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Stationary Increments Harmonizable Stable Fields: Upper Estimates on Path Behaviour

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Abstract

Studying sample path behaviour of stochastic fields/processes is a classical research topic in probability theory and related areas such as fractal geometry. To this end, many methods have been developed for a long time in Gaussian frames. They often rely on some underlying “nice” Hilbertian structure and can also require finiteness of moments of high order. Therefore, they can hardly be transposed to frames of heavy-tailed stable probability distributions. However, in the case of some linear non-anticipative moving average stable fields/processes, such as the linear fractional stable sheet and the linear multi-fractional stable motion, rather new wavelet strategies have already proved to be successful in order to obtain sharp moduli of continuity and other results on sample path behaviour. The main goal of our article is to show that, despite the difficulties inherent in the frequency domain, such kind of a wavelet methodology can be generalized and improved, so that it also becomes fruitful in a general harmonizable stable setting with stationary increments. Let us point out that there are large differences between this harmonizable setting and the moving average stable one. The real-valued harmonizable stable stochastic field X on which we focus is defined on \(\mathbb {R}^d\) through an arbitrary spectral density belonging to a general and wide class of functions. First, we introduce a wavelet-type random series representation of X and express it as the finite sum \(X=\sum _\eta X^\eta \), where the fields \(X^\eta \) are called the \(\eta \)-frequency parts, since they extend the usual low-frequency and high-frequency parts. Moreover, we show the continuity of the sample paths of the \(X^\eta \)’s and X; also, we discuss the existence and continuity of their partial derivatives of an arbitrary order. Thereafter, we obtain several almost sure upper estimates related to: (a) the anisotropic behaviour of generalized directional increments of the \(X^\eta \)’s and X, on an arbitrary fixed compact cube of \(\mathbb {R}^d\); (b) the behaviour at infinity of the \(X^\eta \)’s, of X, and of their partial derivatives, when they exist. We mention that all the results on sample paths obtained in the article are valid on the same event of probability 1; furthermore, this event is “universal”, in the sense that it does not depend, in any way, on the spectral density associated with X.

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Notes

  1. Notice that such a convention will be extensively used in all the rest of our article, without being recalled.

  2. The difference between a norm and a quasi-norm is that for a quasi-norm the triangle inequality is weakened to \( \left| \left| {g+h} \right| \right| \le c\big (\left| \left| {g} \right| \right| +\left| \left| {h} \right| \right| \big ), \) where c is a finite constant strictly bigger than 1.

  3. Therefore, its finite value does not depend on the way the terms of the series are labelled. Moreover, it follows from the Fubini’s theorem that:

    $$\begin{aligned}&\sum _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\big (\Psi _{\alpha ,J}(2^Jt-K)-\Psi _{\alpha ,J}(-K)\big )\varepsilon _{\alpha ,J,K}(\omega )\\&\quad =\sum _{J\in \mathbb {Z}^d}\Big (\sum _{K\in \mathbb {Z}^d}\big (\Psi _{\alpha ,J}(2^Jt-K)-\Psi _{\alpha ,J}(-K)\big )\varepsilon _{\alpha ,J,K}(\omega )\Big ). \end{aligned}$$
  4. Notice that this negligible event does not necessarily coincide with the whole set \(\Omega \setminus \Omega _1^*\). On the other hand, this negligible event may depend on t.

  5. In which B is replaced by \(\widetilde{B}\) and T by 2T.

  6. In which B is replaced by \(\widetilde{B}\), g by \(\partial ^{e_k} g\) and T by 2T.

  7. Notice that when \(\eta =0=(0,\ldots ,0)\), then (2.61) holds for any \( b=(b_1,\ldots ,b_d)\in \mathbb {Z}_+^d\).

  8. That is, for all fixed \(m\in \mathbb {N}\) and \(\theta \in \mathbb {R}\), the random variables \(e^{i\theta }g_m\) and \(g_m\) have the same distribution.

