Statistical inference for exponential functionals of Levy processes

Statistical inference for exp onen tial functionals of L ´ evy pro cesses ∗ Denis Belomestn y University of Duisbur g-Essen The a-L eymann-Str. 9, 45127 Essen, Germany e-mail: denis.belomestny@uni-due.de Vladimir P anov National r ese arch university Higher Scho ol of Ec onomics Shab olovka, 26, Mosc ow, 119049 Russia. e-mail: vpanov@hse.ru Abstract: In this paper, w e consider the exponential functional A ∞ = R ∞ 0 e − ξ s ds of a L´ evy process ξ s and aim to estimate the characteristics of ξ s from the distribution of A ∞ . W e presen t a new approac h, whic h allo ws to statistically infer on the L´ evy triplet of ξ t , and study the theoretical prop- erties of the prop osed estimators. The suggested algorithms are illustrated with numerical sim ulations. Keyw ords and phrases: L´ evy process, exponential functional, general- ized Ornstein-Uhlenbeck process. Con tents 1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Assumptions on the model . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Sub ordinators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 F urther assumptions on ν . . . . . . . . . . . . . . . . . . . . . . 5 3 Estimation of the L ´ evy triplet . . . . . . . . . . . . . . . . . . . . . . . 6 3.1 Estimation of the Laplace exp onen t . . . . . . . . . . . . . . . . 6 3.2 Estimation of a and c . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3 Recov ering the L ´ evy measure ν . . . . . . . . . . . . . . . . . . . 10 4 Sim ulation s tudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Theoretical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6 App endix. Additional pro ofs . . . . . . . . . . . . . . . . . . . . . . . 21 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∗ This research was partially supp orted by the Deutsc he F orsc hungsgemeinsc haft through the SFB 823 “Statistical mo delling of nonlinear dynamic pro cesses” and by Lab oratory for Structural Metho ds of Data Analysis in Predictive Modeling, MIPT, RF governmen t grant, ag. 11.G34.31.0073. 1 D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 2 1. In tro duction F or a L´ evy process ξ = ( ξ t ) t ≥ 0 , the exp onential functional of ξ is defined b y A t = Z t 0 e − ξ s ds, where t ∈ (0 , ∞ ). The main ob ject of this research is the terminal v alue A ∞ := lim t →∞ A t = Z ∞ 0 e − ξ s ds, (1) whic h often (and ev erywhere in this pap er) is called also an exp onen tial func- tional of ξ . The in tegral A ∞ naturally arises in a wide v ariety of financial ap- plications as an in v ariant distribution of the pro cess V t = e − ξ t  V 0 + Z t 0 e ξ s − ds  , (2) see Carmona, P etit, Y or [ 10 ]. F or instance, the pro cess ( 2 ) determines the volatil- it y pro cess in the COGARCH (COntinious Generalized AutoRegressive Condi- tionally Heteroscedastic) mo del introduced b y Kl ¨ upp elb erg et al. [ 17 ]. Note that V t is in fact a partial case of the generalized Ornstein-Uhlen b eck (GOU) process. A comprehensiv e study of the GOU mo del is giv en in the dissertation by Behme [ 2 ]. A ∞ app ears in finance also in other con texts, for instance, in pricing of Asian options, see the monograph by Y or [ 30 ] and the references given by Carmona, P etit, Y or [ 10 ]. As for other fields of applications, A ∞ pla ys a crucial role in studying the carousel systems (see Litv ak and Adan [ 21 ], Litv ak and Zw et [ 22 ]), self-similar fragmentations (see Bertoin and Y or [ 8 ]), and information transmis- sion problems (especially TCP/IP proto col, see Guillemin, Rob ert and Zw art [ 15 ]). F or the detailed discussion of the ph ysical in terpretations, w e refer to Com tet, Mon thus and Y or [ 11 ] and the dissertation b y Month us [ 24 ]. Denote the L´ evy triplet of the pro cess ξ t b y ( c, σ, ν ), i.e., ξ t = ct + σ W t + T t , (3) where T t is a pure jump pro cess with L ´ evy measure ν . The finiteness condition stands that the integral A ∞ is finite if and only if ξ t → + ∞ as t → + ∞ , see Maulik and Zw art [ 23 ] for the proof and Erickson and Maller [ 14 ] for some ex- tensions of this result. Therefore, the in tegral A ∞ is finite if the pro cess ξ t is an y non-degenerated sub ordinator, i.e., any non-decreasing L ´ evy process, or, equiv- alen tly , any non-negative L´ evy process. Nev ertheless, the finiteness condition is fulfilled for other pro cesses also, e.g., for ξ t = − N t + 2 λt , where N t is a Poisson pro cess with intensit y λ . In this pap er, w e mainly fo cus on the case when ξ is a sub ordinator with finite L´ evy measure. In terms of the L´ evy triplet, this means that c > 0, σ = 0, D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 3 ν ( I R − ) = 0 and moreov er a := ν ( I R + ) < ∞ . Supp ose that the pro cess ( 2 ) is observ ed in the time p oints 0 = t 0 < t 1 < ... < t n . T aking into account that the pro cess V t is a Marko v pro cess, and assuming that V 0 has an inv ariant distribu- tion determined b y A ∞ , we conclude that V t 0 , ..., V t n ha v e also the distribution of A ∞ . The main goal of this research is to statistically infer on the L´ evy triplet of ξ from the observ ations V t 0 , ..., V t n . More precisely , we will pursue the fol- lo wing tw o aims: (1) to estimate the drift term c and the parameter a ; (2) to estimate the L´ evy measure ν . T o the b est of our knowledge, the statistical inference for exp onential func- tionals of L ´ evy pro cesses has not been previously considered in the literature. Ho w e v er, some distributional prop erties of the exp onential functionals are well- kno wn, e.g., the in tegro - differential equation b y Carmona, Petit, Y or [ 9 ]. F or the ov erview of theoretical results, we refer to the survey b y Bertoin and Y or [ 8 ]. One distribution prop erty , the recursiv