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

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

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

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

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

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

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

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

We consider partial sums S 0 = 0, S k = X 1 + + 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

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

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

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

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

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

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

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

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

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

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 = 1 1 2 α. (Lo p-condition) Condition (L o p-condition) is optimal. (Rouen 2015) Some functional limit theorems 15 / 34

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

Next onsider self-normalization using block-sums of X k s. Choosing N = [n/m], where m = m(n), define U 2 n = Y 2 1 + + 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

Next onsider self-normalization using block-sums of X k s. Choosing N = [n/m], where m = m(n), define U 2 n = Y 2 1 + + 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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