Some functional (Hölderian) limit theorems and their applications (II)

Transcription

1 Some functional (Hölderian) limit theorems and their applications (II) Alfredas Račkauskas Vilnius University Outils Statistiques et Probabilistes pour la Finance Université de Rouen June 1 5, Rouen (Rouen 2015) Some functional limit theorems 1 / 34

2 Outline of the lecture Functional limit theorems for dependent observaions 1 IP for linear processes Short memory case Long memory case 2 Nearly non-stationary processes (Rouen 2015) Some functional limit theorems 2 / 34

3 Motto of the lecture Essentially, all models are wrong, but some are useful (George E. P. Box) (Rouen 2015) Some functional limit theorems 3 / 34

4 Motto of the lecture Essentially, all models are wrong, but some are useful (George E. P. Box) (Rouen 2015) Some functional limit theorems 3 / 34

5 IP for linear processes A large number of empirical studies show that many financial data series, such as exchange rate returns, stock indices and others, often exhibit some non-standard features like 1 non-gaussianity (the data do not come from a normal, but sometimes from a heavy-tailed distribution); 2 time varying volatility: variance changes over time, with alternating phases of high and low volatility; 3 long-memory dependence: a slow decay of the autocorrelation function. A more general model that captures both long-memory and the heavy-tail feature is a linear process. (Rouen 2015) Some functional limit theorems 4 / 34

6 A stochastic process (Y t ) is a linear process if Y t = µ + j= ψ j ε t j where (ε t ) is a white-noise process and the sequence ψ j, j = 0, ±1, ±2,... is a sequence of real numbers. Thus, a general linear process is a two-sided infinite order MA process. This class of linear models are designed specifically for modelling the dynamic behaviour of time series. These include, moving-average (MA), autoregressive (AR), autoregressive-moving average (ARMA) models, and fractionally integrated ARMA models (FARIMA). The MA(q) and MA( ) are one-sided special cases. The linear process is constructed by taking a weighted average of successive elements of the (ε t ) sequence, which is an operation called,,(linear) filtering. Filtering the white noise process generates a covariance stationary process with a more interesting properties than the white noise process it is built from. (Rouen 2015) Some functional limit theorems 5 / 34

7 We shall consider linear process (X k ) k 0 of the form X k = ψ i ε k i, k = 0, 1,..., (LP-definition-a) i=0 where (ψ i, i Z) is a given sequence of real numbers with ψ i = 0 for i < 0 and (ε i, i Z) is a sequence of independent identically distributed random variables with Eε 0 = 0 and E ε 0 2 <. Throughout we assume that i ψ2 i <. The series in (LP-definition-a) then converge in L 2 and almost surely and the sequence of random variables (X k ) k 0 is stationary and ergodic. (Rouen 2015) Some functional limit theorems 6 / 34

8 We consider partial sums S 0 = 0, S k = X X k, k 1 and polygonal line process ζ n : ζ n (t) = ζ (X ) n (t) := S [nt] + (nt [nt])x [nt]+1, t [0, 1]. Asymptotic properties of ζ n strongly depend on k ψ k : If k ψ k < and k ψ k 0 the linear process (X k ) is said to have short memory. If k ψ k = then the linear process (X k ) is said to have long memory. (Rouen 2015) Some functional limit theorems 7 / 34

9 Observation! Denoting for a sequence a = (a j ), L k a = L k (a j ) = (a j k ) we can write X j = ψ k L k ε j or X j = ε k L k ψ j, k= k= (Rouen 2015) Some functional limit theorems 8 / 34

10 Observation! Denoting for a sequence a = (a j ), L k a = L k (a j ) = (a j k ) we can write X j = ψ k L k ε j or X j = ε k L k ψ j, k= k= Hence or ζ (X ) n (t) = ζ (X ) n (t) = k= k= ψ k ζ (L kε) n (t), t [0, 1] ε k ζ (L kψ) n (t), t [0, 1]. (Rouen 2015) Some functional limit theorems 8 / 34

11 an idea 1 In the case where k ψ k < try to compare k= ψ k ζ (L kε) n (t) with [ k= ψ k ] ζ (L 0ε) n (t) (Rouen 2015) Some functional limit theorems 9 / 34

12 an idea 1 In the case where k ψ k < try to compare k= ψ k ζ (L kε) n (t) with [ k= ψ k ] ζ (L 0ε) n (t) 2 whereas in the case where k ψ k = try to compare k= ε k ζ (L kψ) n (t) with k= γ k ζ (L kψ) n (t), where (γ k ) are i.i.d. zero mean normal random variables with Eγk 2 = Eε2 k. div-proof (Rouen 2015) Some functional limit theorems 9 / 34

