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Definition 1. Let be the convex cone of integrable, monotone non- increasing functions on (0F,π].
Given a stationary sequence Xk with spectral density f, the goal is to estimate f under the assumption{ that} it lies in . We suggest two estima- tors, which are the L2 orthogonal projections onF the convex cone of the periodogram and of the log-periodogram, respectively. F 4 D. ANEVSKI AND P. SOULIER
(i) The L2 minimum distance estimate between the periodogram and is defined as F
(1) fˆn = argmin Q(z), z∈F with π Q(z)= (I (s) z(s))2 ds. n − Z0 This estimator of the spectral density naturally yields a corresponding esti- mator Fˆn of the spectral distribution function F , defined by t (2) Fˆn(t)= fˆn(s) ds. Z0 (ii) The L2 minimum distance estimate between the log-periodogram (often called the cepstrum) and the “logarithm of ” is defined as F (3) f˜n = expargmin Q˜(z), z∈F with π Q˜(z)= log I (s)+ γ log z(s) 2 ds, { n − } Z0 where γ is Euler’s constant. To understand the occurrence of the centering 2 γ, recall that if Xn is a Gaussian white noise sequence with variance σ , − { } 2 2 then its spectral density is σ /(2π) and the distribution of In(s)/(σ /2π) is a standard exponential (i.e., one half of a chi-square with two degrees of freedom), and it is well known that if Z is a standard exponential, then E[log(Z)] = γ and var(log Z)= π2/6 [see, e.g., Hurvich, Deo and Brodsky (1998)]. The− log-spectral density is of particular interest in the context of long-range dependent time series, that is, when the spectral density has a singularity at some frequency and might not be square integrable, though it is always integrable by definition. For instance, the spectral density of an ARFIMA(0, d, 0) process is f(x)= σ2 1 eix −2d, with d ( 1/2, 1/2). It is decreasing on (0,π] for d (0, 1/2)| and− not| square integrable∈ − for d ∈ ∈ (1/4, 1/2). In this context, for technical reasons, we will take In to be a step function changing value at the so-called Fourier frequencies λk = 2πk/n. The paper is organized as follows: in Section 2 we derive the algorithms for the estimators fˆn, Fˆn and f˜n. In Section 3 we derive a lower bound for the asymptotic local minimax risk in monotone spectral density estimation and show that the rate is not faster than n−1/3. In Section 4 we derive the pointwise limit distributions for the proposed estimators. The limit dis- tribution of fˆn (suitably centered and normalized) is derived for a linear MONOTONE SPECTRAL DENSITY ESTIMATION 5 process. The asymptotic distribution is that of the slope of the least concave majorant at 0 of a quadratic function plus a two-sided Brownian motion. Up to constants, this distribution is the so-called Chernoff’s distribution [see Groeneboom and Wellner (2001)] which turns up in many situations in monotone function estimation [see, e.g., Prakasa Rao (1969) for monotone density estimation and Wright (1981) for monotone regression function esti- mation]. The limit distribution for f˜n is derived for a Gaussian process and is similar to the result for fˆn. Section 5 contains a simulation study with plots of the estimators. Section 6 contains the proofs of the limit distribution results (Theorems 5 and 6).
2. Identification of the estimators. Let h be a function defined on a com- pact interval [a, b]. The least concave majorant T (h) of h and its derivative T (h)′ are defined by T (h) = argmin z : z x, z concave , { ≥ } h(v) h(u) T (h)′(t)=min max − . u
Lemma 2. Let g L2([0,π]). Let G be defined on L2([0,π]) by ∈ π G(f)= f g 2 = f(s) g(s) 2 ds. k − k2 { − } Z0 ′ Then arg minf∈F G(f)= T (¯g) .
