Statistical test for some multistable processes

Statistical test for some multistable processes Ronan Le Guével Joint work in progress with A. Philippe Journées MAS 2014

1 Multistable processes First definition : Ferguson-Klass-LePage series Properties of the distributions Second definition: multistable measures 2 How to test the multistability

First definition : Ferguson-Klass-LePage series Consider (E, E, m) a (σ-)finite measure space. Let α (0, 2) and (f t ) t a family of functions such that f t (x) α m(dx) < + : E we can define the stochastic process I (f t ) = f t (x)m(dx) where M is a symmetric α-stable random measure with control measure m: f t is seen as a limit of simple functions.

Let (Γ i ) i 1 be a sequence of arrival times of a Poisson process with unit arrival time, (V i ) i 1 a sequence of i.i.d. random variables with distribution ˆm = m/m(e), (γ i ) i 1 a sequence of i.i.d. random variables with distribution P(γ i = 1) = P(γ i = 1) = 1/2. with the three sequences (Γ i ) i 1, (V i ) i 1, and (γ i ) i 1 independent. Let α (0, 2) and c(α) = (2α 1 Γ(1 α) cos( πα 2 )) 1/α :

We define the multistable processes on an open interval U:

We define the multistable processes on an open interval U: let α : U [c, d] (0, 2) a continuous function, f t (x) a family of functions such that t U, f t (x) α(t) m(dx) < + : E Y (t) := ( α(t) + 2 m(e)) 1/α(t) c(α(t)) γ i Γ 1/α(t) i f t (V i ) is a symmetric multistable process, with kernel f t and stability function α. i=1

Characteristic function: we compute E[e iθy (t) ] = exp θ α(t) f t (x) α(t) m(dx). E

Local structure of multistable processes Let Y be a real stochastic process. Property of localisability : Y admits a tangent process. [Falconer (2002,2003)] Definition A real stochastic process Y = {Y (t) : t R} is h localisable at u if there exists an h R and a non-trivial limiting process Y u (the local form) such that Y (u + rt) Y (u) lim r 0 r h = Y u(t) where convergence occurs in finite dimensional distributions.

Example The Multistable Lévy Motion: + Y (t) = K(α(t)) γ i Γ 1 α(t) i 1 [0,t] (V i ), t [0, 1] i=1 where α is a C 1 function ranging in (1, 2), (V i ) i is i.i.d. uniformly distributed on (0, 1), and the kernel f t (x) = 1 [0,t] (x). 1 The process Y is α(u) -localisable at u [0, 1], with (the Stable Lévy Motion). Y u(t) = L α(u) (t)

Example of a Stable Lévy Motion Trajectory of a α-stable Lévy process, α = 1.9 Trajectory of a α-stable Lévy process, α = 1.1

Example of multistable Lévy process α(t) = 1, 5 + 0, 48 sin(2πt), h(t) = 1 α(t). 0.8 Mouvement de Levy multistable 0.6 0.4 0.2 Y(t) 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 temps α(t) = 1, 02 + 0, 96t. 0.6 Mouvement de Levy multistable 0.4 0.2 0 Y(t) 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 temps

Second definition: multistable measures [Falconer-Liu (2010)]: Let α : R (0, 2). We define the family of stable integrals {I (f ), f F } as a stochastic process indexed by a set F of functions. We specify its finite-dimensional distributions, and apply the Kolmogorov s existence theorem. Here F = {f : f (x) α(x) m(dx) < + }. E Given f 1,..., f d F, we define a probability measure P f1,...,f d by its characteristic function: in R d φ f1,...,f d (θ 1,..., θ d ) := exp E d θ j f j (x) α(x) m(dx), j=1

Characteristic function: by definition, E[e iθi (ft) ] = exp θf t (x) α(x) m(dx). E

Characteristic function: by definition, E[e iθi (ft) ] = exp θf t (x) α(x) m(dx). E For the other definition, E[e iθi (ft) ] = exp E θf t (x) α(t) m(dx).

Link between the definitions We denote by L(t) the Lévy process defined by multistable measures: E[e iθl(t) ] = exp t 0 θ α(x) dx. The idea is to obtain the Ferguson-Klass-LePage representation of this process: for t (0, 1), we have: L(t) = i=1 K(α(V i ))γ i Γ 1/α(V i ) i 1 (Vi t).

Link between the definitions The link is explained by the following decomposition: almost surely, t (0, 1), + i=1 + γ i K(α(t))Γ 1 α(t) i 1 [0,t] (V i ) = γ i K(α(V i ))Γ 1 i where ε(t) = t + 0 i=1 i=1 γ i g i (s)1 [0,s[(V i )ds and g i (t) = K(α(t))Γ 1/α(t) i. α(v i ) 1 [0,t] (V i )+ε(t),

1 Multistable processes First definition : Ferguson-Klass-LePage series Properties of the distributions Second definition: multistable measures 2 How to test the multistability

How to test the multistability We want to know if the observations come from a stable process or from a multistable one. We want to test if α is varying with time. We consider the statistical test: H 0 : α is a constant vs H 1 : α is varying. We consider only the case of the Multistable Lévy motion, so h(t) = 1 α(t) and f (t, u, x) = 1 [0,t](x).