  9. That is satisfying \(\mathbb {E}(G_{J,K})=0\), for all \((J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\).

References

  1. Adler, A.: The Geometry of Random Fields. Wiley, Hoboken (1981)

    MATH  Google Scholar 

  2. Ayache, A., Hamonier, J.: Linear multifractional stable motion: fine path properties. Rev. Mat. Iberoam. 30(4), 1301–1354 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ayache, A., Roueff, F., Xiao, Y.: Linear fractional stable sheets: wavelet expansion and sample path properties. Stoch. Process. Appl. 119(4), 1168–1197 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ayache, A., Taqqu, M.S.: Rate optimality of wavelet series approximations of fractional Brownian motion. J. Fourier Anal. Appl. 9(5), 451–471 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Biermé, H., Lacaux, C.: Hölder regularity for operator scaling stable random fields. Stoch. Process. Appl. 119(7), 2222–2248 (2009)

    Article  MATH  Google Scholar 

  6. Biermé, H., Lacaux, C., Scheffler, H.P.: Multi-operator scaling random fields. Stoch. Process. Appl. 121(11), 2642–2677 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Biermé, H., Richard, F., Rachidi, M., Benhamou, C.L.: Anisotropic texture modeling and applications to medical analysis. ESAIM Proc. Math. Methods Imaging Inverse Probl. 26, 100–122 (2009)

    MATH  MathSciNet  Google Scholar 

  8. Bonami, A., Estrade, A.: Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9(3), 215–236 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cramér, H., Leadbetter, M.: Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications. Wiley, Hoboken (1967)

    MATH  Google Scholar 

  10. Daubechies, I.: Ten Lectures on Wavelets, vol. 61. Society for Industrial Mathematics (1992)

  11. Embrechts, P., Maejima, M.: Self-Similar Processes. Academic Press, Cambridge (2002)

    MATH  Google Scholar 

  12. Khoshnevisan, D.: Multiparameter Processes. Springer, New York (2002)

    Book  MATH  Google Scholar 

  13. Kôno, N., Maejima, M.: Hölder continuity of sample paths of some self-similar stable processes. Tokyo J. Math. 14(1), 93–100 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kôno, N., Maejima, M.: Self-similar stable processes with stationary increments. In: Cambanis, S., Samorodnitsky, G., Taqqu, M.S. (eds.) Stable Processes and Related Topics. Progress in Probability, vol. 25, pp. 275–295. Birkhäuser, Boston (1991)

  15. Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer, New York (1991)

    Book  MATH  Google Scholar 

  16. Lemineur, G., Harba, R., Bretteil, S., Jennane, R., Estrade, A., Bonami, A., Benhamou, C.L.: Relation entre la régularité de fractals 3D et celle de leurs projections 2D; application à l’os trabéculaire. In: 19-ème Colloque du GRETSI (2003)

  17. Lifshits, M.: Gaussian Random Functions. Kluwer, Amsterdam (1995)

    Book  MATH  Google Scholar 

  18. Meerschaert, M., Wang, W., Xiao, Y.: Fernique-type inequalities and moduli of continuity for anisotropic Gaussian random fields. Trans. Am. Math. Soc. 365(2), 1081–1107 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Meyer, Y.: Ondelettes et Opérateurs, vol. 1. Hermann, Paris (1990)

    MATH  Google Scholar 

  20. Meyer, Y.: Wavelets and Operators, vol. 2. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  21. Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Variables. Chapman and Hall, London (1994)

    MATH  Google Scholar 

  22. Xiao, Y.: Sample path properties of anisotropic Gaussian random fields. In: Khoshnevisan, D., Rassoul-Agha, F. (eds.) A Minicourse on Stochastic Partial Differential Equations, pp. 145–212. Springer, New York (2009)

    Chapter  Google Scholar 

  23. Xiao, Y.: Recent developments on fractal properties of Gaussian random fields. In: Barral, J., Seuret, S. (eds.) Further Developments in Fractals and Related Fields, pp. 255–288. Springer, New York (2013)

    Chapter  Google Scholar 

  24. Xue, Y., Xiao, Y.: Fractal and smoothness properties of space-time Gaussian models. Front. Math. China 6(6), 1217–1248 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors are very grateful to the anonymous two referees for their careful reading of the article. This work has been partially supported by ANR-11-BS01-0011 (AMATIS), GDR 3475 (Analyse Multifractale) and ANR-11-LABX-0007-01 (CEMPI).