This result seems to be well known. It is cited, for example, in Mammen [(1991), page 726] but since we have not found a proof, we give one for completeness. Let G : R be an arbitrary functional. It is called Gateaux differen- tiable at theF 7→ point f if the limit ∈ F ′ G(f + th) G(f) Gf (h) = lim − t→0 t exists for every h such that f + th for small enough t. ∈ F Proof of Lemma 2. Denote G(f)= f g 2 and fˆ = T (¯g)′. The k − k2 Gateaux derivative of G at fˆ in the direction h is π G′ (h) = 2 h(t) fˆ(t) g(t) dt. fˆ { − } Z0 By integration by parts, and using that T (¯g)(π) g¯(π)= T (¯g)(0) g¯(0) = 0, for any function of bounded variation h, we have− − π (4) G′ (h)= 2 T (¯g)(t) g¯(t) dh(t). fˆ − { − } Z0 By Lemma 1, the support of the measure dfˆ is included in the closed set T (¯g) =g ¯ , and thus { } π (5) G′ (fˆ)= 2 T (¯g)(t) g¯(t) dfˆ(t) = 0. fˆ − { − } Z0 If h = f fˆ, with f monotone nonincreasing, (4) and (5) imply that − π (6) G′ (f fˆ)= 2 T (¯g)(t) g¯(t) df(t) 0. fˆ − − { − } ≥ Z0 Let f be arbitrary, and let u be the function defined on [0, 1] by u(t)= ∈ F G(fˆ+ t(f fˆ)). Then u is convex and u′(0) = G′ (f fˆ) 0 by (6). Since − fˆ − ≥ u is convex, if u′(0) 0, then u(1) u(0), that is, G(f) G(fˆ). This proves ≥ ≥ ≥ that fˆ= argminf∈F G(f).
2 Since fˆn and log f˜n are the minimizers of the L distance of In and log(In)+ γ, respectively, over the convex cone of monotone functions, we can apply Lemma 2 to derive characterizations of fˆn and f˜n. MONOTONE SPECTRAL DENSITY ESTIMATION 7
Theorem 3. Let fˆn, Fˆn and f˜n be defined in (1), (2) and (3), respec- tively. Then ′ fˆn = T (Fn) ,
Fˆn(t)= T (Fn), f˜ = exp T (F˜ )′ , n { n } where t Fn(t)= In(u) du, Z0 t F˜ (t)= log I (u)+ γ du. n { n } Z0 Standard and well-known algorithms for calculating the map y T (y)′ are the pool adjacent violators algorithm (PAVA), the minimum lower7→ set algorithm (MLSA) and the min–max formulas; cf. Robertson, Wright and Dykstra (1988). Since the maps T and T ′ are continuous operations, in fact the algorithms PAVA and MLSA will be approximations that solve the discrete versions of our problems, replacing the integrals in Q and Q˜ with approximating Riemann sums. Note that the resulting estimators are order-restricted means; the discrete approximations entail that these are approximated as sums instead of integrals. The approximation errors are similar to the ones obtained, for example, for the methods in Mammen (1991) and Anevski and H¨ossjer (2006).
3. Lower bound for the local asymptotic minimax risk. We establish a lower bound for the minimax risk when estimating a monotone spectral den- sity at a fixed point. This result will be proved by looking at parametrized subfamilies of spectral densities in an open set of densities on Rn; the sub- families can be seen as (parametrized) curves in the set of monotone spectral densities. The topology used will be the one generated by the metric
ρ(f, g)= f(x) g(x) dx R | − | Z for f, g spectral density functions on [ π,π]. Note first that the distribution of a stochastic process is not uniquely− defined by the spectral density. To accomodate this, let g be the set of all laws of stationary processes (i.e., the translation invariantL probability distributions on R∞) with spectral den- sity g. Let ε> 0, c1, c2 be given finite constants, and let t0 > 0, the point at which we want to estimate the spectral density, be given. 8 D. ANEVSKI AND P. SOULIER
1 1 Definition 2. For each n Z let := (ε, c1, c2,t0) be a set of mono- tone C1 spectral densities g on∈ [0,π],G such thatG (7) sup g′(t) < 0, |t−t0|<ε
(8) c1 < inf g(t) < sup g(t) < c2. |t−t0|<ε |t−t0|<ε
Theorem 4. For every open set U in 1 there is a positive constant c(U) such that G
2/3 2 lim inf inf sup sup n EL[(Tn g(t0)) ] c(U), n→∞ Tn g∈U L∈Lg − ≥ where the infimum is taken over all functions Tn of the data.
