn j=1 k=1
and as in (4.12) that ∞ n u2 (−iu)ν log φn (u) = − + aν (pn,j ). 2 (α2 − 1)σnν j=1 ν=3
So as in (4.16) there is a constant A with −1 2 3 log φn (u) + u ≤ A|u| 1 − A|u| 2 σn σn
ASYMPOTIC NORMALITY OF SCALING FUNCTIONS
337
whenever A|u| < σn . By the mean value theorem, for A|u| < 12 σn ,
3 −u2 /2 2A|u| φn (u) − e−u2 /2 ≤ e−u2 /2 + φ(u) − e , σn and so −1 3 2A|u|3 −u2 /2 −u2 /2 2A|u| φ(u) −e 1− ≤e σn σn 3 2 4A|u| ≤ e−u /2 σn −1 ≤ Bσn if |u|3 < σn /4A for some constant B. Similarly, if Pn is a reciprocal polynomial, then as in the derivation of (4.18), there are constants A, B > 0 such that φn (u) − e−u2 /2 ≤ Bσn−2 1/2
whenever |u| < Aσn . Finally, if (1.15) is satisfied, then there are constants A, B > 0 with φn (u) − e−u2 /2 ≤ Bσn−3 3/5
whenever |u| < Aσn . To extend these estimates to all of R we need the following result. Lemma 4.2. Suppose that mn is a probability measure, n = 1, 2, . . . , and there is a sequence (βn ) with lim βn = 0 so that φn (u) − e−u2 /2 < βn whenever |u| ≤ A| log βn | for some A > 0. Then 2 limn→∞ βn−1 φn − e−(·) /2 ∞ ≤ 1.
Proof. Take 0 < < 1. Choose n large enough so that 2| log(βn )| < α−2 A2 | log βn |2 . Take any u in R with |u| > A| log βn |. Then for some integer k ≥ 1, α−1 A| log βn | < α−k |u| ≤ A| log βn |. Putting v = α−k |u|, we have v 2 > α−2 A2 | log βn |2 > 2| log(βn )|, and so e−v
2
/2
< βn .
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LOUIS H. Y. CHEN, TIM N. T. GOODMAN, AND S. L. LEE
2 Since |φn (v) − e−v /2 | < βn , recalling (3.4) gives < βn (1 + ). φn (u) ≤ φ(v)
Also e−u
2
/2
< e−v
2
/2
< βn , and so φn (u) − e−u2 /2 < βn (1 + 2).
2 For any u with |u| ≤ A| log βn | we have |φn (u) − e−u /2 | < βn , and thus φn − 2 e−(·) /2 ∞ ≤ βn (1 + 2) for all u ∈ R. The result follows. Proof of Theorem 1.3. Theorem 1.3 follows from Lemma 4.2 and the preceding discussions.