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Correspondence to Antoine Ayache.

Appendices

Appendix 1: Proofs of Proposition 2.6 and Lemma 2.9

Proof of Proposition 2.6

Let us first assume that \(J\in \mathbb {Z}^d\) and show the infinite differentiability of \(\Psi _{\alpha ,J}\) and relation (2.32). We denote by \(\Lambda _{\alpha ,J}\) the integrand in (2.18), that is, for all \(x\in \mathbb {R}^d\) and \(\xi \in \mathbb {R}^d\), we set,

$$\begin{aligned} \Lambda _{\alpha ,J}(x,\xi ):=2^{(j_1+\cdots +j_d)/\alpha }e^{ix\cdot \xi }\,f(2^J\xi )\widehat{\psi }_{0,0}(\xi ). \end{aligned}$$
(4.44)

Observe that \(\Lambda _{\alpha ,J}\) is an infinitely differentiable function on \(\mathbb {R}^d\) with respect to the variable x and that, for any \(b\in \mathbb {Z}_+^d\),

$$\begin{aligned} \partial _x^{b}\Lambda _{\alpha ,J}(x,\xi )=2^{(j_1+\cdots +j_d)/\alpha }\,i^{\mathrm {l}(b)}\xi ^b e^{ix\cdot \xi }\,f(2^{J}\xi )\widehat{\psi }_{0,0}(\xi ). \end{aligned}$$
(4.45)

Thus, in view of a classical rule of differentiation under the integral symbol, in order to show that \(\Psi _{\alpha ,J}\) itself is infinitely differentiable on \(\mathbb {R}^d\) and satisfies (2.32), it is enough to prove that, for any \(b\in \mathbb {Z}^d_+\), there exists \(G_{J}^{b}\in {L}^{ 1 } \left( \mathbb {R}^d \right) \), which does not depend on x, such that the inequality:

$$\begin{aligned} \big |\partial _x^{b}\Lambda _{J}(x,\xi )\big |\le G_{J}^{b}(\xi ), \end{aligned}$$
(4.46)

holds for almost all \(\xi \in \mathbb {R}^d\). Recall that \(\mathcal {K}\) is the compact subset of \(\mathbb {R}\) defined as \(\mathcal {K}:=\big \{\lambda \in \mathbb {R}: 2\pi /3\le |\lambda |\le 8\pi /3\big \}\); also recall that \(\widehat{\psi }_{0,0}\) is a \(C^\infty \) function on \(\mathbb {R}^d\) with a compact support included in \(\mathcal {K}^d\). Thus the smoothness assumption on the function f (that is \((\mathcal {H}_1)\) in Definition 1.1) implies that the supremum \(\big \Vert f(2^j\cdot )\widehat{\psi }_{0,0}(\cdot )\big \Vert _\infty :=\sup _{\xi \in \mathcal {K}^d}\big |f(2^J\xi )\widehat{\psi }_{0,0}(\xi )\big |\) is finite. Then, it turns out that a function \(G_{J}^{b}\), belonging to \({L}^{ 1 } \left( \mathbb {R}^d \right) \) and satisfying (4.46), can simply be obtained by setting, for all \(\xi \in \mathbb {R}^d\),

$$\begin{aligned} G_{J}^{b}(\xi )=2^{(j_1+\cdots +j_d)/\alpha }\Big (\frac{8\pi }{3}\Big )^{\mathrm {l}(b)}\big \Vert f(2^{J}\cdot )\widehat{\psi }_{0,0}(\cdot )\big \Vert _\infty \mathbbm {1}_{\mathcal {K}^d}(\xi ). \end{aligned}$$