e formula for the moments of A ∞ , giv es rise to the approach presented in our paper. This result stands that E  A s − 1 ∞  = ψ ( s ) s E [ A s ∞ ] , (4) where ψ ( s ) is a Laplace exponent of the process ξ , i.e., ψ ( s ) := − log E  e − sξ 1  , and complex s is taken from the area Υ := n s ∈ C : 0 < Re( s ) < θ o with θ := sup  z ≥ 0 : E [ e − z ξ 1 ] ≤ 1  . (5) The recursive form ula ( 4 ) firstly app ear for real s in the pap er b y Maulik and Zw art [ 23 ]. The complete pro of for complex s was giv en recen tly by Kuznetsov, P ardo and Sa vo v [ 20 ]. If ξ t is a subordinator, what is the case under our setup, the parameter θ is equal to infinity . The idea of the pro cedure for solving the first task (estimation of a and c ) is to infer on the parameters of the process ξ from its Laplace exponent. First, making use of ( 4 ), we estimate the Laplace exp onent ψ ( s ) at the p oin ts s = u + i v ∈ Υ, where u is fixed and v v aries on the equidistant grid b etw een εV n and V n (with ε > 0 and V n → ∞ as n → ∞ ) Afterw ards, w e tak e into accoun t that ψ ( u + i v ) = a + c ( u + i v ) − F ¯ ν ( − v ) , u, v ∈ I R, (6) where ¯ ν ( dx ) := e − ux ν ( dx ), and F ¯ ν ( v ) stands for the F ourier transform of the measure ¯ ν , i.e., F ¯ ν ( v ) := R I R + e i v x ¯ ν ( dx ) . It is w orth mentioning that F ¯ ν ( v ) → 0 as v → ∞ , and therefore taking the real and imaginary parts of the left and righ t hand sides of ( 6 ), w e are able to consequently estimate the parameters c and a . With no doubt, the second aim (complete reco v ering of the L´ evy measure) is the most c hallenging task. Since the estimates of the parameters c and a are already obtained, we can estimate by ( 6 ) the F ourier transform F ¯ ν ( v ) for v taken from the equidistant grid [ − V n , V n ]. The last step of this pro cedure, estimation of the L´ evy measure ν , is based on the inv erse F ourier transform form ula, and reveals the main reason for using the complex num b ers in our D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 4 approac h. In fact, one can estimate the function ψ ( · ) in real p oints and then estimate the Laplace transform of the measure ν b y the regression argumen ts. In this case, estimation of ν demands the in verse Laplace transform, whic h is given b y Brom wich in tegral and therefore is in fact m uc h more in volv ed in comparison with the inv erse F ourier transform. The pap er is organized as follows. In the next section, we in tro duce the as- sumptions on this mo del and give some examples. W e formulate the algorithms for estimation the Laplace exp onent ψ ( s ) (Section 3.1), the parameters a and c (Section 3.2), and the L´ evy measure ν (Section 3.3). Next, we pro vide some nu- merical examples in Section 4 and analyze the conv ergence rates of the prop osed algorithms in Section 5. Appendix con tains some related results and additional pro ofs. 2. Assumptions on the mo del 2.1. Sub or dinators In this article, we restrict our attention to the case when the following set of assumptions is fulfilled: (A1) ( c ≥ 0 , σ = 0 , ν ( I R − ) = 0 , a := ν ( I R + ) < ∞ . This set in particularly yields that the pro cess ξ has finite v ariation, i.e., Z I R + ( x ∧ 1) ν ( dx ) < ∞ , (7) and therefore ξ is a non-decreasing L´ evy pro cesses, i.e., a sub ordinator. The detailed discussion of the sub ordination theory as well as v arious examples of suc h pro cesses (Gamma, Poisson, tempered stable, inv erse Gaussian, Meixner pro cesses, etc.), are giv en in [ 1 ], [ 7 ], [ 12 ], [ 26 ], [ 27 ]. Note that in the case of subordinators, the truncation function in the L´ evy- Khinc hine form ula can b e omitted, and therefore the characteristic exp onent of ξ is equal to ψ e ( s ) = log E  e i sξ 1  = i cs + Z ∞ 0  e i sx − 1  ν ( dx ) . (8) Later on, we use a Laplace exp onent of ξ , whic h is defined by ψ ( s ) := − log E  e − sξ 1  = − ψ e (i s ) , and under the assumption (A1) is equal to ψ ( s ) = cs + Z ∞ 0  1 − e − sx  ν ( dx ) (9) = cs + s Z ∞ 0 e − sx ν ( x, + ∞ ) dx. (10) D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 5 Some examples can b e found in Section 4 . In the sequel, we use the fact that the function ψ ( · ) is bounded from ab o ve on the set Υ b y | ψ ( s ) | ≤ c | s | + Z ∞ 0  1 + e − Re( s ) x  ν ( dx ) ≤ c q θ 2 + Im 2 ( s ) + 2 a, and hence the asymptotic b ehavior of the function ψ ( s ) is giv en b y | ψ ( s ) | = O  Im( s )  , Im( s ) → + ∞ . (11) 2.2. F urther assumptions on ν First, we assume the follo wing asymptotic b ehavior of the Mellin transform of the in tegral A ∞ : (A2)   E  A u ◦ +i v ∞     exp {− γ | v |} , as | v | → ∞ with some γ > 0 and u ◦ > 0. Second, w e in tro duce an assumption on the me asure ¯ ν ( dx ) := e − u ◦ x ν ( dx ) : (A3)   ¯ ν ( r )   L ∞ ( I R ) ≤ C for some p ositive r and C . It is worth noting that there is an (indirect) relation betw een the assumptions (A2) and (A3). In fact, joint consideration of ( 4 ) and ( 6 ) yields that F ¯ ν ( − v ) = a + c ( u ◦ + i v ) − ( u ◦ + i v ) m ( u ◦ + i v ) m (( u ◦ + 1) + i v ) , u, v ∈ I R, where m ( s ) := E  A s − 1 ∞  is the Mellin transform. Example 1. F or instance, the set of assumptions (A1) - (A3) fulfills for the class of L´ evy processes with c = 0 and L ´ evy density in the form ν ( x ) = I x> 0 M X j =1 m j X k =1 α j k x k − 1 e − ρ j x with M , m j ∈ N , ρ j > 0, α j k > 0. In fact, assumption (A1) and (A3) ob viously hold; assumption (A2) is c hec k ed in [ 19 ] for an y p ositive u ◦ (p. 658, the pro of of Theorem 1). Example 2. Next, we pro vide an example of the L´ evy pro cess which do esn’t p ossess the prop erty (A2). Consider a subordinator T with drift c > 0 and the L ´ evy densit y ν ( x ) = ab exp {− bx } I { x > 0 } , a, b > 0 , whic h we describ e in details in Section 4 . The exp onential