CONVERGENCE IN DISTRIBUTION OF THE PERIODOGRAM OF CHAOTIC PROCESSES

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1 Stochastics and Dynamics c World Scientific Publishing Company COVERGECE I DISTRIBUTIO OF THE PERIODOGRAM OF CHAOTIC PROCESSES ARTUR O. LOPES and SÍLVIA R. C. LOPES Instituto de Matemática, UFRGS, Av. Bento Gonçalves 95, 95, Porto Alegre, RS, Brazil Received February 22 Revised 7 October 22 In this work we analyze the convergence in distribution sense of the periodogram function (to the spectral density function) based on a time series of a stationary process X t (ϕ T t )(X ) obtained from the iterations of a continuous transformation T invariant for an ergodic probability µ and a continuous function ϕ taking values in R. We only assume a certain rate of convergence to zero for the autocovariance coefficient of the stochastic process, that is, we assume there exist C > and β > 2 such that γ X (h) C h β, for all h, where γ X (h) R (ϕ T h )(x) ϕ(x)dµ(x) ( R ϕ(x)dµ(x)) 2 is the h-autocovariance of the process. Our result applies to the case of exponential decay of correlation (or covariance), as it happens for a continuous expanding transformation T on the circle and a Holder potential ϕ. It can also be applied to the case when the transformation T has a fixed point with derivative equal to one. Keywords: Periodogram Function, Convergence in Distribution, Chaotic dynamics, Stochastic Processes.. Introduction Here we consider the stochastic process {X t } t obtained from the iterations of a continuous transformation T (not necessarily invertible) from the unit interval (or the circle) to itself and µ an ergodic probability invariant under T. This stochastic and stationary process {X t } t is given by X t (ϕ T t )(X ) ϕ(t t (X )) (ϕ T )(X t ), for t, (.) where ϕ is a continuous map ϕ : [, ) R and X is distributed over [, ) according to µ. Considering the identification of the circle z S with x [, ) by z e xi, from now on we can use either one of the two forms T : S S or T : [, ) [, ). We consider µ a probability absolutely continuous with respect to the Lebesgue measure on the unit interval with density φ, that is, dµ(x) φ(x)dx. alopes@mat.ufrgs.br and slopes@mat.ufrgs.br

2 A.O. Lopes and S.R.C. Lopes We shall assume a certain rate of convergence to zero for the autocovariance function of such stochastic processes. We will denote by γ X ( ) the autocovariance function of the process {X t } t, that is, γ X (h) E µ (X h X ) E µ (X ) 2, for h. The spectral density function of the process {X t } t defined in the expression (.) is given by f X (λ) γ X (h)e iλh, for λ (, ], (.2) h where γ X ( ) is the autocovariance function of the process. In this work we analyze the convergence in distribution sense, to the spectral density function f X ( ), of the periodogram function associated to a time series T h (x ), h, obtained from a x chosen with probability one according to the measure µ. This periodogram function is given by where f (λ) I(λ k ) f (λ k )f (λ k ), ϕ(t t (x ))e iλt, λ (, ], (.3) t f ( ) indicates the complex conjugate of f ( ) and λ k k, for k,,,, is the k-th discrete Fourier frequency. ote that the periodogram function depends on x and (large). Our main theorem shows a mathematical proof that one obtains an approximation of the spectral density function f X ( ) by means of the periodogram (see [2]). See also remark 2.. We assume, without loss of generality, that E µ (X ) ϕ(x)dµ(x) ϕ(x)φ(x)dx. If we consider a non-invertible transformation T, for the negative integers h, the values γ X (h) do not make sense, but there is a general procedure to solve this problem going to the natural extension of T (see [6]). We assume here that there exists such natural extension and we refer the reader to section 5.3 in [7] for the procedure of extending γ X (h) for negative values of h. In [] several examples of such general procedure are also presented. Another alternative is not consider the natural extension and then take an expression like π h γ X(h) cos(λh) instead of h γ X(h)e iλh in the definition of the spectral density function. We prefer to use here the expression with terms of the form e ihλ.