13 Short memory case Lemma (almost Beveridge-Nelson decomposition a ) a generalization of Peligrad, Utev lemma Let (E, ) be a separable Banach space. Let (a i ) i Z be a collection of linear bounded operators on E satisfying i Z a i <.Assume that (Z n,i, n N, i Z) is a collection of random elements with values in E satisfying the following conditions: (i) sup n N,i Z E Z n,i <, P (ii) For every fixed i, j Z it holds Z n,i Z n,j 0. n Then for each n N the series i Z a iz n,i converge a.s. and for every index l Z, the following convergence P a i Z n,i AZ n,l 0 n i Z holds, where A = i Z a i. (Rouen 2015) Some functional limit theorems 10 / 34

14 i a i < entails the summability of (a i ) i I in L(E) and justifies the definition of A. We legitimate the existence of i a iz ni by noting that E i I a i Z n,i = i I a i E Z n,i <, whence i a i Z n,i is almost surely finite, so i I a i(z n,i ) is almost surely convergent in E. (Rouen 2015) Some functional limit theorems 11 / 34

15 Now let Y n = i a iz n,i and le us prove the convergence to zero of P n,ε := P( Y n AZ n,e > ε) for arbitrary ε > 0. Once fixed such an ε, the summability of i a i combined with the condition provides for any positive δ a finite subset K N, such that for every n 1, a i E Z n,i Z n,e < δε. i N\K Starting from the splitting Y n AZ n,e = a i (Z n,i Z n,e ) i N = a i (Z n,i Z n,e ) i N\K + i K a i (Z n,i Z n,e ), (Rouen 2015) Some functional limit theorems 12 / 34

16 we easily obtain ( ) P n,ε 2 a i E Z n,i Z n,e + P a i Z n,i Z n,e > ε ε 2 i N\K i K ( ) 2δ + P a i max Z n,j Z n,e > ε j K 2 i K 2δ + ( P Z n,j Z n,e > ε ), 2τ j K where we put τ := i I a i, recalling that τ > 0. Now from condition and the finiteness of K we obtain : lim sup P n,ε 2δ. n As this limsup does not depend on the arbitrary positive δ, it is in fact a null limit, which was to be proved. (Rouen 2015) Some functional limit theorems 13 / 34

17 Theorem (FCLT in C[0, 1] for short memory linear process) Let (X k ) k 0 be the linear process defined by (LP-definition-a) a and assume that the filter (ψ i ) i 0 is such that ψ i < and A := ψ i > 0. i=0 Put bn 2 = A 2 Eε 2 0n. Then holds in the space C[0, 1]. W bn 1 D ζ n n i=0 a X j = k=0 ψ kε j k, (ε j )- i.i.d., Eε j = 0, σ 2 = Eε 2 j (0, ) (Rouen 2015) Some functional limit theorems 14 / 34

18 Theorem (LIP for short memory linear process) Let (X k ) k 0 be the linear process defined by (LP-definition-a) and assume that (a i ) i 0 satisfies condition (A) a i < and A := a i > 0. i=0 i=0 Put bn 2 = A 2 neε 2 0. Then for 0 < α < 1/2, the convergence holds in the space H o α provided W bn 1 D ζ n n lim t tp P( ε 0 > t) = 0, where p = α. (Lo p-condition) Condition (L o p-condition) is optimal. (Rouen 2015) Some functional limit theorems 15 / 34

19 Normalization b n contains unknown components A(Eε 2 0 )1/2. Could self-normalization help? Theorem If ε 0 DNA, then where Vn 1 D S n N (0, n µ2 1/µ 2 2), µ 1 = ψ k, µ 2 2 = ψk 2. k=0 k=0 (Rouen 2015) Some functional limit theorems 16 / 34

20 Next onsider self-normalization using block-sums of X k s. Choosing N = [n/m], where m = m(n), define U 2 n = Y Y 2 N, where Y j = X i, j = 1, 2,..., N. (j 1)m<i jm (Rouen 2015) Some functional limit theorems 17 / 34

21 Next onsider self-normalization using block-sums of X k s. Choosing N = [n/m], where m = m(n), define U 2 n = Y Y 2 N, where Y j = X i, j = 1, 2,..., N. (j 1)m<i jm Theorem (block-selfnormalized CLT) If i a i <, and ε 1 DAN, then Un 1 D S n n N(0, 1) provided that m and m/n 0 as n. (Rouen 2015) Some functional limit theorems 17 / 34

22 Set [nt] ζ n (t) = X i + (nt [nt])x [nt]+1, t [0, 1] i=1 in the Banach space C[0, 1] Theorem (block-selfnormalized CLT) If i a i <, and ε 1 DAN, then Un 1 D ζ n n W in the space C[0, 1], as n. provided that m and m/n 0 as n. (Rouen 2015) Some functional limit theorems 18 / 34