functional of this pro cess has a densit y k ( x ) = C 1 x b (1 − cx ) ( a/c ) − 1 I { 0 < x < 1 /c } D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 6 with some C 1 > 0, see [ 9 ]. In other words, the exp onential functional has a distribution B ( α + 1 , β + 1) /c , where B stands for Beta distribution with pa- rameters α = b and β = a/c − 1. The Mellin transform of the function k ( x ) in the half -plane Re( s ) > − α is given by m ( s ) = C 2 ( α, β ) c s Γ( α + s ) Γ( α + β + 1 + s ) , where C 2 ( α, β ) > 0, see T able 1 from [ 13 ]. T aking in to account the following asymptotical behavior of the Gamma function | Γ( u + i v ) | = exp  − π 2 v +  u − 1 2  ln v + O (1)  , v → ∞ , see (9) from [ 19 ], we conclude that the exp onential functional of the pro cess T has a p olynomial decay of the Mellin transform. More precisely , | m ( s ) |  C 3 v − a/c − 1 with some C 3 > 0. 3. Estimation of the L´ evy triplet In the sequel, supp ose that the pro cess ( 2 ) is observed at the time p oints 0 = t 0 < t 1 < ... < t n . Assuming that V 0 has a stationary inv ariant distribution, w e get that the v alues A ∞ ,k := V t k , k = 1 ..n ha v e the distribution of the integral A ∞ . 3.1. Estimation of the L aplac e exp onent The first step of the estimation pro cedure is to construct the estimate of the function ψ ( s ) in the complex p oints s = u + i v , where u is fixed and v v aries. The reason for such choice of s is clear from the further steps of the algorithm. The estimator of ψ ( s ) is based on a recursive form ula for the s − th (complex) momen t of A ∞ : E  A s − 1 ∞  = ψ ( s ) s E [ A s ∞ ] , (12) In [ 9 ], this form ula is prov ed for real p ositive s suc h that ψ ( s ) > 0 and E [ A s ∞ ] < ∞ . The case of infinite mathematical exp ectations is carefully discussed in [ 23 ]. The case of complex s is considered in [ 20 ], where one can find also some generalizations of the formula ( 12 ) for integrals with resp ect to the Brownian motion with drift. In particular, applying Theorem 2 from [ 20 ], we get that ( 12 ) holds for an y s ∈ Υ. In the situation when ξ t is a sub ordinator, the set Υ coincides with the p ositive half-plane (equiv alently , the parameter θ is equal to infinit y), b ecause it follows from ( 10 ) that E  e − sξ 1  = − ψ ( − s ) = − cs − s Z I R + e sx ν ( x, + ∞ ) dx < 0 , s > 0 . D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 7 Motiv ated by ( 12 ), we no w presen t the the first t wo steps in the estimation pro cedure. Let the v alues α 1 , ..., α M comp ose the equidistant grid with the step ∆ > 0 on the set [ ε, 1], where ε > 0 and the sequence V n tends to infinity . First, w e estimate A s ∞ for s = u + i α m V n , m = 1 ..M and s = ( u − 1) + i v m , m = 1 ..M where u := u ◦ ∈ ( − 1 , θ ) satisfies the assumptions (A2) and (A3). Theoretical studies (see Section 5 ) show that the optimal choice is V n = κ log( n ) with κ < 1 / (2 γ ), provided that the Assumptions (A1)-(A3) hold. The estimator of A s ∞ is defined by b E n [ A s ∞ ] = 1 n n X k =1 A s ∞ ,k . (13) Next, w e define an estimate of ψ ( · ) at the p oints ( u + i α m V n ) b y ˆ ψ n ( u + i α m V n ) = ( u + i α m V n ) b E n h A ( u − 1)+i α m V n ∞ i b E n h A u +i α m V n ∞ i , m = 1 ..M . (14) The performance of this estimator is later sho wn in Section 4 , see in particularly Figures 1 and 3 . The quality of ˆ ψ n ( · ) is theoretically studied in Theorem 5.1 , whic h stands that under the assumptions (A1) and (A2) and the following con- dition on V n Λ n := V n exp { cV n } p log V n = o  r n log( n )  , n → ∞ , it holds for n large enough P ( sup v ∈ [ εV n ,V n ]    ˆ ψ n ( u + i v ) − ψ ( u + i v )    ≤ β Λ n r log( n ) n ) > 1 − αn − 1 − δ , (15) with some p ositive α , β and δ . 3.2. Estimation of a and c In Section 2.1 , we presen t the representation ( 9 ) for the Laplace exp onent of the process ξ . Substituting now the complex argumen t z = u + i v , we get ψ ( u + i v ) = c ( u + i v ) − Z I R + e − i v x ¯ ν ( dx ) + Z I R + ν ( dx ) = c ( u + i v ) − F ¯ ν ( − v ) + a, u, v ∈ I R, (16) where a := R I R + ν ( dx ) and ¯ ν ( dx ) := e − ux ν ( dx ). The general idea of the pro- cedure describ ed b elow is to estimate the Laplace exp onent ψ ( · ) at the p oints s = u + i v , where u is fixed at v v aries (see Section 3.1 ), and afterw ards to use ( 16 ) for consequent estimation of the parameters. D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 8 T aking imaginary and real of b oth hand sides in ( 16 ), we get Im ψ ( u + i v ) = cv − Im F ¯ ν ( − v ) , (17) Re ψ ( u + i v ) = cu − Re F ¯ ν ( − v ) + a. (18) By Riemann - Leb esque lemma, F ¯ ν ( − v ) → 0 as v → + ∞ , see, e.g., [ 16 ]; note that the rates of this conv ergence are assumed in (A3). Therefore, lo oking at ( 17 ), w e conclude that Im ψ ( u + i v ) is a (asymptotically) linear in v function, and the parameter c can be interpreted a slop e parameter. Next, from ( 18 ), it follo ws that Re ψ ( u + i v ) tends to ( cu + a ) as v → + ∞ . These observ ations lead to the following optimization problems ˜ c n := arg min c Z I R + w n ( v )  Im ˆ ψ n ( u + i v ) − cv  2 dv (19) ˜ a n := arg min a Z I R + w n ( v )  Re ˆ ψ n ( u + i v ) − ˜ c n u − a  2 dv , (20) where the weigh ting function is chosen in the form w n ( v ) = w ( v /V n ) /V n with an integrable non-negativ e function w ( · ) supp orted on [ ε, 1]. Under this choice of V n , w e can rewrite ( 19 ) as follows: ˜ c n := arg min c Z 1 ε w ( α )  Im ˆ ψ n ( u + i αV n ) − cαV n  2 dv ˜ a n := arg min a Z 1 ε w ( α )  Re ˆ ψ n ( u + i αV n ) − ˜ c n u − a  2 dv , In practice, w e first get the estimates of the Laplace exp onent at the points s = u + i α m V n (see abov e) and define an estimate of the parameter c b y ˆ c n := arg min c M X m =1 w ( α m )  Im ˆ ψ n ( u + i α m V n ) − cα