3 Periodogram of Chaotic Processes Assumption. We will assume that T is such that for the given ϕ : [, ) R continuous map there exist C > and β > 2 such that the autocovariance function of the process {X t } t has the rate of convergence to zero given by γ X (h) C h β, for all h. (.4) This paper proceeds in the following way: Section presents some general definitions related to the chaotic process of the form (.) while Section 2 presents the mathematical proof of the convergence in distribution sense of the periodogram function I( ), under our main assumption, given by the expression (.4), imposed on the autocovariance function γ X ( ). Considering X t (ϕ T t )(X ) as above, where X is distributed over [, ) according to the ergodic probability µ, then from Birkhoff s theorem let x [, ) be a fixed number chosen according to µ such that for all continuous function g (or indicator functions of the interval) lim j g(t j (x )) g(x)dµ(x). We want to prove the convergence in distribution of the periodogram (to the function f X ( )) based on a time series {X t } t of the process {X t } t given by (.) beginning with µ-almost everywhere x. The theorem below describes this property in a precise way. In the sequel, δ y denotes the Dirac delta measure in y. Theorem.. Let {X t } t be the stationary zero mean process given by X t (ϕ T t )(X ). Let x be in a set of µ-probability one and let γ X ( ) be the autocovariance function of the process {X t } t such that the assumption given by (.4) holds. Let f X ( ) be the spectral density function of the process {X t } t. Then, in the distribution sense lim I(λ k )δ λk f X (λ), for λ (, ], where I( ) is the periodogram function defined by (.3), λ k k, for k {,,, }, is the k-th discrete Fourier frequency and δ λk is the Dirac delta measure with mass at λ k. Our main assumption includes the case where γ X ( ) exponentially decays to zero, that is, when there exists < λ < such that γ X (h) E µ (X h X ) C λ h, for h, (.5)

4 A.O. Lopes and S.R.C. Lopes where C is a positive constant and E µ (X t ) E µ [(ϕ T t )(X )]. In general, a transformation T of the circle with an indifferent fixed point (see [3],[4],[9] and [2]) defines a stochastic process with polynomial (not exponential) decay of correlation and the results presented here can also be applied. The β > 2 consider here is equivalent to γ > 4 in the notation of [3]. We observe that we are not considering general fractionally integrated processes (see []). In [] several methods for estimating the natural parameter of Manneville-Pomeau processes with short and long dependence are analyzed. In this case the periodogram method is a good way to find out the velocity of decay of the autocorrelation function of the process. The periodogram function is an unbiased estimator for the spectral density function f X ( ), even though it is not consistent (see [2]). In applications, it is very useful to have an approximation, as close as one wants, of the function f X ( ) by means of the periodogram function which is in general easy to obtain. We refer the reader to Figures 4 and 5 in [8] for a good geometrical description of the above property. 2. Convergence in Distribution of the Periodogram In this section we will show the convergence in distribution of the periodogram function. First we need Lemma 2.. Lemma 2.. Let f X ( ) be the spectral density function of the process {X t } t defined by the expression (.). Let f X,r ( ) be the truncated r-spectral density function of the process {X t } t given by f X,r (λ) γ X (h)e iλh, for λ (, ], (2.) for r. Then, for all ɛ >, there exists r such that for all K r, f X (λ) f X,K (λ) < ɛ, for all λ (, ], where g means the infinity norm of the function g. Proof: Let ɛ be a positive fixed constant. For all K, let us consider the spectral density function for all h such that h K, that is, the function