5. Convergence in the time domain. From Theorems 1.1 and 1.2 we can deduce the convergence of m n and φn to the Gaussian function G in the time domain only in the weak sense of (1.7). In this section we shall show that under mild assumptions on (rn,j ) in Theorem 1.2, both m n and φn have a “nice” shape, which ensures that the convergence is uniform. We consider two possibilities for the shape. For a continuous function ψ, we say ψ is bell-shaped if ψ ≥ 0, limx→±∞ ψ(x) = 0, and there are two points α < β such that ψ is convex on (−∞, α] and [β, ∞) and concave on [α, β]. We say that ψ is logconcave if it is supported on a closed interval, ψ > 0, and log ψ is concave on its interior. Neither of these properties implies the other. We note that in both cases there is a point γ such that ψ is increasing on (−∞, γ] and decreasing on [γ, ∞). We also note that logconcavity is equivalent to total positivity of order 2, which says that for any x1 < x2 and y1 < y2 , ψ(x1 − y1 ) ψ(x1 − y2 ) ψ(x2 − y1 ) ψ(x2 − y2 ) ≥ 0. The following lemma shows that for a sequence of bell-shaped or logconcave functions, asymptotic normality implies uniform convergence. The result was stated in [5] for the case of logconcave functions, but no proof was given. ∞ Lemma 5.1. Suppose that (gn ) is a sequence of continuous functions with −∞ gn = 1, which are either bell-shaped or logconcave, and for each x ∈ R, x x lim (5.1) gn = G. n→∞
−∞
−∞
Then gn converges to G uniformly on R. Proof. By (5.1), for any interval I ⊂ R, lim (5.2) gn = G. n→∞
Take > 0. Then
n→∞
−
−
lim
−3
gn =
I
I
G,
lim
−
−3
n→∞
−
gn =
G. −
− G < − G, we have −3 gn < − G for large enough n. Similarly, for large 3 enough n, gn < − G. So for large enough n, there are points −3 < an < − < Since
−3
ASYMPOTIC NORMALITY OF SCALING FUNCTIONS
339
bn < < cn < 3 with gn (an ) < gn (bn ) > gn (cn ). For any such n, maxx∈R gn (x) occurs only for x ∈ (−3, 3). For if maxx∈R gn (x) = gn (α) for α ≤ −3, then gn (α) > gn (an ) < gn (bn ) > gn (cn ), which contradicts the shape of gn . Similarly, maxx∈R gn (x) cannot occur for x ≥ 3. Again take > 0. Choose δ > 0 such that |G(x) − G(y)| < whenever |x − y| < δ. δ 1 < ∞. Then Take a function B ≥ 0 with support in [0, δ], 0 B = 1, and ||B||
∞
lim
n→∞
−∞
B(x − a)gn (x)dx =
∞
−∞
B(x − a)G(x)dx
uniformly in a ∈ R. To see this, choose A > 0 so that N so that
|u|>A
|B(u)|du < , and choose
< for all n > N, u ∈ [−A, A]. | gn (u) − G(u)| Then for all n > N, ∞ ∞ B(x − a)gn (x)dx − B(x − a)G(x)dx −∞ −∞ ∞ ∞ −iau gn (u)du − G(u)du e e−iau B(u) B(u) = −∞ −∞ A |B(u)| | gn (u)|du + | gn (u) − G(u)| |B(u)|du ≤ −A |u|>A 1 ), |B(u)| |G(u)|du < (2 + ||B|| + |x|>A
∞ on noting that | gn (u)| ≤ −∞ gn (u)du = 1. Take z < 0. Choose N so that for all n > N, gn is increasing on (−∞, z] and ∞ ∞ B(x − a)gn (x)dx − B(x − a)G(x)dx < −∞
−∞
for all a ∈ R. For y ≤ z, n > N , ∞ B(x − y + δ)gn (x)dx = −∞
y
B(x − y + δ)gn (x)dx
y−δ y
≤
B(x − y + δ)gn (y)dx y−δ
= gn (y) Also for n > N, ∞ −∞
B(x − y + δ)gn (x)dx >
∞
−∞ ∞
∞
−∞
B = gn (y).
B(x − y + δ)G(x)dx −
>
−∞
B(x − y + δ)G(y)dx − 2
= G(y) − 2.
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LOUIS H. Y. CHEN, TIM N. T. GOODMAN, AND S. L. LEE
Thus gn (y) > G(y)−2 for all n > N. Similarly, for y +δ ≤ z, gn (y) < G(y)+2 for all n > N. Thus gn converges to G uniformly on (−∞, z − δ]. A similar argument holds for z > 0, and so gn converges to G uniformly outside any open interval containing 0. Once again take > 0 and choose δ > 0 so that |G(x)−G(y)| < 2 for |x−y| ≤ 2δ. Choose N so that for n > N , |gn (x) − G(x)| < 2 for all |x| ≥ δ, and max gn (x) occurs only for x in (−δ, δ). Take any n > N and x in (−δ, δ). Then either gn (x) ≥ gn (−δ) or gn (x) ≥ gn (δ). Now gn (δ) > G(δ) − 2 > G(x) − and similarly gn (−δ) > G(x) − . Thus gn (x) > G(x) − . So we have shown that for any > 0, there exists an integer N such that for all n > N and all x ∈ R, gn (x) > G(x) − . Now suppose that gn does not converge uniformly to G on R. Then there is a number k > 0 and a sequence (xn ) with lim xn = 0 so that for arbitrarily large n, (5.3)
gn (xn ) > G(xn ) + k and log gn (xn ) > log G(xn ) + k.