Proof. Let k be a fixed real valued continuously differentiable function, with support [ 1, 1] such that k(t) dt = 0, k(0)=1 and sup k(t) 1. Then, since k′ is continuous− with compact support, k′
E (T g (t ))2λ(θ) dθ. ≥ θ n − θ 0 ZΘ MONOTONE SPECTRAL DENSITY ESTIMATION 9
Then, by the Van Trees inequality, we obtain
2 1 (9) Eθ(Tn gθ(t0)) λ(θ) dθ , − ≥ I (θ)λ(θ) dθ + I˜(λ) ZΘ n where R I (θ)= 1 tr( M −1(g )M (∂ g ) 2) n 2 { n θ n θ θ } is the Fisher information matrix [cf. Dzhaparidze (1986)] with respect to the parameter θ of a Gaussian process with spectral density gθ, and for any even nonnegative integrable function φ on [ π,π], Mn(φ) is the Toeplitz matrix of order n − π M (φ) = φ(x) cos((i j)x) dx. n i,j − Z−π For any n n nonnegative symmetric matrix A, define the spectral radius of A as × ρ(A)=sup utAu utu = 1 , { | } where ut denotes transposition of the vector u, so that ρ(A) is the the largest eigenvalue of A. Then, for any n n matrix B, × tr(AB) ρ(A) tr(B). ≤ If φ is bounded away from zero, say φ(x) a> 0 for all x [ π,π], then ≥ ∈ − ρ(M −1(φ)) a−1. n ≤ By the Parseval–Bessel inequality, π tr( M (φ) 2) n φ2(x) dx. { n } ≤ Z−π Thus, if g is bounded below, then In(θ) is bounded by some constant times π n k2((t t )/h) dt = nh k2(t) dt. − 0 Z−π Z In order to get an expression for I˜(λ), let λ0 be an arbitrary density on 1 θ ( 1, 1), and define the prior density on Θ = ( δh, δh) as λ(θ)= δh λ0( δh ). Then− − δh (λ′(θ))2 1 1 λ′ (u)2 I I˜(λ)= dθ = 0 du = 0 . λ(θ) δ2h2 λ (u) δ2h2 Z−δh Z−1 0 Finally, plugging the previous bounds into (9) yields, for large enough n,
E 2 1 sup sup L(Tn(t0) g(t0)) −2 −2 , g∈U L∈Lg − ≥ nhc3 + I0δ h 10 D. ANEVSKI AND P. SOULIER which, if h = n−1/3, becomes 2 −2/3 sup sup EL[ Tn(t0) g(t0) ] c4n , g∈U L∈Lg { − } ≥ for some positive constant c4. This completes the proof of Theorem 4.
4. Limit distribution results. We next derive the limit distributions for fˆn and f˜n under general assumptions. The main tools used are local limit distributions for the rescaled empirical spectral distribution function Fn and empirical log-spectral distribution function F˜n, respectively, as well as max- imal bounds for the rescaled processes. These will be coupled with smooth- ness results for the least concave majorant map established in Anevski and H¨ossjer (2006), Theorems 1 and 2. The proofs are postponed to Section 6.
4.1. The limit distribution for the estimator fˆn.
Assumption 1. The process Xi, i Z is linear with respect to an i.i.d. sequence ε , i Z with zero mean{ and∈ unit} variance, that is, { i ∈ } ∞ (10) Xk = ajεk−j, Xj=0 where the sequence a satisfies { j} ∞ (11) (j1/2 a + j3/2a2) < . | j| j ∞ Xj=1 Remark 1. Condition (11) is needed to deal with remainder terms and apply the results of Mikosch and Norvaiˇsa (1997) and Brockwell and Davis (1991). It is implied, for instance, by the simpler condition ∞ (12) j3/4 a < . | j| ∞ Xj=1 It is satisfied by most usual linear time series such as causal invertible ARMA processes.