23 Lemma Let (a i ) i Z satisfies i Z a i <. Assume that (Z (j) n,i, n, j N, i Z) is a collection of random variables satisfying the following conditions: ( ) (j) 1/2 (i) sup n N, i Z E j N [Z n,i ]2 < ; (j) (ii) for any i, l Z, j N (Z n,i Z (j) n,l )2 P 0. n Then with any l Z it holds ( ( n := j N where A = i Z a i. i=1 ) a i Z (j) 2 ) 1/2 ( n,i A 2 [Z (j) n,l ]2) 1/2 j 1 P n 0, (Rouen 2015) Some functional limit theorems 19 / 34

24 IP for long memory LP For 1/2 < β < 1, let (X k ) k 0 be the linear process X k = ψ j ε k j, j=0 (LP-definition-b) with ψ 0 = 1, ψ j = l(j) j β, j 1 where l is a positive non decreasing normalized slowly varying function and (ε j, j Z) is a sequence of i.i.d. random variables with Eε 0 = 0 and E ε 0 2 is finite. (Rouen 2015) Some functional limit theorems 20 / 34

25 In this farmework one needs to compare div U n := k ε k b nk with V n := k γ k b nk To this aim one can use the so called ζ 3 metric: { ζ 3 (X, Y ) = sup Ef (X ) Ef (Y ) : 3 k=0 } sup f (k) (x) 1. x Lemma If the following two conditions 1 lim n sup j Z b nj = 0 2 lim sup n j Z b2 nj < are satisfied, then lim ζ 3(U n, V n ) = 0. n (Rouen 2015) Some functional limit theorems 21 / 34

26 Put H := 3 2 β. (Hurst exponent) Let (S n ) and (ζ n ) be the partial sums and partial sums process built on (X k ) k 0. Put with b n = n H l(n)c β ( E ε 2 0 ) 1/2, c β := (1 β) 2 where x + := max(0; x). 0 ( x 1 β (x 1) 1 β ) 2 + dx, (norming sequence) (Rouen 2015) Some functional limit theorems 22 / 34

27 Theorem (IP for long memory LP) For 0 < α < H, the weak-hölder convergence bn 1 D ζ n is obtained in the following cases. W H in the space H o n α (IP LM case) 1 For 0 < α < H 1/2, (IP LM case) holds true if Eε 2 0 <. 2 For α = H 1/2, (IP LM case) holds true if lim (t ln t t)2 P( ε 0 > t) = 0 3 For H 1/2 < α < H, (IP LM case) holds true if lim t tp P( ε 0 > t) = 0, where p = 1 H α. (Rouen 2015) Some functional limit theorems 23 / 34

28 The slowly varying function l is said normalized if for every δ positive, t δ l(t) is ultimately increasing and t δ l(t) is ultimately decreasing. Two remarks. The variance σ 2 n of S n is asymptotically equivalent to b 2 n. Therefore the FCLT holds as well with b n replaced by σ n. The necessity of condition lim t tp P( ε 0 > t) = 0, where p = 1 H α. remains an open and interesting question. (Rouen 2015) Some functional limit theorems 24 / 34

29 Array of linear processes Let for each n 1 X nj = k=0 ψ nk ε j k, j = 1,..., n, where (ε k ) are i.i.d. zero mean random variables, Eε 2 k = σ2 (0, ). Assume that for each n but ψ k < k lim n ψ k =. k How it reflects in asymptotic behaviour of polygonal line process ζ n build on (X nj, j = 1,..., n); n 1? (Rouen 2015) Some functional limit theorems 25 / 34

30 At least two interesting cases are: and ψ nk = φ k n, 0 < φ n < 1, lim n φ n = 1 ψ nk = k dn, d n > 1, lim n d n = 1. In the first case we deal with so called nearly non-stationary processes. The second type is called nearly long memory processes. (Rouen 2015) Some functional limit theorems 26 / 34

31 Let W = (W t, t [0, 1]) be a standard Brownian motion. Let for γ > 0, J γ be the integrated Ornstein-Uhlenbeck process: J γ (t) := t 0 Theorem U γ (s) ds, 0 t 1, where U γ (s) = s 0 e γ(s r) dw (r). Suppose that X nj = j=0 e γj/n ε k j, j = 1,..., n; n 1, with γ > 0. Then n 3/2 D ζ n γ n in the space C[0, 1]. If in moreover for 0 < α < 1/2, lim t t 1/(1/2 α) P( ε 0 > t) = 0, then n 3/2 ζ n D n J in the space Ho α. (Rouen 2015) Some functional limit theorems 27 / 34