Estimation of h We need to estimate the function h, with the observation of one trajectory of Y. We define the sequence (Y k,n ) k Z,N N by Y k,n = Y ( k + 1 N ) Y ( k N ).

Estimation of h We need to estimate the function h, with the observation of one trajectory of Y. We define the sequence (Y k,n ) k Z,N N by Y k,n = Y ( k + 1 N ) Y ( k N ). Let t 0 R. We introduce an estimate of h(t 0 ) with 1 ĥ N (t 0 ) = n(n) log N [Nt 0 ]+ n(n) 2 1 k=[nt 0 ] n(n) 2 log Y k,n where (n(n)) N N is a sequence taking even integer values.

Estimation of h Theorem Under technical conditions on the general kernel f (t, u, x), if N lim N + n(n) = +, then r > 0, t 0, lim E ĥ N (t 0 ) h(t 0 ) r = 0. N +

Estimation of h Theorem Under technical conditions on the general kernel f (t, u, x), if N lim N + n(n) = +, then r > 0, t 0, lim E ĥ N (t 0 ) h(t 0 ) r = 0. N + Theorem For a Lévy process (f (t, u, x) = 1 [0,t] (x)), when lim n(n) = + and lim N + N + n(n) N = 0, t 0 (0, 1), n(n) ( log N(ĥ N (t 0 ) h(t 0 )) + E[ln S α(t0 ) ]) L N (0, E[Y 2 ]) where Y ln S α(t0 ) E[ln S α(t0 ) ].

Estimation of h, case of the Lévy motion α(t) = 1, 5 + 0, 48 sin(2πt), N = 20000, n(n) = 500. 1 Estimation de H, Levy Motion 0.95 0.9 0.85 0.8 1/alpha(t) 0.75 0.7 0.65 0.6 0.55 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 temps

Test of the multistability The hypotheses are : vs H 0 : (t 1, t 2 ) (0, 1) 2, α(t 1 ) = α(t 2 ). H 1 : (t 1, t 2 ) (0, 1) 2, α(t 1 ) α(t 2 ). We use the following statistic, for one t 0 : I N = Under (H 0 ), E[I N ] 0. 1 0 ĥ N (t) ĥ N (t 0 ) 2 dt. Under (H 1 ), E[I N ] 1 0 h(t) h(t 0) 2 dt > 0.

Results under H 0 Let σ 2 = E[Y 2 ] = Var(ln S α(t0 ) E[ln S α(t0 ) ]). We have the following convergence: 1 n(n)(log N) 2 ĥ N (t) ĥ N (t 0 ) 2 dt d σ 2 (1 + χ 2 (1)). 0

Proof Let µ = E[ln S α(t0 ) ]. We have n(n) ( log N(ĥ N (t 0 ) h(t 0 )) µ) d N (0, σ 2 ) so with Z N (t) = ( ) n(n) log N(ĥ N (t) h(t)) µ(t) N (0, σ 2 ) and h(t) = h(t) h(t 0 ), µ(t) = µ(t) µ(t 0 ), n(n)(log N) 2 1 0 ĥ N (t) ĥ N (t 0 ) 2 dt = 1 0 Z N (t) Z N (t 0 ) + n(n)((log N) h(t) + µ(t) ) 2 dt.

Under (H 0 ), we expand the square, Proof 1 0 Z N (t) Z N (t 0 ) 2 dt = 1 We are now able to control each term: 1 0 Z N(t) 2 dt P σ 2, 1 0 Z N(t)dt P 0, Z N (t 0 ) 2 σ 2 χ 2 (1). Finally, 0 1 Z N (t) 2 dt 2Z N (t 0 ) 1 n(n)(log N) 2 ĥ N (t) ĥ N (t 0 ) 2 dt d σ 2 (1 + χ 2 (1)). 0 0 Z N (t)dt+ Z N (t 0 ) 2.

Results under H 1 The rejection region of the test is n(n)(log 1 N)2 R c = { ˆσ 2 0 ĥ N (t) ĥ N (t 0 ) 2 dt > q β } with q β the quantile of the distribution 1 + χ 2 (1).

Results under H 1 The rejection region of the test is n(n)(log 1 N)2 R c = { ˆσ 2 0 ĥ N (t) ĥ N (t 0 ) 2 dt > q β } with q β the quantile of the distribution 1 + χ 2 (1). Statistical power for a Lévy process (multistable measures) Theorem (ε N ) N such that lim ε N = 0 and N + α = min α(u) and α = max α(u), lim ε N log N = +, if N + ( ) log P 1 Rc lim inf N + Nε N log N 2( 1 1 α α ).

Results under H 1 Statistical power for a Lévy process ( ) log P 1 Rc lim inf = +. N + log log N