m V n  2 (21) = P M m =1 w ( α m ) α m Im ˆ ψ n ( u + i α m V n ) V n · P M m =1 w ( α m ) α 2 m . (22) Afterw ards, w e estimate the parameter a by ˆ a n := arg m in a M X m =1 w ( α m )  Re ˆ ψ n ( u + i α m V n ) − ˆ c n u − a  2 (23) = P M m =1 w ( α m ) Re ˆ ψ n ( u + i α m V n ) P M m =1 w ( α m ) − ˆ c n u. (24) W e sho w empirical and theoretical prop erties of the estimators ˆ a n and ˆ c n b elo w, see Figure 2 and Theorem 5.3 . Similar to ( 15 ), w e prov e that under the D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 9 c hoice V n = κ log( n ) with κ < 1 / (2 γ ), it holds P ( | ˜ c n − c | ≤ ζ 1 log − ( r +2) ( n ) ) > 1 − αn − 1 − δ , and P ( | ˜ a n − a | ≤ ζ 2 log − ( r +1) ( n ) ) > 1 − αn − 1 − δ , with ζ 1 , ζ 2 > 0, and α , δ introduced abov e. Constants γ and r inv olved in this statemen t are comming from assumptions (A2) and (A3) resp. Moreov er, we pro v e Theorem 5.4 , whic h stands that this rate for c is optimal one in the class A of the mo dels satisfying the assumptions (A1) - (A3). More precisely , we sho w that lim n →∞ inf ˜ c ∗ n sup A P ( | ˜ c ∗ n − c | ≥ ζ 3 log − ( r +2) ( n ) ) > 0 , where ζ 3 < ζ 1 is some p ositive constant, the supremum is taken o v er all mo dels from A , and infimum - o ver all p ossible estimates of the parameter c . W e summarize the steps discussed ab ov e in the follo wing algorithm. D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 10 Algorithm 1: Estimation of a and c Data : n observ ations A ∞ , 1 , ..., A ∞ ,n of the integral A ∞ = R I R + exp {− ξ s } ds , where ξ = ( ξ t ) t ≥ 0 is a L´ evy process with unknown L´ evy triplet ( c, 0 , ν ). T ak e V n = κ log( n ) with κ < 1 / (2 γ ), fix ε ∈ (0 , 1) and u > − 1. T ak e the v alues α 1 , ..., α M on the equidistant grid on the set [ ε, 1] with a step ∆. Define a function w ( · ) ≥ 0 supp orted on [ ε, 1]. Denote v m,n := α m V n . 1. Estimate A s ∞ for s = u j + i v m,n , m = 1 ..M , where u 1 = u and u 2 = u − 1 b E n  A u j +i v m,n ∞  = 1 n n X k =1 A u j +i v m,n ∞ ,k , m = 1 ..M , j = 1 , 2 . 2. Estimate ψ ( u + i v m,n ) b y ˆ ψ n ( u + i v m,n ) = ( u + i v m,n ) b E n h A ( u − 1)+i v m,n ∞ i b E n h A u +i v m,n ∞ i , m = 1 ..M . 3. Estimate c by the solution of the optimization problem ( 21 ), whic h is explicitly giv en b y ˆ c n := P M m =1 w ( α m ) α m Im ˆ ψ n ( u + i v m,n ) V n · P M m =1 w ( α m ) α 2 m . 4. Estimate a by the solution of the optimization problem ( 23 ), whic h is explicitly giv en b y ˆ a n := P M m =1 w ( α m ) Re ˆ ψ n ( u + i v m,n ) P M m =1 w ( α m ) − ˆ c n u. 3.3. R e c overing the L´ evy me asur e ν As the result of the algorithm describ ed b elow, we obtain the estimates ˆ c n and ˆ a n of the parameters c and a . In this subsection, we presen t the algorithm for estimation the L´ evy measure ν . First, we take p oints s = u + i α m V n , where α m , m = 1 ..M , b elong to the in terv al [ − 1 , 1]. The construction of the estimates ˆ ψ n ( s ) remains the same as in Section 3.1 . Next, lo oking at ( 16 ), we define an estimate F ¯ ν ( − v ) for v = D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 11 α m V n , m = 1 ..M by ˆ F ¯ ν ( − v ) = − ˆ ψ n ( u + i v ) + ˆ c n ( u + i v ) + ˆ a n . (25) The last step is to reco ver the measure ν from the estimator of the F ourier transform of the measure ¯ ν . Motiv ated b y the inv erse F ourier transform form ula, w e propose the following nonparametric estimator of the measure ν : ˜ ν ( x ) = 1 2 π e ux Z I R e i v x ˆ F ¯ ν ( − v ) K ( v h n ) dv , (26) where K is a regularizing k ernel supp orted on [ − 1 , 1] and h n is a sequence of bandwidths whic h tends to 0 as n → ∞ . The formal description of the algorithm is giv en below. Algorithm 2: Estimation of ν Data : n observ ations A ∞ , 1 , ..., A ∞ ,n of the integral A ∞ = R I R + exp {− ξ s } ds , where ξ = ( ξ t ) t ≥ 0 is a L´ evy process with unknown L´ evy triplet ( c, 0 , ν ). The estimates ˆ a n and ˆ c n are described in Algorithm 1. T ak e the v alues α 1 , ..., α M on the equidistant grid on the set [ − 1 , 1] with a step ∆. Denote v m,n := α m V n . Define a regularizing k ernel K supp orted on [ − 1 , 1], and a (large enough) n umber h . 1-2 The first tw o steps coincide with giv en in Algorithm 1. 3. Estimate F ¯ ν ( − v m,n ) for ¯ ν ( dx ) = e − ux ν ( dx ) b y ˆ F ¯ ν ( − v m,n ) = − ˆ ψ n ( u + i v m,n ) + ˆ c n · ( u + i v m,n ) + ˆ a n , m = 1 ..M . 4. Estimate ν by ˆ ν ( x ) = e ux ∆ 2 π M X m =1 e i v m,n x ˆ F ¯ ν ( − v m,n ) K ( v m,n h ) . Some theoretical and practical asp ects of this algorithm are discussed in Sections 4 and 5 . Remark 3.1. It is a worth mentioning that the estimation algorithms 1 and 2 c an b e applie d to mor e gener al situation when the pr o c ess T t is a differ enc e b etwe en two sub or dinators, i.e., T t = T + t + T − t , wher e T + and T − ar e the pr o c esses of finite variation with L ´ evy me asur es ν + and ν − c onc entr ate d on I R + and I R − r esp. In fact, in this c ase, the formula ( 16 ) stil l holds with ν ( dx ) = I I { x > 0 } ν + ( dx ) + I I { x < 0 } ν − ( dx ) . D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 12 Ther efor e, the c onse quent estimation of c , a and the F ourier tr ansform of the me asur e e − ux ν ( dx ) , as wel l as the estimation of ν ar e stil l p ossible. The or etic al r esults under the assumptions (A2) and (A3) r emain the same. The example fr om Se ction 2.2 c an b e natur al ly extende d to ν ( x ) = I x> 0 M X j =1 m j X k =1 α j k x k − 1 e − ρ j x + I x< 0 ˜ M X j =1 ˜ m j X k =1 ˜ α j k x k − 1 e − ˜ ρ j x with M , ˜ M , m j , ˜ m j ∈ N , ρ j , ˜ ρ j > 0 , α j k , ˜ α j k > 0 . Note that Assumption (A2) is alr e ady che cke d in The or em 1 fr om [ 19 ]. 