Choose points 0 < a < a + h < a + 2h < 1. Then 2G(a + h) > G(a) + G(a + 2h) and 2G(−a − h) > G(−a) + G(−a − 2h). So for large enough n, (5.4) (5.5)
2gn (a + h) > gn (a) + gn (a + 2h), 2gn (−a − h) > gn (−a) + gn (−a − 2h).
Next choose 0 < 2δ < a so that |G(x) − G(y)| < k/3 whenever |x − y| ≤ δ. For large enough n, xn + 2δ < a and xn + δ/2 > 0. Since gn → G uniformly on [δ/2, ∞) and |G(xn + δ) − G(xn + 2δ)| < k/3, we have for large enough n, (5.6)
|gn (xn + δ) − gn (xn + 2δ)|
−a, xn + 2δ < a, and (5.4), (5.5), and (5.8) are satisfied. Let α, β be such that gn is convex on (−∞, α] and [β, ∞) and concave on [α, β]. By (5.4) and (5.5), β > a and α < −a. So gn is concave on [−a, a], which contradicts (5.8). Next suppose that gn is logconcave. A similar (but simpler) argument to that above shows that (5.8) can be replaced by 2 log gn (x + δ) < log gn (xn ) + log gn (xn + 2δ), which again gives a contradiction. the uniform convergence of gn to G on R and the condition ∞ We remark that ∞ g = 1 = G imply that gn → G in Lp (R) as n → ∞ for all p, 1 ≤ p ≤ n −∞ −∞ ∞. Since convergence in L1 (R) implies (5.1), the converse of Lemma 5.1 also holds.
ASYMPOTIC NORMALITY OF SCALING FUNCTIONS
341
From Lemma 5.1 we now derive the uniform convergence of m n to G under an extra condition on the numbers (rn,j ) as in Theorem 1.4. Theorem 5.2. We assume the conditions of Theorem 1.4. Then
k − µn lim σn an,k − G (5.9) =0 n→∞ σn uniformly over k in Z. Proof. Since all rn,j lie in the sector | arg z| ≤ π3 , it follows that the matrix (an,i−j ) is totally positive of order 2. Hence an,k ≥ 0 and (5.10)
k = 1, . . . , n − 1, n = 1, 2, . . . .
a2n,k ≥ an,k−1 an,k+1 ,
For n = 1, 2, . . . , we define ψn as follows. Without loss of generality we may assume an,0 an,n = 0, and it follows from (5.10) that an,k > 0, k = 0, . . . , n. We define ψn on [−µn /σn , (n − µn )/σn ] to be the piecewise linear function with knots (j − µn )/σn , j = 0, . . . , n, satisfying j − µn = log(σn an,j ), j = 0, 1, . . . , n. ψn σn From (5.10), ψn is concave on [−µn /σn , (n − µn )/σn ] . We now extend ψn to a continuous concave function on (α, β), where α = −(µn + 1)/σn , β = (n − µn + 1)/σn , and lim ψn (x) = lim ψn (x) = −∞. x→β −
x→α+
For n = 1, 2, . . . , we define
gn (x) =
eψn (x) , α < x < β, 0 otherwise.
Clearly, gn is logconcave, and j − µn gn = σn an,j , σn
j = 0, 1, . . . , n.