The spectral density of the process X is given by { i} ∞ 2 1 iju f(u)= aje . 2π j=0 X Unfortunately, there is no explicit condition on the coefficients a that im- j plies monotonicity of f, but the coefficients aj are not of primary interest here. MONOTONE SPECTRAL DENSITY ESTIMATION 11
The limiting distribution of the estimator will be expressed in terms of the so-called Chernoff distribution, that is, the law of a random variable ζ 2 defined by ζ = argmaxs∈R W (s) s , where W is a standard two-sided Brownian motion. See Groeneboom{ − and} Wellner (2001) for details about this distribution.
Theorem 5. Let Xi be a linear process such that (10) and (11) hold E 8 { } and [ε0] < . Assume that its spectral density f belongs to . Assume f ′(t ) < 0 at∞ the fixed point t . Then, as n , F 0 0 →∞ L n1/3(fˆ (t ) f(t )) 2 πf 2(t )f ′(t ) 1/3ζ. n 0 − 0 → {− 0 0 }
4.2. The limit distributions for the estimator f˜n. In this section, in order to deal with the technicalities of the log-periodogram, we make the following assumption.
Assumption 2. The process Xk is Gaussian. Its spectral density f is monotone on (0,π] and can be expressed{ } as f(x)= 1 eix −2df ∗(x), | − | with d < 1/2 and f ∗ is bounded above and away from zero and there exists a constant| | C such that for all x,y (0,π], ∈ x y f(x) f(y) C | − |. | − |≤ x y ∧ Remark 2. This condition is usual in the long memory literature. Simi- lar conditions are assumed in Robinson (1995), Assumption 2, Moulines and Soulier (1999), Assumption 2, Soulier (2001), Assumption 1 (with a typo). It is used to derive covariance bounds for the discrete Fourier transform ordi- nates of the process, which yield covariance bounds for nonlinear functionals of the periodogram ordinates in the Gaussian case. It is satisfied by usual long memory processes such as causal invertible ARFIMA(p,d,q) processes with possibly an additive independent white noise or AR(1) process.
Recall that π f˜ = expargmin log f(s) log I (s)+ γ 2 ds, n { − n } f∈F Z0 where γ is Euler’s constant and In is the periodogram, defined here as a step function n 2 2π i2kπ[nt/2π]/n In(t)= In(2π[nt/2π]/n)= Xke . n k=1 X
12 D. ANEVSKI AND P. SOULIER
Theorem 6. Let Xi be a Gaussian process that satisfies Assumption 2. Assume f ′(t ) < 0 at{ the} fixed point t (0,π). Then, as n , 0 0 ∈ →∞ 4 ′ 1/3 L π f (t ) n1/3 log f˜ (t ) log f(t ) 2 − 0 ζ. { n 0 − 0 } → 3f(t ) 0 Corollary 7. Under the assumptions of Theorem 6,
n1/3 f˜ (t ) f(t ) L 2 π4f 2(t )f ′(t )/3 1/3ζ. { n 0 − 0 } → {− 0 0 } Remark 3. This is the same limiting distribution as in Theorem 5, up −1/3 to the constant 3 π> 1. Thus the estimator f˜n is less efficient than the estimator fˆn, but the interest of f˜n is to be used when long memory is suspected, that is, the spectral density exhibits a singularity at zero, and the assumptions of Theorem 5 are not satisfied.
5. Simulations and finite sample behavior of estimators. In this section we apply the nonparametric methods on simulated time series data of sums of parametric models. The algorithms used for the calculation of fˆn and f˜n are the discrete versions of the estimators f,ˆ f˜n, that are obtained by doing isotonic regression of the data (λk,In(λk)), k = 1,..., [(n 1)/2] where { ˆd ˆ − } λk = 2πk/n. For instance, the discrete version fn of fn is calculated as n ˆd 2 fn = argmin (In(λk) z(λk)) . z∈F − Xk=1 ˜ ˜d Note that the limit distribution for fn is stated for the discrete version fn. The simulations were done in R, using the “fracdiff” package. The code is available from the corresponding author upon request.