4. Sim ulation study Example 1. Consider the sub ordinator T t with the L´ evy densit y ν ( x ) = ab exp {− bx } I { x > 0 } , a, b > 0 . (27) F or this sub ordinator, the integral A ∞ is finite for an y σ , see [ 9 ]. The Laplace exp onen t of ξ is giv en b y ψ ( z ) = z  c − 1 2 σ 2 z + a b + z  . (28) As for the distribution prop erties of A ∞ , the densit y function of A ∞ satisfies the follo wing differen tial equation − σ 2 2 x 2 k 00 ( x ) +  σ 2 2 (3 − b ) + c  x − 1  k 0 ( x ) +  (1 − b )  σ 2 2 + c  − a + b x  k ( x ) = 0 , (29) see [ 9 ]. Some t ypical situations are given b elow: 1. In the case c = 0 , σ = 0 (pure jump pro cess), this equation has a solution k 1 ( x ) = C x b e − ax I { x > 0 } , (30) and therefore A ∞ d G ( b + 1 , a ), where G ( α , β ) is a Gamma distribution with shape parameter α and rate β . 2. If c > 0 , σ = 0 (pure jump pro cess with drift), then k 2 ( x ) = C x b (1 − cx ) ( a/c ) − 1 I { 0 < x < 1 /c } . (31) In this situation A ∞ d B ( b +1 , a/c ) /c , where B ( α, β ) is a Beta - distribution. D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 13 −30 −10 10 30 54.670 54.675 54.680 54.685 54.690 54.695 Im(s) Re −30 −10 10 30 −40 −20 0 20 40 Im(s) Im −30 −10 10 30 55 60 65 70 75 Im(s) Abs Fig 1 . Plots of the oretic al (blue dashe d) and empiric al (re d solid) L aplac e exponents for Exam- ple 1. Gr aphs present r e al, imaginary p arts and absolute values. In spite of visual distinction in the r e al p arts, the differenc e b etwe en the or etic al and empiric al L aplac e exp onents is quite smal l. 3. In the case c 6 = 0 , σ 6 = 0, the equation ( 29 ) also allo ws for the closed form solutions. Assuming for simplicity σ 2 / 2 = 1, c = − ( b + 1), w e get the solution of ( 29 ) in the following form: k 3 ( x ) = C x b − 1 / 2 exp  1 2 x  I µ  1 2 x  , (32) where we denote by I µ the mo dified Bessel function of the first kind, µ = p a + 1 / 4, and the constan t c is later c hosen to guarantee the condition R ∞ 0 k 3 ( x ) dx = 1. F or the n umerical study , w e assume that the data follo ws the mo del ( 1 ) where the pro cess ξ t is defined by ( 3 ) with c = 1 . 8, σ = 0, and the sub ordinator T t has a L´ evy densit y in the form ( 27 ) with a = 0 . 7, b = 0 . 2. The v alues of the integral A ∞ are sim ulated from the Beta-distribution, see ( 31 ). On the first step, we estimate A s ∞ for s = u + i v with u = 29 and u = 30 and v from the equidistant grid b etw een − 30 and 30. Next, we estimate the Laplace exp onen t b y the formula ( 14 ). One can visually compare the prop osed estimator and the theoretical v alue ( c + a/ ( b + s )) ∗ s looking at Figure 1 . Estimation of the parameters c and a is provided by ( 22 ) and ( 25 ) resp. The b o xplots of this estimates are presented on Figure 2 . Example 2. Consider the comp ound Poisson pro cess ξ t = − log q N t X k =1 η k ! , D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 14 1 3 5 10 1.798 1.800 1.802 c 1000*n 1 3 5 10 0.60 0.65 0.70 0.75 0.80 a 1000*n Fig 2 . Boxplots for the estimates of c and a for differ ent values of n b ase d on 25 simulation runs. where q ∈ (0 , 1) is fixed, N t is a Poisson pro cess with intensit y λ and η k are i.i.d. random v ariables with a distribution L. It is a w orth mentioning that the in tegral A ∞ allo ws the representation A ∞ = Z ∞ 0 q − ξ t dt = ∞ X n =0 q − n ( T n +1 − T n ) , where T n is the jump time T n = inf { t : N t = n } . Note that if η k tak es only p ositiv e v alues then − ξ t is a sub ordinator. F or the ov erview of the prop erties of the integral A ∞ in the particular case L ≡ 1 (that is, ξ t is a Poisson pro cess up to a constant), we refer to [ 8 ]. Fix some p ositive α and consider the case when L is the standard Normal distribution truncated on the interv al ( α, + ∞ ). The density function of L is giv en b y p L ( x ) = p ( x ) / (1 − F ( α )) , where p ( · ) and F ( · ) are p df and cdf of the standard Normal distribution. In this case, the Laplace exponent of ξ t is equal to ψ ( s ) = λ " 1 − 1 − F ( α + (log q ) s ) 1 − F ( α ) exp ( − (log q ) 2 s 2 2 )# , where the function F ( · ) in the complex p oint z can b e calculated from the error function: F ( z ) := 1 2  erf  z √ 2  + 1  , where erf ( z ) = 2 √ π Z z 0 e − s 2 ds. D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 15 −4 0 2 4 0.4 0.6 0.8 1.0 v X1 Re −4 0 2 4 −0.4 −0.2 0.0 0.2 0.4 v X1 Im −4 0 2 4 0.4 0.6 0.8 1.0 1.2 v X1 Abs Fig 3 . Plots of the oretic al (blue dashed) and empiric al (re d solid) Laplac e exponents for Example 2. Gr aphs pr esent re al, imaginary and absolute values. F or v ∈ [ − 3 , 3] the curves ar e visual ly indistinguishable. In this example, we aim to estimate the L ´ evy measure of the pro cess ξ t , whic h is equal to ν ( dx ) = λ 1 − F ( α ) p ( x ) I { x > α } dx. F or the numerical study , we tak e q = 0 . 5 , α = 0 . 1, and λ = 1. First, we estimate the Laplace exp onent by ( 14 ). The qualit y of estimation at the complex p oints s = u + i v with u = 1 and v ∈ [ − 5 , 5] can b e visually c heck ed on Figure 3 . Next, w e pro ceed with the estimation of the F ourier transform of the measure ¯ ν ( x ) := e − ux ν ( x ) of the L´ evy measure by applying ( 25 ). F or the last step of the Algorithm 2, reconstruction of the L´ evy measure by ( 26 ), w e follow [ 4 ] and tak e the so-called flat-top k ernel, which is defined as follo ws: K ( x ) =        1 , | x | ≤ 0 . 05 , exp  − e − 1 / ( | x |− 0 . 05) 1 −| x |  , 0 . 05 < | x | < 1 , 0 , | x | ≥ 1 . The qualit y of the resulted estimation is giv en on Figure 4 . 