As in Theorem 1.2, we define measures mn , n = 1, 2, . . . , by mn ({k}) = an,k ,
k = 0, 1, . . . , n,
and it follows that (mn ) is asymptotically normal. We note that for k ∈ Z,
k−µn σn
−∞
dm n =
k
an,j ,
j=0
where we put an,j = 0 for j > n. It follows from (5.15) that as n → ∞, x x gn − dm n = O(σn−1 ) −∞
−∞
∞ uniformly in x. We can then apply Lemma 5.1 to the sequence of functions gn / −∞ gn to show that this sequence converges to G on R. Hence gn converges uniformly to G on R, which by (5.15) gives (5.9).
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LOUIS H. Y. CHEN, TIM N. T. GOODMAN, AND S. L. LEE
We now consider the uniform convergence of the normalized mn -scaling functions φn to G. Theorem 5.3. Assume the conditions of Theorem 1.5. Then φn → G as n → ∞ uniformly on R. Proof. It follows from the work of Goodman and Micchelli (see [8]) and the properties of totally positive matrices (see [2]) that the functions φn , and hence φn , are bell-shaped. The result then follows from Theorem 1.1, Theorem 1.2, and Lemma 5.1. We remark that if the set of all rn,j lies in Re z ≥ 0, then the condition that it also lies in Dγ for some γ ∈ [0, π2 ) is equivalent to requiring that for some β ∈ [0, π2 ) the set of all rn,j lying outside the sector | arg z| ≤ β is bounded and bounded away from zero. In [4], Chui and Wang consider convergence (φn ) as in
nof the sequence k Theorem 5.3 under the assumption that the polynomial k=0 an,k z is reciprocal and all rn,j are real and positive. They also assume that for n = 1, 2, . . . , rn,j = 1 for at least Kn values of j for some fixed K > 0. They prove convergence in Lp , 1 ≤ p < ∞, which we have noted is weaker than uniform convergence. We shall finish the paper by considering the order of uniform convergence for both the measures and the corresponding scaling functions. We first need to extend concepts of bell-shaped and logconcave to discrete measures. Suppose m is a probability measure on Z with m({j}) = aj , j ∈ Z. We say m is bell-shaped if there are integers k ≤ such that 2aj ≤ aj−1 + aj+1 , 2aj ≥ aj−1 + aj+1 ,
j ≤ k − 1 and j ≥ + 1, k ≤ j ≤ .
We say m in logconcave if a2j ≥ aj−1 aj+1 ,
j ∈ Z.
Lemma 5.4. For n = 1, 2, . . . , let mn be a probability measure on {0, 1, . . . , n} given by mn ({k}) = an,k , k = 0, 1, . . . , n, which is either bell-shaped or logconcave, with mean µn and standard derivation σn . Suppose that for some K > 0 and r ≥ 1, −u2 /2 −r (5.11) n (u) − e ≤ Kσn for |u| ≤ Kσn . m Then as n → ∞, (5.12)
k − µn −s max σn an,k − G = O(σn ), k=0,... ,n σn
where s = min{ 2r , 23 }. ∞ Proof. Take a nonnegative function N with support in [−1, 1], −∞ N = 1, 1 < ∞, and for some A > 0, N (u)| ≤ A(1 + |u|)−3r−1 , |N
u ∈ R.
Take 0 < δ < 1/2. Let B1 (x) := δ −1 N (x/δ) and B2 (x) := δ −1 (x/δ − 4). Then B1 N ∞ ∞ and B2 have supports on [−δ, δ] and [3δ, 5δ], respectively, and −∞ B1 = −∞ B2 = 1. So ∞ ∞ B1 G > G(δ), B2 G < G(3δ). −∞
−∞
343
ASYMPOTIC NORMALITY OF SCALING FUNCTIONS
and hence
∞
−∞
B1 G −
∞
−∞
B2 G > G(δ) − G(3δ) > |G (δ)|2δ > |G (δ)|2δ 2 > |G (1/2)|2δ 2 .