5 Periodogram of Chaotic Processes f X,K ( ) given by the expression (2.). Then, f X (λ) E µ (X h X )e iλh h γ X (h)e iλh h γ X (h)e iλh + h K f X,K (λ) + h >K γ X (h)e iλh h >K γ X (h)e iλh. (2.2) ow, since there exist C > and β > 2 such that, for all h, we have γ X (h) C h β then h γ X(h) converges. The last term in the above expression goes to zero when r goes to infinity since h γ X(h)e iλh converges. Therefore, given ɛ >, there exists such r. Therefore, the Lemma 2. is proved. We will consider, in the sequel, several numbers such as, q and r that will go to infinity. It will be very important the order we take them, that is, which number goes to infinity first, and then which one will be the next, etc... However, the value K will be much larger than all of them. Lemma 2. says that h > K has small order in the computation of the spectral density function. Let x be a point in a set of µ-probability one. From now on we will denote T t (x ) x t and X t ϕ(t t (x )) ϕ(x t ). From expression (.3) one has, for fixed, that I(λ k ) 4π 2 s,t X t X s e iλ k(t s) 4π 2 s s+h X s X s+h e iλ kh, (2.3) where in the last equality we change variable t s by h. Therefore, for each s fixed, the range of h is s h s. We want to prove the convergence in distribution of the periodogram (to the function f X ( )) based on a time series {X t } t of the process {X t } t given by (.) beginning with a µ-almost everywhere point x. The proof of Theorem. will be given after some lemmas. In order to have the convergence in distribution sense we will show that for any smooth function g : S R lim I(λ k )δ λk, g f X, g f X (λ)g(λ)dλ γ X (h)e iλh g(λ)dλ, h

6 A.O. Lopes and S.R.C. Lopes where I(λ k )δ λk, g g(x) I(λ k )δ λk I(λ k ) 4π 2 2 s s+h X s X s+h e iλ kh. Remark 2.. Given x, in order to determine the (approximated) value f X (x), one takes a continuous function g such that it has support in a small neighborhood of x (see pages in [8]) and apply the theorem for such g. Changing the point x one can graph the (approximated) function f X (x). Let r {} be a fixed value such that Lemma 2. holds. Then, we can write I(λ k )δ λk, g + 4π 2 2 4π 2 2 4π 2 2 s s+h s s+h s s+h h >r X s X s+h e iλ kh X s X s+h e iλ kh (2.4) (2.5) X s X s+h e iλ kh. (2.6) Lemma 2.2 below is crucial and we shall prove that the expression (2.6) goes to zero, when. Lemma 2.2. Given ɛ >, there exists r such that, for all K > r and for x [, ) µ-almost everywhere, there exists {} such that 4π 2 2 s s+h K > h >r X s X s+h e iλ kh < ɛ, for all >.

7 Periodogram of Chaotic Processes Proof: Given ɛ >, let ɛ be such that ɛ 4π2 2M ɛ. Given r and K fixed, the function v(x) ϕ(x)ϕ(t h (x)) e iλ kh r< h <K r< h <K ϕ(x)ϕ(t h (x)) is continuous. If r is large enough one has, for this ɛ > and K > r, that v(x)dµ(x) < ɛ 3. (2.7) Here we use again that h γ X(h) converges, since the autocovariance function of order h of the process {X t } t goes to zero with order of convergence h β, with β > 2. ow we fix r and K (much more larger than r). For such fixed function v( ), we want to estimate the µ-measure of bad points x given by P ɛ 3 µ ( { x ; sup > s v(t s (x )) v(x)dµ(x) > ɛ } ). 3 From Theorem 3, part, in [5], as σ f ( δ, δ) o(δ β ) as δ (since Assumption given by (.4) holds), then P ɛ 3 o( (β ) ) as. Therefore, as β > 2, P ɛ 3 <. Then, from Borel-Cantelli Lemma, one has that ( { µ x ; sup v(t s (x )) v(x)dµ(x) > ɛ } 3 > s i.o. ), that, is, for any x µ-almost everywhere, there exists (x ) > such that x / P ɛ 3, for all >. Hence, sup > s v(t s (x )) v(x)dµ(x) ɛ 3. (2.8) From the expressions (2.7) and (2.8), for all > and for all >, we have v(t s (x )) ϕ(t s (x ))ϕ(t s+h (x )) s s r< h <K < 2ɛ 3.

