Limit Theorems for a Class of Additive Functionals of Symmetric Stable Process and Fractional Brownian Motion in Besov-Orlicz Spaces

We use the Potter’s Theorem and the tightness criterion in Besov-Orlicz spaces, recently proved, to generalize some limit theorem for occupation times problem of certain self-similar process, namely symmetric stable process of index 1<α≤ and fractional Brownian motion of Hurst parameter 0<H<1. We give also strong approximation version of our limit theorem, more precisely, we show L-estimate version.


INTRODUCTION
In the present study, we are interested in the limit theorems of a class of continuous additive functionals of some self-similar process, namely stable process and fractional Brownian motion. The interesting properties such as self similarity and stationarity of increments make these processes good candidates as models for different phenomena, related to financial mathematics and telecommunications.
Most of the estimates in this paper contain unspecified positive constants. We use the same symbol C for these constants, even when they vary from one line to the next. We first collect some facts about these processes. Let It is known from Boylan (1964) and Barlow (1988) that X α admits a continuous local time process {L (t, x); t≥0, x∈ ℝ } satisfying the scaling property: where, " L " means the equality in the sense of the finitedimensional distributions and the occupation density formula: for any bounded or nonnegative Borel function f. Moreover, in Marcus and Rosen (1992) and in Ait Ouahra and Eddahbi (2001), for each T > 0 fixed, there Science Publications JMSS exists a constant 0 < C < ∞ such that for any integer p ≥ 1, all 0 ≤ t, s ≤ T and all x; y∈ ℝ : with stationary increments and covariance function: However, the increments of fBm are not independent except in the Brownian motion case ( 1 H 2 = , Bm for brevity). The dependence structure of the increments is modeled by a parameter H. fBm is self-similar with exponent τ = H and his local time satisfies the occupation density formula and the scaling property. Notice that the α-SSP is self-similar with exponent 1 τ = α . Geman and Horowitz (1980) proved that the local time of fBm exists and has a.s. Holder continuous modification of order γ 0 -ε in space and of order 1-H-ε in time for any ε > 0 and γ 0 = min precisely, it is proved by Xiao (1997), that for each T > 0 fixed, there exists a constant 0 < C < ∞ such that for any integer p ≥ 1, all 0 ≤ t, s ≤ T and all x, y∈ ℝ : t s x y for any 0 For an excellent summary of fBm, the reader is referred to Mandelbrot and Ness (1968) and Samorodnitsky and Taqqu (1994).

Remark 1
• Notice that for 1 H 2 = (respectively α = 2), B H (respectively X α ) is a Bm • The α-SSP has independent increments, contrary to fBm which does not have independent increments, except for the special case of the Bm • B H has a.s. Holder continuous modification of order β < H but X α is just cadlag Throughout this study, we use the same symbol and we denote {L(t, x); t ≥ 0, x ∈ R} its local time. Then, for each T > 0 fixed, there exists a constant 0 < C < ∞ such that for any integer p ≥ 1, all 0 ≤ t, s ≤ T and all x, y∈ ℝ Equation 1-3: where, 0 1 2 − τ δ = δ = τ for α-SSP and 0 0 1 0 2 − τ < δ < δ = < γ + τ for fBm.
Self-similar process arise naturally in limit theorems of random walks and other stochastic process. Many authors have studied the limit theorems of the process Equation 4: and g∈C β ∩L 1 ( ℝ ) with compact support.
We cite Yamada (1986; for Bm Shieh (1996) for fBm (τ = H) and Fitzsimmons and Getoor (1992) for α-SSP . All these results are established in the space of continuous functions. Ouahra and Eddahbi (2001) extended the results of Fitzsimmons and Getoor (1992) to Holder spaces and Ouahra et al. (2002) in Besov spaces and recently, Ouahra et al. (2011) in Besov-Orlicz spaces. The result of Shieh (1996) was extended by Ouahra and Ouali (2009) in Besov spaces. The objective of the present study is to study in Besov-Orlicz spaces, the limit theorem of the process (4), where f has the form l, , (see the definition of l, K γ ± below). We recall the following definition which will be useful in the sequel.

Definition 1
A measurable function U : + + → ℝ ℝ is regularly varying at infinity in (Karamata's sense), with a real exponent r, if for all t positive: If r = 0, we call U slowly varying function denoted by l. We see that U(x) = x r l(x).
We are interested in the behavior of l at + ∞, then we can assume for example that l is bounded on each interval of the form [0, a]; (a > 0).
The following theorem called Potter's Theorem has played a central role in the proof of our results, (Bingham et al., 1989).

Theorem 1
• If l is slowly varying function, then for any chosen constants A > 1 and ξ > 0, there exists X = X(A, ξ) such that: • If further, l is bounded away from 0 and ∞ on every compact subset of [0,+∞[, then for every ξ>0, there exists A' = A'(ξ) >1 such that:

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• If U is regularly varying function with exponent r ∈ ℝ , then for any chosen A > 1 and ξ > 0, there exists X = X(A, ξ) such that: The monograph by Seneta (1976) contains a very readable exposition of the basic theory of regularly varying functions on ℝ .
The following proposition is the main result of this section. It is a consequence of a simple computation integral.
The remainder of this study is organized as follows: we present some basic facts about Besov-Orlicz spaces. We give the proof of our main result. Finally, we state and prove strong approximation version of our limit theorem.