Example 1. The first example consists of sums of several AR(1) pro- cesses. Let X be a stationary AR(1) process, that is, for all k Z, { k} ∈ Xk = aXk−1 + εk, with a < 1. This process has spectral density function f(λ) = (2π)−1σ2 1 iλ −| 2| 2 2 | − ae for π λ π, with σ = var(ε1) and and thus f is decreasing on | (1)− ≤ ≤(p) [0,π]. If X ,...,X are independent AR(1) processes with coefficients aj such that a < 1, j = 1,...,p, and we define the process X by | j| p (j) Xk = Xk , Xj=1 then X has spectral density f(λ) = (2π)−1 p σ2 1+ a eiλ −2 which is de- j=1 j | j | creasing on [0,π], since it is a sum of decreasing functions. Assuming that P MONOTONE SPECTRAL DENSITY ESTIMATION 13 we do not know how many AR(1) processes are summed, we have a non- parametric problem: estimate a monotone spectral density. Figure 1 shows a plot of the periodogram, the true spectral density and the nonparametric estimators fˆn and f˜n for simulated data from a sum of three independent
Fig. 1. The spectral density (red), the periodogram (black), the estimates fˆn (green) and f˜n (yellow), for n = 100, 500, 1,000 and 5,000 data points, for Example 1.
Fig. 2. Left plot: spectral density (black), pointwise mean of estimates fˆn (red) and 95% confidence intervals (green). Right plot: spectral density (black), pointwise mean of the estimates f˜n (red) and 95% confidence intervals (green), for n = 1,000 data points, for Example 1. 14 D. ANEVSKI AND P. SOULIER
Table 1 MISE values for Example 1
MISE n = 100 n = 500 n = 1,000 n = 5,000
In 9.59 12.96 13.67 14.25 fˆn 6.38 5.48 4.76 2.95 f˜n 9.11 8.52 7.27 4.26
Table 2 MISE values for Example 2
MISE n = 100 n = 500 n = 1,000 n = 5,000
In 1.80 1.99 2.02 2.07 fˆn 0.710 0.520 0.432 0.305 f˜n 1.12 0.803 0.659 0.472
AR(1) processes with a1 = 0.5, a2 = 0.7, a3 = 0.9. Figure 2 shows the point- wise means and 95% confidence intervals of fˆn and f˜n for 1,000 realizations.
Example 2. The second example is a sum of an ARFIMA(0, d, 0) pro- cess and an AR(1) process. Let X(1) be an ARFIMA(0, d, 0)-process with −1 2 iλ −2d 0 Table 1 shows mean square root of sum of squares errors (comparing with the true function), calculated on 1,000 simulated samples of the times series of Example 1. Table 2 shows the analog values for Example 2. Both estimators fˆn and f˜n seem to have good finite sample properties. As indicated by the theory, f˜n seems to be less efficient than fˆn. 6. Proofs of Theorems 5 and 6. Let Jn be the integral of the generic preliminary estimator of the spectral density, that is the integral of In or of MONOTONE SPECTRAL DENSITY ESTIMATION 15 Fig. 3. The spectral density (red), the periodogram (black), the estimates fˆn (green) and f˜n (yellow), for n = 100, 500, 1,000 and 5,000 data points, for Example 2. Fig. 4. Left plot: spectral density (black), pointwise mean of estimates fˆn (red) and 95% confidence intervals (green). Right plot: spectral density (black), pointwise mean of the estimates f˜n (red) and 95% confidence intervals (green), for n = 1,000 data points, for Example 2. log(In), let K denote F or the primitive of log f, respectively, and write (13) Jn(t)= K(t)+ vn(t). 