8 A.O. Lopes and S.R.C. Lopes This is not enough. As r and K are fixed, one has and lim lim h s s h r< h <K ϕ(t s (x ))ϕ(t s+h (x )) r< h <K ϕ(t s (x ))ϕ(t s+h (x )). Therefore, given ɛ 3 >, there exists 2 {} such that, for all > 2, h ϕ(t s (x ))ϕ(t s+h (x )) s r< h <K < ɛ 6 and Since s s h s s+h ϕ(t s (x ))ϕ(t s+h (x )) ϕ(t s (x ))ϕ(t s+h (x )) r< h <K < ɛ 6. ϕ(t s (x ))ϕ(t s+h (x )) r< h <K r< h <K ϕ(t s (x ))ϕ(t s+h (x )) s s> h s r< h <K s< h ϕ(t s (x ))ϕ(t s+h (x )) < r< h <K < 2ɛ 3 + ɛ 6 + ɛ 6 ɛ. Since M sup λ [,) g(λ), one has, for large, that 4π 2 s s+h r< h <K X s X s+h e iλ kh < + 4π 2 M ɛ < 2M 4π 2 ɛ ɛ. Therefore, Lemma 2.2 is proved. 4π 2 ɛ <

9 Periodogram of Chaotic Processes Remark 2.2. Consider ɛ fixed and r as in Lemma 2.. We can truncate the infinite sum defining the spectral density function in K according to Lemma 2.. ow in Lemma 2.2, for ɛ ɛ/3 take r > r and K > r. In this way we obtain in Lemma 2.2 a value and all values >, in the future, will be larger than such. The values r and K will be fixed in the future but we still have to impose some more restrictions. The above lemma will imply, as we will see later, that the values h such that r < h < K have small order in the estimation of the values of the periodogram function. ow we return to the proof of the main theorem. We have to analyze the contribution of the values h < r. From Lemma 2.2, the equality in expression (2.4), when, can be rewritten as I(λ k )δ λk, g 4π 2 2 h s s+h X s X s+h e iλ kh + o(). (2.9) ow, we fix q and let us define B,, B q a partition of the unit interval. The intervals B j, j {,, q}, are such that B i B j φ, for all i, j {,, q}, i j, and such that [ ) q j B j [, ), where B j j q, j+ q, for j {,, q}. Remark 2.3. The value X s+h ϕ(t s+h )(x ) is defined above for positive s + h. However, the value h can be negative. In order to avoid a heavy notation, when h is negative, the expression ϕ(x s )ϕ(t h (x s )) in the sum below will mean ϕ(x s+h )ϕ(t h (x s+h )). Let α j be a fixed interior point of B j, for j {,, q}. Then, the expression (2.9) can be rewritten as + I(λ k )δ λk, g 4π 2 2 h 4π 2 4π 2 4π 2 + o(), s s+h q j s s+h s:xs B j s s+h q j s s+h X s X s+h e iλ kh + o() X s X s+h e iλ kh + o() [ ϕ(xs )ϕ(t h (x s )) ϕ(α j )ϕ(t h (α j )) ] e iλ kh #[x s B j ]ϕ(α j )ϕ(t h (α j ))e iλ kh (2.)

10 A.O. Lopes and S.R.C. Lopes where X s ϕ(x s ) ϕ(t s (X )) and x s T s (x ). ote that the restriction s + h, with h < r and r fixed, is a mild assumption because the number of s such that h < s < h is of the same order as. By Birkoff s Theorem lim # [ x s B j s + h ] lim #[x s B j ] µ(b j ), given ɛ > take large enough such that #[x s B j ] µ(b j ) 4π 2 2M rqm2 2 ɛ, for any j {,, q}, where M sup λ [,] g(λ) and M 2 sup x S ϕ(x). Suppose goes to infinity faster than q. ow, we will show the following claim: Claim 2.. Given ɛ >, for and r large enough, the absolute value of the expression (2.) can be written as q 4π 2 #[x s B j ]ϕ(α j )ϕ(t h (α j ))e iλ kh < s j s+h < 4π 2 q µ(b j )ϕ(α j )ϕ(t h (α j ))e iλ kh + for large. j + + ɛ, (2.) Proof. Observe that using Birkoff s Theorem, for large, 4π 2 4π 2 + 4π 2 s s+h q #[x s B j ]ϕ(α j )ϕ(t h (α j ))e iλ kh j j j q µ(b j )ϕ(α j )ϕ(t h (α j ))e iλ kh + q 4π 2 ɛ 2M rqm2 2 ϕ(α j ) ϕ(t h (α j ))