The Functional Framework
We will present a brief survey of Besov-Orlicz spaces. For more details, we refer the reader to Boufoussi (1994) and Ciesielski et al. (1993).
Let ( Ω, ∑, µ) be a σ-finite measure space. We denote by L p (Ω), 1 ≤ p < +∞, the space of Lebesgue integrable real valued functions f on Ω with exponent p endowed with the norm: The Orlicz space L Mβ(dµ) (Ω) corresponding to the Young function M β (x) = e |x|β -1, β ≥ 1 is the Banach space of real valued measurable functions f on Ω endowed with the norm: This norm is equivalent to the norm of Luxemburg (1955) given by: In case of (Ω, ∑, P) being a probability space, the Orlicz norm become: In this study, we use the following equivalence norm in M (d ) L ( ) β µ Ω , (see for example Ciesielski et al. (1993): : where, ||g|| ∞ = sup x∈A |g(x)|. These last two results and the Potter's Theorem have played a central role in the proof of our limit theorem.
The modulus of continuity of a Borel function f: for any 0 < µ < 1 and v > 0. Let {ϕ n = ϕ j,k , j ≥ 0, k = 1,…2 j } be the Schauder basis. The decomposition and the coefficients of continuous functions f on [0, 1] in this basis are respectively given as follows: It is known from Ciesielski et al. (1993) that the subspace ,v ,0 M , B µ β ω ∞ corresponds to sequences (f j , k ) j,k such that: For the proof of our results, we need the following tightness criterion in the subspace ≤ . This complete the proof of Corollary 1.

Limit Theorems
In order to establish our limit theorem, we need the following regularities.

Remark 4
This regularity is similar to that given in Ouahra and Eddahbi (2001) for fractional derivatives of local time of α-SSP and in Ouahra and Ouali (2009) for fBm case.

Proof of Lemma 1
We treat only the case We estimate I 1 and I 2 separately. Estimate of I 1 . Since l is bounded on each compact in + ℝ , it follows from (3) that: Now we return to estimate I 2 .
Potter's Theorem with 0 < ξ < γ implies the existence of A(ξ) > 1 such that: Combining this fact with (1), we obtain: The proof of Lemma 1 is done. We prove, in the same way as before the following result. It will be useful to prove the tightness in Theorem 3.

Corollary 2
Let T>0 fixed and 0 < γ < δ. There exists a constant 0 < C < ∝ such that for all 0 ≤ t, s ≤ T, all x∈ ℝ and n large enough:

Proof
We treat only the case u L s, u L s, L s, n n n n du u 1 l(n u) ( ) l(n ) We estimate J 1 and J 2 separately. Estimate of J 1 : It follows from (3) that: Now we return to estimate J 2 .
. l , l , 1 n n M (dP) Now we are ready to state the main result of this section.

Theorem 3
where g ∈ C β ∩ L 1 ( ℝ ) with compact support for some γ < β. Then as n → + ∞, the sequence of process: converges in law to the process:

Remark 6
Notice that even if f is not a fractional derivative of some function g, the limiting process is fractional derivative of local time.

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where, S = supp(g). Thanks to Remark 5, for n large enough, we have: Case of fBm. By analogous arguments using in Fitzsimmons and Getoor (1992), (Remark 3.18), in the case of α-SSP, we obtain the convergence of the finitedimensional distributions of the processes n t A . The tightness follows easily as in the case of α-SSP. This with Theorem 2 completes the proof of Theorem 3.

Remark 7
Our limit theorems in the case of fBm are new even in the space of continuous functions.

Strong Approximation
We give strong approximation, L p -estimate, of Theorem 3. Our main result in this paragraph reads.

Theorem 4
Let f be a Borel function on ℝ satisfying Equation 8: for some k > 0. Then, for any sufficiently small ε > 0 and any integer p ≥ 1, when t goes to infinity, we have: In order to prove Theorem 4, we shall first state and prove some technical lemmas.
In the same way, using (3) for s = 0 and the fact that L(0, x) = 0 a.s., we get the following lemma.

Proof
We have:

Lemma 6
Under same conditions as in Theorem 4. For any sufficiently small ε > 0 and any integer p ≥ 1, when t goes to infinity, we have:

Lemma 7
Let f be a Borel function on R satisfying (8) for some k > 0. Then, for any sufficiently small ε > 0 and any integer p ≥ 1, when t goes to infinity, we have:

Proof
We have by (8) In particular, if we take l 1 ≡ , we get: The proof of Lemma 7 is done. Now, we return to the proof of Theorem 4.

Proof of Theorem 4
This theorem is an immediate consequence of Lemma 6 and Lemma 7.

Remark 8
• In case f is the fractional derivative of some function g, the analogous results of Theorem 4 appeared in Ouahra and Ouali (2009). On the other hand, the a.s. estimate of Theorem 4 is given in Csaki et al. (2000) for special Bm case • We should point out that in this paper we only study the L p -estimate of our limit theorems. This is enough for the purpose of this study. We will study the a.s. estimates in future work and apply this idea to study the law of the iterated logarithm of stochastic process of the form t l, s 0 K (Y )ds γ τ ∫

ACKNOWLEDGMENT
This study was completed while the first authors was visiting the Department of Statistics and Probability at Michigan State University (DSPMSU). This author would like to express his sincere thanks to the staff of DSPMSU for generous supported and hospitality, especially Prof. Yimin Xiao. He was also supported by Fulbright Visiting Scholar grant 2012-2013.