32 Theorem Suppose that X nj = n(1 φ n ). Then j=0 φj/n n 1/2 (1 φ n )ζ n n ε k j, j = 1,..., n; n 1, with φ n 1, D W in the space C[0, 1]. n If moreover for a p > 2, lim t t p P( ε 0 > t) = 0, then n 1/2 (1 φ n )ζ n D n W in the space H β provided that lim inf n (1 φ n )n 1 2β/(p 2) > 0. (Rouen 2015) Some functional limit theorems 28 / 34

33 Maximal inequality Lemma Let (η j ) j 0 be a sequence of i.i.d. random variables, with Eη 0 = 0 and E η 0 q < for some q 2. Suppose φ n = 1 γn n, where γ n and γ n /n 0, as n. Define z k = k j=1 φk j n η j.then there exists an integer n 0 (q) 1 depending on q only, such that for every n n 0 (q), γ n > γ n0 (q), and every λ > 0, ( ) P max z k > λ 4C qe q E η 0 q 1 k n λ q n q/2 γ 1 q/2 n, (1) where C q is the universal constant in the Rosenthal inequality of order q. Choosing λ = n 1/2 γn 1/q 1/2 τ for arbitrary τ > 0 provides: ( ) max z k = O P n 1/2 γn 1/q 1/2. 1 k n (Rouen 2015) Some functional limit theorems 29 / 34

34 Proof The idea of the proof relies on the following observation. For a < k b, z k = φ k n k j=1 φ j n η j φ a n k j=1 φ j n η j. Here { k j=1 φ j n η j, a < k b} is a martingale adapted to its natural filtration and if we repeat this procedure with regularly spaced bounds a and b, we keep the structure of a geometric sum for the coefficients φ a n. To profit of these two features we are lead to the following splitting: n = MK, max z k = 1 k n max 1 m M max z k, (m 1)K<k mk where M and K (not necessarily integers) depend on n in a way which will be precised later. (Rouen 2015) Some functional limit theorems 30 / 34

35 Applying this splitting we obtain first: ( ) P max z k > λ P 1 k n 1 m M φ (m 1)K n k max 1 k mk j=1 n η j > λ. φ j Then using Markov s and Doob s inequalities at order q gives ( ) P max z k > λ φ q(m 1)K n T m 1 k n λ q where T m := E 1 m M 1 j mk To bound T m, we treat separately the special case q = 2 with a simple variance computation and use Rosenthal inequality in the case q > 2. In both cases, the following elementary estimate is useful. 1 j mk φ jq n [mk] = φn [qmk] j=1 φ [mk]q jq n [mk] 1 = φ [qmk] n j=0 φ jq n φ j n η j (2) φ [qmk] n 1 φ q n φ qmk n 1 φ n (Rouen 2015) Some functional limit theorems 31 / 34

36 recalling that 0 < φ n < 1, whence, 1 j mk φ jq n Now in the special case q = 2, we have k T m = Var φ j n η j = Eη0 2 so by (3), j=1 n γ n φ qmk n. (3) 1 j mk φ 2j n, T m n γ n φ 2mK n Eη 2 0. (4) When q > 2, we apply Rosenthal inequality which gives here T m C q ( Eη0 2 ) q/2 q/2 φ 2j n + E η 0 q 1 j mk 1 j mk φ jq n. As q > 2, ( Eη0) 2 q/2 E η0 q. Also we may assume without loss of generality that n γ n 1, so n γ n ( ) n q/2. γ n (Rouen 2015) Some functional limit theorems 32 / 34

37 Then using (3), we obtain T m 2C q E η 0 q n q/2 γn q/2 φ qmk n. (5) Note that (4) obtained in the special case q = 2 can be included in this formula by defining C 2 := 1/2. Going back to (2) with this estimate, we obtain ( ) P max z k > λ 1 k n 2C q E η 0 q n q/2 γ q/2 n λ q 1 m M 2C q E η 0 q n q/2 γn q/2 λ q Mφ Kq n. φ Kq n Now, choosing K = n γ n, we see that φ Kq n converges to e q, so for n n 0 (q), φ Kq n 2e q. Then (1) follows by pluging this upper bound in the inequality above and noting that M = γ n. (Rouen 2015) Some functional limit theorems 33 / 34

38 Remark Under assumptions of Lemma 29 there exists such constant c q depending on q only, such that for every n 1 and every λ > 0 ( ) P max z k > λ c qe η 0 q 1 k n λ q n q/2 γ 1 q/2 n. (Rouen 2015) Some functional limit theorems 34 / 34