11 4π 2 j j Periodogram of Chaotic Processes q µ(b j )ϕ(α j )ϕ(t h (α j ))e iλ kh + + q ɛ 2M j rqm2 2 M2 2 4π 2 q µ(b j )ϕ(α j )ϕ(t h (α j ))e iλ kh + < 4π 2 + j ɛ 2rq 2M rq < q µ(b j )ϕ(α j )ϕ(t h (α j ))e iλ kh + + ɛ. This proves Claim 2.. As + I(λ k )δ λk, g 4π 2 2 h 4π 2 4π 2 q j s s+h s:xs B j s s+h q j from Claim (2.), we can write I(λ k )δ λk, g + h 4π 2 4π 2 q j xs B j s s+h j s s+h X s X s+h e iλ kh + o() [ ϕ(xs )ϕ(t h (x s )) ϕ(α j )ϕ(t h (α j )) ] e iλ kh #[x s B j ]ϕ(α j )ϕ(t h (α j ))e iλ kh + o(), [ϕ(x s )ϕ(t h (x s )) ϕ(α j )ϕ(t h (α j ))]e iλ kh q µ(b j )ϕ(α j )ϕ(t h (α j ))e iλkh + o(). (2.2)

12 A.O. Lopes and S.R.C. Lopes In Lemma 2.3 we shall prove that the first term in the expression (2.2) can be taken as small as we want if and q are large enough. Lemma 2.3. Given ɛ >, 4π 2 xs B j s s+h r h q j ( xs B j s s+h <h r [ϕ(x s )ϕ(t h (x s )) ϕ(α j )ϕ(t h (α j ))]e iλ kh + [ϕ(x s+h )ϕ(t h (x s+h )) ϕ(α j )ϕ(t h (α j ))]e iλ kh ) < ɛ, (2.3) for q sufficiently large but fixed and for all large enough. Proof. In order to avoid a heavy notation we consider just the part with h > in expression (2.3). The reasoning for the case h < is similar. Since ϕ is a continuous map on the compact set [, ) it is uniformly continuous. The transformation T is also continuous, so ϕ( )ϕ(t h ( )) is also a uniformly continuous function. Therefore, for fixed h, for all ɛ >, there exists δ h such that for all x, y with x y < δ h, then ϕ(x)ϕ(t h (x)) ϕ(y)ϕ(t h (y)) < ɛ. Since h r is uniformly bounded one can find a uniform δ that works for all h such that h r. Take ɛ 4π2 2M ɛ > and q large enough such that q < δ. Therefore, if length of B j < δ, for all j {,, q}, we have ϕ(x)ϕ(t h (x)) ϕ(y)ϕ(t h (y)) < ɛ, for all x, y B j and for any h Z such h < r. Therefore, 4π 2 q j xs B j s s+h [ϕ(x s )ϕ(t h (x s )) ϕ(α j )ϕ(t h (α j ))]e iλ kh 4π 2 q j xs B j s s+h ϕ(x s )ϕ(t h (x s )) ϕ(α j )ϕ(t h (α j ))

13 Periodogram of Chaotic Processes 4π 2 q j xs B j s s+h h <r ɛ 4π 2 s s+h h <r ɛ, for x s and α j such that x s α j < δ, where the double summation, in the above expression, has terms all of them less than ɛ. Therefore, where 4π 2 q j xs B j s s+h [ϕ(x s )ϕ(t h (x s )) ϕ(α j )ϕ(t h (α j ))]e iλ kh < < ( + )M 4π 2 ɛ < 2M 4π 2 ɛ ɛ, M Therefore, Lemma 2.3 is proved. sup g(λ). λ (,] ow, using Lemma 2.3, the whole expression (2.2) can be rewritten as I(λ k )δ λk, g 4π 2 + o(). j q µ(b j )ϕ(α j )ϕ(t h (α j ))e iλ kh (2.4) The following lemma proves that the expression (2.4) goes to f X,r, g, when. Lemma 2.4. For r fixed, 4π 2 j when q and q. q µ(b j )ϕ(α j )ϕ(t h (α j ))]e iλkh f X,r, g, Proof. Given ɛ >, note that for fixed, with much larger than q, where q is also large, one has q µ(b j )ϕ(α j )ϕ(t h (α j ))e iλkh ( ) ϕ(x)ϕ(t h (x))dµ(x) e iλ kh j