5. Theoretical study Theorem 5.1. Consider the mo del ( 1 ) with L´ evy pr o c ess ξ in the form ( 3 ) satisfying the assumptions (A1) - (A3). L et the se quenc e V n tend to ∞ and mor e over satisfy the assumption Λ n := V n exp { γ V n } p log V n = o  r n log( n )  , n → ∞ , (33) D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 16 0 2 4 6 8 0.0 0.1 0.2 0.3 z Y Re 0 2 4 6 8 −1.0e−17 5.0e−18 2.0e−17 z Y Im Fig 4 . Plots of the L ´ evy me asur e (blue dashe d line) and its estimate (r e d solid) depicte d for r e al (left) and imaginary (right) parts. Note that the values on the right plot ar e quite smal l. wher e the c onstant γ is intr o duc e d in (A2). Then ther e exists a set W n such that P {W n } > 1 − αn − 1 − δ (with some p ositive α and δ ) and W n ⊂ ( sup v ∈ [ εV n ,V n ]    ˆ ψ n ( u + i v ) − ψ ( u + i v )    ≤ β Λ n r log( n ) n ) (34) wher e β > 0 and u = u ◦ was intr o duc e d in (A2). Remark 5.2. The c ondition ( 33 ) fulfil ls for instanc e for V n = κ log ( n ) with κ < 1 / (2 γ ) . Pr o of. 1. Denote J ( s ) :=  b E n [ A s ∞ ] − E [ A s ∞ ]  / E [ A s ∞ ] , where s = u + i v . In this notation,      ψ ( s ) − ˆ ψ n ( s )      =      s E  A s − 1 ∞  E [ A s ∞ ] − b E n  A s − 1 ∞  b E n [ A s ∞ ] !      =      s E  A s − 1 ∞  E [ A s ∞ ]      ·      J ( s ) − J ( s − 1) 1 + J ( s )      . (35) By ( 12 ), the first term is equal to | ψ ( s ) | , and therefore by ( 11 ) it is b ounded by C 1 Im( s ) for Im( s ) large enough with some C 1 > 0. As for the second term, we firstly note that      J ( s ) − J ( s − 1) 1 + J ( s )      ≤ | J ( s ) | + | J ( s − 1) | 1 − | J ( s ) | . D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 17 The aim of the further pro of is to sho w that the right hand side in the last inequalit y is bounded b y p log( n ) /n on a probabilit y set with desired properties. 2. Prop osition 6.2 yields that there exists such set W n of probability mass larger than 1 − αn − 1 − δ , suc h that it holds on this set sup s : Im( s ) ∈ I n    b E n [ A s ∞ ] − E [ A s ∞ ]    . p log( V n ) log ( n ) /n, n → ∞ , (36) where α and δ are p ositive, and I n := [ εV n , V n ]. In fact, direct application of Prop osition 6.2 with a weigh ting function w ∗ ( x ) := log − 1 / 2 ( e + | x | ) giv es sup v ∈ I n    b E n  A u +i v ∞  − E  A u +i v ∞     ≤ sup v ∈ I n     w ∗ ( v ) inf x ∈ I n w ∗ ( x )    ·     b E n  A u +i v ∞  − E  A u +i v ∞      i ≤ p log ( e + V n ) · sup v ∈ I n    w ∗ ( v )  b E n  A u +i v ∞  − E  A u +i v ∞      . p log( V n ) log( n ) /n, n → ∞ . 3. F ormula ( 36 ) in particularly means the follo wing inequlity holds on the set W n sup s : Im( s ) ∈ I n | J ( s ) | . exp { γ V n } p log( V n ) log( n ) /n n → ∞ , . (37) It is worth mentioning that under the assumption ( 33 ), sup s : Im( s ) ∈ I n | J ( s ) | → 0 as n → ∞ . (38) Substituting ( 37 ) in to ( 35 ) and taking in to account ( 11 ) and ( 38 ), w e arrive at the follo wing bound for the quality of the estimate ˆ ψ n ( s ):    ψ ( s ) − ˆ ψ n ( s )    . V n exp { γ V n } p log( V n ) log ( n ) /n, whic h holds on the set W n . This observ ation completes the pro of. Theorem 5.3. Consider the setup of The or em 5.1 and take V n = κ log( n ) with κ < 1 / (2 γ ) . Then it holds W n ⊂ ( | ˜ c n − c | ≤ ζ 1 log r +2 ( n ) ) and W n ⊂ ( | ˜ a n − a | ≤ ζ 2 log r +1 ( n ) . ) , (39) wher e s is intr o duc e d in (A3), the set W n is define d in The or em 5.1 , and ζ 1 , ζ 2 > 0 . D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 18 Pr o of. 1. First note that the estimate ( 19 ) can b e rewriten as ˜ c n = Z ∞ 0 w ∗ n ( v ) Im ˆ ψ n ( u + i v ) dv , where w ∗ n ( v ) = w n ( v ) v R w n ( y ) y 2 dy = 1 V 2 n w ∗  v V n  , where w ∗ ( x ) = ( w ( x ) x ) /  R w ( y ) y 2 dy  . Next, consider the following “theoretical coun terpart” of the estimate ˜ c n : ¯ c n := Z ∞ 0 w ∗ n ( v ) Im ψ ( u + i v ) dv , and note that | ˆ c n − c | ≤ | ˆ c n − ¯ c n | + | ¯ c n − c | . (40) The first summand in the righ t hand side of ( 40 ) is bounded on the set W n for n large enough: | ˆ c n − ¯ c n | ≤ A Λ n r log( n ) n 1 V n , where A := V n β     R w ∗ n ( v ) v dv R w ∗ n ( v ) v 2 dv     =      R 1 ε w ∗ ( v ) v dv R 1 ε w ∗ ( v ) v 2 dv      (41) do esn’t depend on n . As for the second term, using R w ∗ n ( v ) v dv = 1, we get | ¯ c n − c | =     Z ∞ 0 w ∗ n ( v ) h Im ˆ ψ n ( u + i v ) dv − cv i dv     =     Z ∞ 0 w ∗ n ( v ) Im F ¯ ν ( − v ) dv     . Applying Lemma 6.3 with w ∗ n ( v ) = V − 2 n w ∗ 1 ( v /V n ), w e get    Z ∞ 0 w ∗ n ( v ) F ¯ ν ( v ) dv    . V − ( r +2) n , n → ∞ . (42) Substituting ( 41 ) and ( 42 ) into ( 40 ), and b earing in mind our c hoice of V n , w e complete the proof of the first em b edding in ( 39 ). 2. Without limitations w e can assume that R I R + w n ( v ) dv = R 1 ε w ( v ) dv = 1. The second embedding directly follows from Theorem 5.1 and the first part of D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 19 this proof, because | ˜ a n − a | =      " Z I R + w n ( v ) Re ˆ ψ n ( u + i v ) dv − ˜ c n u # − " Z I R + w n ( v )  Re ψ ( u + i v ) + Re F ¯ ν ( − v )  dv − cu #      ≤ | ˜ c n − c | u +      Z I R + w n ( v )  Re ˆ ψ n ( u + i v ) − Re ψ ( u + i v )  dv      +      Z I R + w n ( v ) Re F ¯ ν ( − v ) dv      . ζ 1 log r +2 ( n ) + β Λ n r log( n ) n + λ log r +1 ( n ) . 1 log r +1 ( n ) , n → ∞ , where λ > 0. Note that here we use the inequality      Z I R + w n ( v ) Re F ¯ ν ( − v ) dv      . log − ( r +1) ( n ) , whic h follo ws b y applying Lemma 6.3 to w n ( v ) = V − 1 n w n ( v /V n ). This completes the proof. Theorem 5.4. L et A b e a set of functions that satisfy assumptions (A1) - (A3). Then it holds lim n →∞ inf ˜ c ∗ n sup A P ( | ˜ c ∗ n − c | ≥ ζ 3 log − ( r +2) ( n ) ) > 0 , wher e ζ 3 is some p ositive c onstant, the supr emum is taken over al l mo dels fr om A , and infimum - over al l p ossible estimates of the p ar ameter c . Pr o of. W e follo w the general reduction scheme, whic h can b e found in [ 18 ] and [ 28 ]. Consider a class of L ´ evy processes A that satisfies the assumptions (A1)- (A3). There exist t w o L´ evy pro cess ξ 0 and ξ 1 from A , having L´ evy triplets ( c 0 , 0 , ν 0 ), ( c 1 , 0 , ν 1 ), Laplace exp onents φ 0 , φ 1 , exp onential functionals with densities p 0 , p 1 and Mellin transforms M 0 , M 1 , such that it holds sim ultaneously 1. the L ´ evy triplets are related by the follo wing iden tities: c 0 − c 1 = 2 δ, ν 0 ( x ) − ν 1 ( x ) = 2 δ K 0 h ( x ) , (43) where δ > 0, K h ( x ) = h − 1 K  h − 1 x  for an y x ∈ I R and some h > 0, and K ∈ L 1 ( C ) satisfy F K ( z ) = − 1 for z with Re( z ) ∈ [ − 1 , 1] and p olynomical deca y | F K ( z ) | . | R e ( z ) | − η as | z | → ∞ . D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 20 2. the density of one of the functionals, say the first one, decays at most p olynomially , i.e., there exists m ∈ N such that p 0 ( x ) & (1 + x ) − 2 m , x → + ∞ . 