16 D. ANEVSKI AND P. SOULIER Let dn 0 be a deterministic sequence, and define the rescaled process and rescaled↓ centering (14) v˜ (s; t )= d−2 v (t + sd ) v (t ) , n 0 n { n 0 n − n 0 } (15) g (s)= d−2 K(t + sd ) K(t ) K′(t )d s . n n { 0 n − 0 − 0 n } Consider the following conditions: (AH1) There exists a stochastic processv ˜( ; t ) such that · 0 (16) v˜ ( ; t ) L v˜( ; t ), n · 0 → · 0 in D( , ), endowed with the topology generated by the supnorm metric−∞ on compact∞ intervals, as n . (AH2) For each ε,δ > 0 there is a finite τ→∞such that v˜ (s; t ) (17) lim supP sup n 0 > ε < δ, g (s) n→∞ |s|≥τ n v˜(s; t ) (18) P sup 0 ε < δ. s2 |s|≥τ (AH3) There is a constant A< 0 such that for each c> 0, 2 (19) lim sup gn(s) As = 0; n→∞ |s|≤c | − | (AH4) For each a R and c,ε> 0 ∈ (20) P (˜v(s; t )(s) v˜(0; t )+ As2 as ε s for all s [ c, c]) = 0. 0 − 0 − ≥ | | ∈ − If there exists a sequence dn such that these four conditions hold, then, defin- 2 ing the process y by y(s) =v ˜(s; t0)+ As , by Anevski and H¨ossjer [(2006), Theorems 1 and 2] as n , it holds that →∞ L (21) d−1 T (J )′(t ) K′(t ) T (y)′(0), n { n 0 − 0 } → where T (y)′(0) denotes the slope at zero of the smallest concave majorant of y. 6.1. Proof of Theorem 5. The proof consists in checking conditions (AH1)– (AH4) with Jn = Fn and K = F . - It is proved in Lemma 8 below that (16) holds with d = n1/3 andv ˜( ; t ) n · 0 the standard two-sided Brownian motion times π2/6. - If f ′(t ) < 0, then (19) holds with A = 1 f ′(t ) and d 0 an arbitrary 0 2 0 p n deterministic sequence. ↓ - Lemma 9 shows that (17) holds and the law of iterated logarithm yields that (18) holds for the two-sided Brownian motion. MONOTONE SPECTRAL DENSITY ESTIMATION 17 - Finally, (20) also holds for the two-sided Brownian motion. Thus (21) holds with the process y defined by 1 ′ 2 √ y(s)= 2 f (t0)s + 2πf(t0)W (s). The scaling property of the Brownian motion yields the representation of T (y)′(0) in terms of Chernoff’s distribution. Lemma 8. Assume the process Xn is given by (10), that (11) holds and that E[ε8] < . If d = n−1/3,{ then} the sequence of processes v˜ ( ; t ) 0 ∞ n n · 0 defined in (14) converges weakly in C([ c, c]) to √2πf(t0)W where W is a standard two-sided Brownian motion. − Proof. For clarity, we omit t0 in the notation. Write ε v˜n(s) =v ˜n(s)+ Rn(s) with t0+dns (ε) −2 (ε) v˜n (s)= dn f(u) In (u) 1 du, t0 { − } (22) Z n 2 (ε) 1 iku In (u)= εke , n k=1 X t0+dns (23) R = d− 2 r (u) du, r (u)= I (u) f(u)I(ε)(u). n n n n n − n Zt0 −1 ε Note that (2π) In is the periodogram for the white noise sequence εi . We π 2 2 { } first treat the remainder term Rn. Denote = g : −π g (u)f (u) du < . Equation (5.11) (with