14 A.O. Lopes and S.R.C. Lopes This is true since, for fixed h and h < r, q µ(b j )ϕ(α j )ϕ(t h (α j )) j ɛ 2r ɛ. ϕ(x)ϕ(t h (x))dµ(x), when q. Therefore, given ɛ >, if q is large then, for all h < r, q µ(b j )ϕ(α j )ϕ(t h (α j )) ϕ(x)ϕ(t h (x))dµ(x) < ɛ. j Since ϕ(x)ϕ(t h (x))dµ(x) γ X (h), for fixed h, and one has 4π 2 4π 2 f X,r (λ k ) j ( 4π 2 for large q. Therefore, one has 4π 2 γ X (h)e iλkh, q µ(b j )ϕ(α j )ϕ(t h (α j ))e iλkh ) ϕ(x)ϕ(t h (x))dµ(x) e iλ kh ɛ, j e iλ kh q µ(b j )ϕ(α j )ϕ(t h (α j ))e iλkh f X,r (λ k ) 4π 2 ɛ, (2.5) for large q, and much larger than q. Expression (2.5) suggests to take Then, I(λ k )δ λk, g ɛ 2 M ɛ >. f X,r (λ k ) + 4π 2 ɛ + o()

15 for large. Then, for large, one has I(λ k )δ λk, g This proves Lemma 2.4. f X,r (λ k ) + + ɛ + o() g(λ)f X,r (λ)dλ + + ɛ + o(), Periodogram of Chaotic Processes f X,r (λ)g(λ)dλ < o(). (2.6) ow we shall prove Theorem.. Considering the expression (2.6) and Lemma 2., for given ɛ/3 >, one has I(λ k )δ λk, g for large enough. This proves our main theorem. Acknowledgments f X (λ)g(λ)dλ ɛ, The authors were partially supported by CPq-Brazil and by FAPERGS-RS- Brazil. S.R.C. Lopes was also partially supported by Pronex Fenômenos Críticos em Probabilidade e Processos Estocásticos (Convênio FIEP/MCT/CPq 4/ 96/ 923/). A.O. Lopes was also partially supported by Pronex Sistemas Dinâmicos (Projeto o /996-4) and Instituto do Milênio. References. S. Borovkova, R. Burton and H. Dehling, Limit Theorems for Functionals of Mixing Processes with Applications to U-Statistics and Dimension Estimation, Transactions of the American Mathematical Society 353 (2) P.J. Brockwell and R.A. Davis, Time Series: Theory and Methods (Springer- Verlag, 99), 2nd. edition. 3. A.M. Fisher and A.O. Lopes, Exact bounds for the polynomial decay of correlation, f -noise and the CLT for the equilibrium state of a non-hölder potential, onlinearity 4 (2) S. Isola, Renewal sequences and intermitency, Journal of Statistical Physics 97 (999) A. G. Kachurovskii, The rate of convergence in ergodic theorems, Russian Mathematical Surveys 5 (996) G. Keller, Equilibrium States in Ergodic Theory (Cambridge Press, 998).

16 A.O. Lopes and S.R.C. Lopes 7. A.O. Lopes and S.R.C. Lopes, Parametric Estimation and Spectral Analysis of Chaotic Time Series, Advances in Applied Probability 3 (998) A.O. Lopes, S.R.C. Lopes and R. R. Souza, Spectral analysis of expanding onedimensional chaotic transformations, Random and Computational Dynamics 5 (997) C. Maes, F. Redig, F. Takens, A.V. Moffaert and E. Verbitski, Intermittency and Weak Gibbs State. Preprint from the Department of Mathematics, University of Groningen, The etherlands B. Olbermann, S.R.C. Lopes and A.O. Lopes, Estimation of Parameters in Manneville- Pomeau Processes. Preprint from Inst. Mat., UFRGS, Brazil 22.. V.A. Reisen and S.R.C. Lopes, Some Simulations and Applications of Forecasting Long Memory Time Series Models, Journal of Statistical Planning and Inference 8 (999) L.S. Young, Recurrence times and rates of mixing, Israel Journal of Mathematics (999)