3. M 0 ( s ) and M 1 ( s ) coincide on the lines s = u ( k ) + i v , k = 1 , 2 , for u (1) = 3 / 2, u (2) = m + 3 / 2, and any v . Moreo ver, the asymptotics of the Mellin transforms along these lines is given b y (A2), i.e., | M j ( u ( k ) + i v ) |  exp {− γ ( k ) | v |} , as v → ∞ , with some γ ( k ) > 0 , k = 1 , 2 , j = 0 , 1 . Let the exp onential functionals of these L ´ evy pro cesses hav e distribution laws P 0 and P 1 . χ 2 n (1 | 0) := χ 2 ( P ⊗ n 1 | P ⊗ n 0 ) ≤ exp  nχ 2 ( P 1 | P 0 )  − 1 , see Lemma 5.5 from [ 5 ]. The aim is to show that there exist a constant C > 0 suc h that χ 2 n (1 | 0) < C ; after that the desired result will immediately follow, see P art 2 and especially Theorem 2.2 from [ 28 ]. Our c hoice of the mo dels leads to the follo wing estimate of the chi-squared distance betw een P 1 and P 2 : χ 2 (1 | 0) := χ 2 ( P 1 | P 0 ) = Z I R + ( p 1 ( x ) − p 0 ( x )) 2 p 0 ( x ) . Z I R + (1 + x 2 m ) ( p 1 ( x ) − p 0 ( x )) 2 dx. (44) By Lemma 6.4 w e get that χ 2 (1 | 0) . ∆(0) + ∆( m ) , where ∆( · ) := Z ∞ −∞ | M 0 ( · + 1 / 2 + i v ) − M 1 ( · + 1 / 2 + i v ) | 2 dv . (45) Note that by ( 4 ) and our assumptions, M 0 ( s − 1) − M 1 ( s − 1) = φ 0 ( s ) − φ 1 ( s ) s M 0 ( s ) , (46) where s = u ( k ) + i v , k = 1 , 2. By our choice of the L´ evy measures ( 43 ) and the represen tation of the Laplace exp onent ( 9 ), we get φ 0 ( s ) − φ 1 ( s ) = ( c 0 − c 1 ) s + Z I R +  1 − e − sx  ν 0 ( dx ) − Z I R +  1 − e − sx  ν 1 ( dx ) = 2 δ s + Z I R + [ ν 0 ( dx ) − ν 1 ( dx )] − [ F ν 0 (i s ) − F ν 1 (i s )] = 2 δ s + 2 δ Z I R + K 0 h ( x ) dx − 2 δ F K 0 h (i s ) D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 21 Next, w e tak e in to account that F K 0 h ( y ) = i y F K h ( y ) = i y F K ( y h ) for any y ∈ C . Therefore R I R + K 0 h ( x ) dx = F K 0 h (0) = 0 and moreov er φ 0 ( s ) − φ 1 ( s ) = 2 δ s (1 + F K (i sh )) . (47) Substituting ( 47 ) into ( 46 ), we arrive at M 0 ( s − 1) − M 1 ( s − 1) = 2 δ (1 + F K (i sh )) M 0 ( s ) , and therefore ∆( · ) = δ Z I R    1 + F K  − v + i u ( k )  h     2 ∗    M 0  u ( k ) + i v     2 dv , where k = 1 if · = 0 and k = 2 if · = m . By our assumptions on the k ernel K , w e get ∆( · ) . δ Z | v | > 1 /h e − γ ( k ) | v | dv = δ γ ( k ) e − γ ( k ) /h , and therefore χ 2 (1 | 0) . δ γ ∗ e − γ ∗ /h , with γ ∗ := min n γ (1) , γ (2) o . If we choose δ = h s +2 and h = log − 1 ( n ) γ ∗ / (1 + ε ) for any (small) ε > 0, the χ 2 - div ergence is b ounded by χ 2 (1 | 0) = ( γ ∗ ) s +2 (1 + ε ) s +2 log − ( s +2) ( n ) n 1+ ε . log( C + 1) n for an y C > 0 an n large enough. Therefore, χ 2 n (1 | 0) ≤ exp { nχ 2 (1 | 0) } − 1 ≤ C, and the statement of the theorem follo ws. 6. App endix. Additional pro ofs Lemma 6.1 (Exp onential inequalities for dep endent sequences) . L et ( G k , k ≥ 1) b e a se quenc e of c enter e d r e al-value d r andom variables on the pr ob ability sp ac e (Ω , F , P ) . Assume that 1. G k is a str ongly mixing se quenc e with the mixing c o efficients satisfying α G ( n ) ≤ ¯ α 0 exp {− ¯ α 1 n } , n ≥ 1 , ¯ α 0 > 0 , ¯ α 1 > 0; (48) 2. sup k ≥ 1 | G k | ≤ M a.s. for some p ositive M ; D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 22 3. the quantities ρ k := E h G 2 k | 2 log G k | 2(1+ ε ) i , k = 1 , 2 , . . . , ar e finite for al l k with some smal l ε > 0 . Then ther e is a p ositive c onstant C 1 dep ending on ¯ α := ( ¯ α 0 , ¯ α 1 ) such that P ( n X k =1 G k ≥ β ) ≤ exp  − C 1 β 2 nv 2 + M 2 + M β log 2 n  . for al l β > 0 and n ≥ 4 , wher e v 2 ≤ sup k E [ G 2 k ] + C 2 sup k ρ k with C 2 > 0 . Pr o of. The pro of directly follo ws from Theorem A.1 and Corollary A.2 from [ 6 ]. The next result gives the uniform probabilistic inequality for the empirical pro cess. This result is an analogue of Prop osition A.3 from [ 6 ], whic h giv es the uniform inequalit y for the case when u = 0 (see b elow). F or similar results in i.i.d. case, see [ 25 ]. Prop osition 6.2. L et Z j , j = 1 , . . . , n, b e a stationary se quenc e of r andom variables. Define ϕ n ( v ) := 1 n n X j =1 exp { ( u + i v ) Z j } , wher e u ∈ I R + is fixe d and v ∈ I R varies. L et ϕ ( v ) b e a char acteristic function of the c orr esp onding stationary distribution. L et also w b e a p ositive monotone de cr e asing Lipschitz function on R + such that 0 < w ( z ) ≤ 1 p log( e + | z | ) , z ∈ R . (49) Supp ose that the fol lowing assumptions hold: (A1) r andom variables e Z j p ossess finite absolute moments of or der p > 2 . (A2) Z j is a str ongly mixing se quenc e with the mixing c o efficients satisfying α Z ( n ) ≤ ¯ α 0 exp {− ¯ α 1 n } , n ≥ 1 , ¯ α 0 > 0 , ¯ α 1 > 0 . (50) Then ther e ar e δ 0 > 0 and ζ 0 > 0 , such that the ine quality P  r n log n k ϕ n − ϕ k L ∞ ( R ,w ) > ζ  ≤ B ζ − p n − 1 − δ 0 . (51) holds for any ζ > ζ 0 and some p ositive c onstant B not dep ending on ζ and n. D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 23 Pr o of. Denote W 1 n ( v ) := w ( v ) n n X j =1  e ( u +i v ) Z j I  e uZ j < Ξ n  − E h e ( u +i v ) Z I  e uZ < Ξ n  i , W 2 n ( v ) := w ( v ) n n X j =1  e ( u +i v ) Z j I  e uZ j ≥ Ξ n  − E h e ( u +i v ) Z I  e uZ ≥ Ξ n  i , where Z is a random v ariable with stationary distribution of Z j . The main idea of the pro of is to show that P ( |W 1 n ( v ) | > ζ r log n n ) ≤ e B 1 ζ − p n − 1 − δ 0 , (52) P ( |W 2 n ( v ) | > ζ r log n n ) ≤ e B 2 ζ − p n − 1 − δ 0 , (53) with Ξ n = ... and some positive e B 2 and e B 2 . Step 1. The aim of the first step is to show ( 52 ). The pro of follows the same lines as the proof of Proposition A.3 from [ 6 ]. 