a typo in the normalization)G { in Mikosch and Norvaiˇsa∞} (1997) states that if (11) and E[ε8] < hold, thenR 0 ∞ π (24) √n sup g(x)rn(x) dx = oP (1). g∈G Z−π Define the set ˜ = kn( ,s)f : n N,s [ c, c] . Since f is bounded, we have that k2 (u, s)Gf 2(u{) du· < , so∈ ˜ ∈ and− we} can apply (24) on ˜, which n ∞ G ⊂ G G shows that Rn converges uniformly (over s [ c, c]) to zero. We next show that R ∈ − (25) v˜(ε)(s) L √2πf(t )W (s), n → 0 as n , on C(R), where W is a standard two-sided Brownian motion. →∞ Since εk is a white noise sequence, we set t0 = 0 without loss of generality. Straightforward{ } algebra yields (26) v˜(ε)(s)= d−2 γˆ (0) 1 F (d s)+ S (s) n n { n − } n n 18 D. ANEVSKI AND P. SOULIER with n n −1 2 γˆn(0) = n εj , Sn(s)= Ck(s)εk, Xj=1 Xk=2 k−1 dns 3/2 −1/2 iju Ck(s)= dn αj(s)εk−j, αj(s)= dn f(u)e du. −dns Xj=1 Z Since εj is a white noise sequence with finite fourth moment, it is easily checked{ that} 2 n var(ˆγn(0)) = var(ε0), (27) dns −2 −1 −1/2 sup dn f(u) du γˆn(0) 1 = OP (dn n )= OP ( dn) s∈[−c,c] 0 | − | Z p so that the first term in (26) is negligible. It remains to prove that the sequence of processes Sn converges weakly to a standard Brownian motion. We prove the convergence of finite dimension distribution by application of the Martingale central limit theorem [cf., e.g., Hall and Heyde (1980), Corollary 3.1]. It is sufficient to check the following conditions: n (28) lim E[C2(s)] = 2πf 2(0)s, n→∞ k Xk=2 n (29) lim E[C4(s)] = 0. n→∞ k Xk=2 By the Parseval–Bessel identity, we have ∞ dns 2 −1 2 2 αj (s) = 2πdn f (u) du 4πf (0)s. −dns ∼ j=X−∞ Z Since α (s) 2f(0)√d , this implies that 0 ∼ n n n−1 ∞ E[C2(s)] = (1 j/n)α2(s) α2(s) 2πf 2(0)s. k − j ∼ j ∼ Xk=2 Xj=1 Xj=1 This proves condition (28). For the asymptotic negligibility condition (29), we use Rosenthal’s inequality [cf. Hall and Heyde (1980), Theorem 2.12], k−1 k−1 2 4 −2 4 −2 2 −2 E[Ck ] cstn αj (s)+ cstn αj (s) = O(n ), ≤ ! Xj=1 Xj=1 MONOTONE SPECTRAL DENSITY ESTIMATION 19 n E 4 −1 implying k=1 [Ck (s)] = O(n ), which proves (29). To prove tightness, we compute the fourth moment of the increments of Sn. Write P n k−1 S (s) S (s′)= n−1/2 α (s,s′)ε ε , n − n j k−j k Xk=1 Xj=1 with ′ dns −dns ′ −1/2 iju −1/2 iju αj(s,s )= dn f(u)e du + dn f(u)e du. ′ Zdns Z−dns By Parseval’s inequality, it holds that n α2(s,s′) C s s′ . j ≤ | − | Xj=1 ′ 4 Applying again Rosenthal’s inequality, we obtain that E[ Sn(s) Sn(s ) ] is bounded by a constant times | − | n n 2 −1 4 ′ 2 ′ ′ 2 n αj (s,s )+ αj (s,s ) C s s . ! ≤ | − | Xj=1 Xj=1 Applying [Billingsley (1968), Theorem 15.6] concludes the proof of tightness. Lemma 9. For any δ> 0 and any κ> 0, there exists τ such that v˜ (s) (30) lim supP sup | n | > κ δ. n→∞ s ≤ |s|≥τ | | Proof. Without loss of generality, we can assume that f(t0) = 1. Recall (ε) (ε) (ε) thatv ˜n =v ˜n + Rn andv ˜n (s)= F (dns)ζn + Sn(s), wherev ˜n and Rn are defined in (22) and (23), ζ = d−2(ˆγ (0) 1) and S is defined in (26). Then n n n − n v˜n(s) Sn(s) P sup | | > κ P sup ζn F (dns)/s > κ/3 + P sup | | > κ/3 s≥τ s ≤ s≥τ | | s≥τ s R (s) + P sup | n | > κ/3 . s s≥τ The spectral density is bounded, so F (dns)/s Cdn for all s. Since var(ζn)= −1 ≤ O(dn ), by (27) and the Bienayme–Chebyshev inequality, we get P −1 2 sup ζn F (dns)/s > κ O(dn dn)= O(dn). s≥τ | | ≤ 20 D. ANEVSKI AND P. SOULIER Let s , j 0 be an increasing sequence. Then we have the bound { j ≥ } ∞ Sn(s) P sup | | > κ P( Sn(sj) > κsj) s≥s0 s ≤ | | Xj=0 ∞ + P sup Sn(s) Sn(sj−1) > κsj−1 . sj−1≤s≤sj | − | Xj=1 From (28), we know that var(Sn(s)) = O(s). Thus ∞ P( S (s ) > κs ) cstκ−2 s−1. | n j | j ≤ j Xj=0 Xj=0 −1 Thus if the series sj is summable, this sum can be made arbitrarily small by choosing s0 large enough. It was shown in the proof of Lemma 8 that E[ S (s) S (s′) 4] C s s′ 2. | n − n | ≤ | − | By Billingsley (1968), Theorem 15.6 [or more specifically Ledoux and Tala- grand (1991), Theorem 11.1], this implies that 2 P C(sj sj−1) sup Sn(s) Sn(sj−1) > κsj−1 2−2 . sj−1≤s≤sj| − | ≤ κ sj−1 ρ Thus choosing sj = (s0 + j) for some ρ> 1 implies that the series is con- vergent and S (s) P sup | n | > κ = O(s−1), s 0 s≥s0 which is arbitrarily small for large s0. To deal with the remainder term R , we prove that P(sup R (s) /s > n s≥s0 | n | s0)= oP (1) by the same method as that used for Sn. Thus we only need to obtain a suitable bound for the increments of Rn. By definition of Rn, we have, for s Hence, E[ R (s′) R (s) 2] Cd n(s′ s)2. | n − n | ≤ n − The rest of the proof is similar to the proof for Sn. This concludes the proof of (30). 6.2. Sketch of proof of Theorem 6. The proof consists in checking condi- tions (AH1)–(AH4) with J and K now defined by J (t)= t log I (s)+ n n n 0 { n γ ds and K(t)= t log f(2π[ns/2π]/n) ds. Let λ = 2kπ/n denote the so- 0 k R called} Fourier frequencies. For t [0,π], denote k (t) = [nt/2π]. Denote R ∈ n ξ = log I (λ )/f(λ ) + γ, k { n k k } where γ is Euler’s constant. Then k (t) 2π n v (t)= J (t) K(t)= ξ + (t λ )ξ . n n − n j − kn(t) kn(t) Xj=1 The log-periodogram ordinates ξj are not independent, but sums of log-peri- odogram ordinates, such as the one above, behave asymptotically as sums of independent random variables with zero mean and variance π2/6 [cf. Soulier (2001)], and bounded moments of all order. Thus, for t0 (0,π), the process −2 −1/3 ∈ v˜n(s; t0)= dn vn(t0 + dns) vn(t0) with dn = n converges weakly in D( , ) to{ the two-sided− Brownian} motion with variance 2π4/3. 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The asymptotic behavior of monotone regression estimates. Ann. Statist. 9 443–448. MR0606630 Centre for Mathematical Sciences Department of Mathematics Box 118 University Paris X SE-22100 Lund Batimentˆ G, Bureau E18 Sweden 200 avenue de la Republique´ E-mail: [email protected] 92000 Nanterre Cedex France E-mail: [email protected]