Also for j = 1, 2, ∞ Bj dm n − Bj G = n (u) − G(u))du Bj (−u)(m −∞ −∞ −∞ j (−u)|du (|m n (u)| + G(u))| B ≤
∞
∞
|u|≥Kσn
+ ≤2 = ≤
2 δ
K σnr
|u|>Kσn
Kσn
−Kσn
j (u)|du |B
(δu)|du + |N
|u|≥Kδσn
(u)|du + |N
K σnr
K δσnr
∞
(δu)|du |N
−∞ ∞
−∞
(u)|du |N
C C + r δ(Kδσn )3r δσn
for some C > 0. Choosing δ = cσnβ−1 for some 13 ≤ β < 1, and c > 1, gives ∞ ∞ D (5.13) Bj dm n − Bj G ≤ r+β−1 cσn −∞ −∞ for some D > 0. Then ∞ ∞ D B1 dm n > B1 G − r+β−1 cσn −∞ −∞ ∞ 1 D 2 B2 G + G > 2δ − r+β−1 2 cσn −∞ ∞ |G ( 12 )|2c2 2D (5.14) > B2 dm n + − r+β−1 . 2−2β σn cσn −∞ Now for n = 1, 2, . . . , choose a continuous function gn , which is bell-shaped or logconcave as mn is bell-shaped or logconcave, respectively, and satisfies j−µ gn = σn an,j , j = 0, 1, . . . , n. (5.15) σn If mn is logconcave, then this can be done as in the proof of Theorem 5.2, while if mn is bell-shaped we can take gn to be simply the piecewise linear interpolant. that if, for some constant b, gn ≥ b on the support of Bj , j = 1 or 2, then Note ∞ B dm n bounds the product of b and a Riemann sum for Bj over its support with −∞ j
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LOUIS H. Y. CHEN, TIM N. T. GOODMAN, AND S. L. LEE
interval length σn−1 . This Riemann sum equals a Riemann sum for N over [0, 1] with 1 interval length δ −1 σn−1 , which differs from 0 N by O(δ −2 σn−2 ). Thus, by the uniform boundedness of gn , we have Bj dm n ≥ b + O(δ −2 σn−2 ), and similarly the result holds with ≥ replaced by ≤ . Thus if gn (x) ≤ gn (y) for all x ∈ [−δ, δ], y ∈ [3δ, 5δ], we have ∞ ∞ ∞ a a B2 dm n + B2 dm n + 2 2 = B1 dm n ≤ 2 δ σ c σn2β −∞ −∞ −∞ n for a fixed constant a. Choosing β = 2/3 and c large enough, this would contradict (5.14), and so there are points −δ < bn < δ, 3δ < cn < 5δ with gn (bn ) > gn (cn ). Similarly, we can choose bn so that there is a point an in (−5δ, −3δ) with gn (an ) < gn (bn ). As in the proof of Lemma 5.1, it follows from the shape of gn that the maximum of gn (x) occurs only for x in (−5δ, 5δ). So we have shown that for a constant a, −1/3 1/3 maximum of gn (x) occurs for x in (−aσn , aσn ) for n = 1, 2, . . . . −1/3 Now take δ = σnβ−1 for some 1/3 ≤ β < 1 and γ ≥ aσn + δ. Let B(x) = −1 −1 δ N (δ (x − γ)) so that B has support on [γ − δ, γ + δ]. As in (5.13) ∞ ∞ D Bdm n − BG ≤ r+β−1 σn −∞ −∞ −1/3
for some D > 0. Since gn is decreasing on [aσn , ∞), for a constant b > 0, ∞ b Bdm n − 2 2 gn (γ − δ) ≥ δ σn −∞ ∞ b D ≥ BG − 2 2 − r+β−1 σ δ σn −∞ n b D ≥ G(γ + δ) − 2β − r+β−1 . σn σn Since |G (τ )| < 1 for all τ in R, |G(x) − G(y)| ≤ |x − y| for all x, y ∈ R. So G(γ + δ) ≥ G(γ − δ) − 2δ, and so gn (γ − δ) ≥ G(γ − δ) −
b σn2β
−
D σnr+β−1
− 2δ.
Similarly, gn (γ + δ) ≤ G(γ + δ) + −1/3
Thus for all x ≥ aσn
b σn2β
+
D σnr+β−1
+ 2δ.
+ 2δ,
|gn (x) − G(x)| ≤
b σn2β
+
D σnr+β−1
+
For r ≥ 4/3, put β = 1/3 to give −2
|gn (x) − G(x)| = 0(σn 3 ).
2 σn1−β
.
ASYMPOTIC NORMALITY OF SCALING FUNCTIONS
345
For 1 ≤ r ≤ 4/3, put β = 1 − r/2 to give −r
|gn (x) − G(x)| = 0(σn 2 ). −1
Similarly, the result holds for x ≤ −aσn 3 − 2δ. Thus for a constant b > a, −1
sup{|gn (x) − G(x)| : |x| ≥ bσn 3 } = O(σn−s )
(5.16)
for s as in the statement of Lemma 5.4. Note that for any δ > 0 and x, y ∈ (−δ, δ), |G(x) − G(y)| ≤ |G (0)|δ|x − y| ≤ δ|x − y|.
(5.17)
−1
−1
Take any x ∈ (−bσn 3 , bσn3 ). Then either −1
−1
gn (x) ≥ gn (bσn 3 ) or gn (x) ≥ gn (−bσn 3 ). Suppose the former. Then −1
−1
gn (x) ≥ gn (bσn 3 ) > G(bσn 3 ) − O(σn−s ) −1
> G(x) − O(σn−s ) − 2(bσn 3 )2 . The same holds similarly for the latter case. Thus −1
sup{gn (x) − G(x) : |x| ≤ bσn 3 } = O(σn−s ).
(5.18)
Now note, as in the proof of Lemma 5.1, that if gn is bell-shaped, then for all large enough n, gn is concave on [− 23 , 23 ]. Since concavity implies logconcavity, gn is logconcave on [− 23 , 23 ] for all large enough n. By (5.16) and the mean value theorem,
2 −1 = O(σn−s ). sup | log gn (x) − log G(x)| : bσn 3 ≤ |x| ≤ 3 −1
−1
Take 0 ≤ x ≤ bσn 3 and n so large that bσn 3 ≤ 29 . Then −1
−1
log gn (x) ≤ 2 log gn (x + bσn 3 ) − log gn (x + 2bσn 3 ) −1
−1
≤ 2 log G(x + bσn 3 ) − log G(x + 2bσn 3 ) + O(σn−s ) −1
≤ log G(x) + | log G(x + bσn 3 ) − log G(x)| −1
−1
+| log G(x + bσn 3 ) − log G(x + 2bσn 3 )| + O(σn−s ) −2
≤ log G(x) + O(σn 3 ) + O(σn−s ). −1
A similar argument holds for −bσn 3 ≤ x ≤ 0, and applying the mean value theorem gives (5.19)
−1
sup{G(x) − gn (x) : |x| ≤ bσn 3 } = O(σn−3 ).
Combining (5.16), (5.18), and (5.19) and recalling (5.15) then gives the result. Proof of Theorem 1.4. Theorem 1.4 follows from Proposition 4.1 and Lemma 5.4.
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LOUIS H. Y. CHEN, TIM N. T. GOODMAN, AND S. L. LEE
To consider the order of uniform convergence for the scaling functions, we need the following analogue of Lemma 5.4. This can be proved in a similar manner to Lemma 5.4, but the proof is simpler, in particular because there is no restriction on the range of u as in (5.11). Lemma 5.5. Suppose that (gn ) is a sequence of continuous functions, which are ∞ 2 gn (u) − e−u /2 ∞ < αn for either bell-shaped or logconcave with −∞ gn = 1 and n = 1, 2, . . . , where limn→∞ αn = 0. Then as n → ∞, 1
gn − G ∞ = O(αn2 ). Proof of Theorem 5.5.
1.5.
Theorem 1.5 follows from Theorem 1.3 and Lemma
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