1.1. Consider the sequence A k = e k , k ∈ N and cov er each in terv al [ − A k , A k ] b y M k = ( b 2 A k /γ c + 1) disjoint small interv als Λ k, 1 , . . . , Λ k,M k of the length γ . Let v k, 1 , . . . , v k,M k b e the cen ters of these in terv als. W e hav e for an y natural K > 0 max k =1 ,...,K sup A k − 1 < | v |≤ A k |W 1 n ( v ) | ≤ max k =1 ,...,K max 1 ≤ m ≤ M k sup v ∈ Λ k,m |W 1 n ( v ) − W 1 n ( v k,m ) | + max k =1 ,...,K max n 1 ≤ m ≤ M k : | v k,m | >A k − 1 o |W 1 n ( v k,m ) | . Hence for any p ositive λ , P max k =1 ,...,K sup A k − 1 < | v |≤ A k |W 1 n ( v ) | > λ ! ≤ P sup | v 1 − v 2 | <γ |W 1 n ( v 1 ) − W 1 n ( v 2 ) | > λ/ 2 ! + K X k =1 X n 1 ≤ m ≤ M k : | v k,m | >A k − 1 o P ( |W 1 n ( v k,m ) | > λ/ 2) . (54) The aim of the next tw o steps is to get the upp er b ounds for the summands in the right hand side, where λ is taken in the form λ = ζ p (log n ) /n with arbitrary large enough ζ . D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 24 1.2. W e pro ceed with the first summand in ( 54 ). It holds for any v 1 , v 2 ∈ R |W 1 n ( v 1 ) − W 1 n ( v 2 ) | ≤ | w ( v 1 ) − w ( v 2 ) | × max v     W 1 n ( v ) w ( v )     ·     W 1 n ( v 1 ) w ( v 1 ) − W 1 n ( v 2 ) w ( v 2 )     × max v [ w ( v )] ≤ 2 Ξ n | w ( v 1 ) − w ( v 2 ) | + 1 n n X j =1 h    e ( u +i v 1 ) Z j − e ( u +i v 2 ) Z j    I  e uZ j < Ξ n  i +    E h e ( u +i v 1 ) Z − e ( u +i v 2 ) Z  I  e uZ < Ξ n  i    ≤ | v 1 − v 2 | Ξ n   2 L w + 1 n n X j =1 | Z j | + E | Z |   , (55) where L ω is the Lipschitz constant of w and Z is a random v ariable distributed b y the stationary la w of the sequence { Z j } . Next, the Mark ov inequalit y implies P    1 n n X j =1 h | Z j | − E | Z | i > c    ≤ c − p n − p E       n X j =1 h | Z j | − E | Z | i       p for any c > 0 . Using now Y oko yama inequality [ 29 ] and taking into accoun t the assumptions of the contin uity of momen ts of Z j and the assumption 1 from Lemma 6.1 , we get E       n X j =1 h | Z j | − E | Z | i       p ≤ C p ( ¯ α ) n p/ 2 , where C p ( ¯ α ) is some constant dep ending on ¯ α = ( ¯ α 0 , ¯ α 1 ) and p . Returning to our c hoice of γ and λ , which in particularly yields that γ = λ/ζ = p (log n ) /n, w e obtain from ( 55 ) P n sup | v 1 − v 2 | <γ |W 1 n ( v 1 ) − W 1 n ( v 2 ) | > λ/ 2 o ≤ P    1 n n X j =1 h | Z j | − E | Z | i > ζ 2Ξ n − 2 L w − 2 E | Z |    ≤ B 0 c p ( ¯ α )  ζ / (2Ξ n ) − 2 L w − 2 E | Z |  − p n − p/ 2 ≤ B 1 ζ − p Ξ p n n − p/ 2 with some constan ts B 0 , B 1 not dep ending on ζ and n, provided ζ is large enough. 1.3. No w w e turn to the second term on the righ t-hand side of ( 54 ). Applying Lemma 6.1 with G k = n Re  W 1 n ( u k,m )  and β = nλ , w e get P  | Re  W 1 n ( v k,m )  | > λ/ 4  ≤ K , D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 25 where K := exp − B 3 λ 2 n B 2 Ξ 2 n w 2 ( A k − 1 ) log 2(1+ ε ) (Ξ n w ( A k − 1 )) + λ log 2 ( n )Ξ n w ( A k − 1 ) ! with some constan ts B 2 and B 3 dep ending only on the characteristics of the pro cess Z . Similarly , applying the same result with G k = n Im  W 1 n ( u k,m )  , w e conclude that P  | Im  W 1 n ( v k,m )  | > λ/ 4  ≤ K , and therefore X {| v k,m | >A k − 1 } P ( |W 1 n ( v k,m ) | > λ/ 2) ≤ ( b 2 A k /γ c + 1) K . Set no w γ = p (log n ) /n and λ = ζ p (log n ) /n and note that under our c hoice of Ξ n , Ξ 2 n w 2 ( A k − 1 ) log 2(1+ ε ) (Ξ n w ( A k − 1 )) & λ log 2 ( n )Ξ n w ( A k − 1 ) . Therefore, X {| v k,m | >A k − 1 } P ( |W 1 n ( v k,m ) | > λ/ 2) . A k r n log( n ) exp − B ζ 2 log( n ) w 2 ( A k − 1 )Ξ 2 n log 2(1+ ε ) ( w ( A k − 1 )) ! , n → ∞ with some constant B > 0 . Fix θ > 0 suc h that B θ > 1 and compute X {| v k,m | >A k − 1 } P ( |W 1 n ( v k,m ) | > λ/ 2) . r n log( n ) exp n k − θ B ( k − 1) − B ( k − 1)( ζ 2 (log n/ Ξ n ) − θ ) o . r n log( n ) e k (1 − θ B ) e − B ( k − 1)( ζ 2 (log n/ Ξ n ) − θ ) . Since ζ 2 (log n/ Ξ n ) > θ , w e arriv e at K X k =2 X {| v k,m | >A k − 1 } P ( |W n ( v k,m ) | > λ/ 2) . r n log( n ) e − B ( ζ 2 (log n/ Ξ n ) − θ ) h K X k =2 e k (1 − θ B ) i . log − 1 / 2 ( n ) exp n − B ζ 2 (log n/ Ξ n ) + log ( n ) o . D.Belomestny and V.Panov/Statistic al infer ence for exp onential functionals 26 T aking large enough ζ > 0, we get ( 52 ). Step 2 . No w w e are concentrated on ( 53 ). The idea of the pro of given b elow w as published in [ 3 ], Prop osition 7.4. Consider the sequence R n ( v ) := 1 n n X j =1 e ( u +i v ) Z j I  e uZ j ≥ Ξ n  . By the Marko v inequalit y w e get | E [ R n ( u )] | ≤ E  e uZ j  P  e uZ j ≥ Ξ n  ≤ Ξ − p n E  e uZ j  E  e upZ j  = o  p (log n ) /n  Set ν k = 2 k , k ∈ 1 , 2 , ... , then it holds ∞ X k =1 P n max j =1 ..η k +1 e uZ j ≥ Ξ η k o ≤ ∞ X k =1 η k +1 P { e uZ ≥ Ξ η k } ≤ E e puZ ∞ X k =1 η k +1 Ξ − p η k < ∞ . By the Borel-Cantelli lemma, P n max j =1 ..η k +1 e uZ j ≥ Ξ η k for infinitely many k o = 0 . F rom here it follows that R n ( u ) − E R n ( u ) = o  p (log n ) /n  . This completes the proof. Lemma 6.3. L et the me asur e ¯ ν b e such that k ¯ ν ( r ) k ∞ ≤ C 1 for some p ositive C 1 , the weighting function w n admits the pr op erty w n = V − k n w ( v /V n ) for some k > 0 and function w satisfying kF w ( u ) /u r ( · ) k L 1 ≤ C 2 with some C 2 > 0 . Then    Z ∞ 0 w n ( v ) F ¯ ν ( v ) dv    . V − ( r + k ) n , n → ∞ . Pr o of. 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