Let us now prove that parts (i) and (ii) of the proposition hold. For the sake of simplicity, we restrict to the case where \(x=(x_1,\ldots , x_d)\in \mathbb {R}_+^d\); the other cases can be treated similarly. It easily follows from (2.32), (2.3) and (2.4) that, for every \(T\in (0,+\infty )\), \(J\in \mathbb {Z}^d\) and \(x\in \mathbb {R}_+^d\),

$$\begin{aligned} \left| {\partial ^{b}\Psi _J(x)} \right| =2^{(j_1+\cdots +j_d)/\alpha } \left| {\int _{\mathcal {K}^d}\left( \prod _{l=1}^d e^{i(1+T+x_l)\xi _l}\,{{\widehat{\Phi }}}_l(\xi _l)\right) f(2^{J}\xi )\mathrm d\xi } \right| , \end{aligned}$$
(4.47)

where \({\widehat{\Phi }}_l(\xi _l):=e^{-i(1+T)\xi _l}\,\xi _l^{b_l}\,\widehat{\psi ^1}(\xi _l)\). Next, we set \(R_{J}(\xi ):=f(2^{J}\xi )\prod _{l=1}^d{\widehat{\Phi }}_l(\xi _l)\), for all \(\xi \in \big (\mathbb {R}\setminus \{0\}\big )^d\). Observe that, similarly to \(\widehat{\psi ^1}\) (see the beginning of Sect. 2), \({\widehat{\Phi }}_l\) is a \(C^\infty \) function on \(\mathbb {R}\) having a compact support included in \(\mathcal {K}\). Thus, using the condition \((\mathcal {H}_1)\) in Definition 1.1, it turns out that the partial derivative \(\partial ^{(p_*,\ldots ,p_*)}R_J\) is a well-defined continuous function on \(\big (\mathbb {R}\setminus \{0\}\big )^d\) having a compact support included in \(\mathcal {K}^d\). Hence, integrating by parts in (4.47), we obtain that

$$\begin{aligned} \left| {\partial ^{b}\Psi _J(x)} \right|= & {} 2^{(j_1+\cdots +j_d)/\alpha } \left| {\int _{\mathcal {K}^d}\left( \left( \partial ^{(p_*,\ldots ,p_*)}R_{J}\right) (\xi )\prod _{l=1}^d \frac{e^{i(1+T+x_l)\xi _l}}{(1+T+x_l)^{p_*}}\right) \mathrm d\xi } \right| \nonumber \\\le & {} c_1 \frac{2^{(j_1+\cdots +j_d)/\alpha }}{\prod _{l=1}^d(1+T+x_l)^{p_*}} \sup _{\xi \in \mathcal {K}^d} \left| { \left( \partial ^{(p_*,\ldots , p_*)}R_{J}\right) (\xi )} \right| , \end{aligned}$$
(4.48)

where the constant \(c_1>0\) is the Lebesgue measure of \(\mathcal {K}^d\). On the other hand, using the Leibniz formula, we get, for every \(\xi \in \big (\mathbb {R}\setminus \{0\}\big )^d\), that

$$\begin{aligned} \left( \partial ^{(p_*,\ldots ,p_*)} R_J\right) (\xi )=\sum _{p_1=0}^{p_*}\cdots \sum _{p_d=0}^{p_*}\left( \partial ^{(p_1,\ldots ,p_d)} f\right) (2^{J}\xi )\prod _{l=1}^d\left( {\begin{array}{c}p_*\\ p_l\end{array}}\right) 2^{j_l p_l}{\widehat{\Phi }}_l ^{(p_*-p_l)}(\xi _l), \end{aligned}$$
(4.49)

where \({\widehat{\Phi }}_l ^{(p_*-p_l)}\) is the derivative of order \(p_*-p_l\) of \({\widehat{\Phi }}_l\). In view of (4.48), it turns out that for deriving (2.33), it is enough to show that

$$\begin{aligned} \sup _{J\in \mathbb {Z}_+^d} \sup _{\xi \in \mathcal {K}^d}\Big \{ \left( 2^{-j_1}+\cdots +2^{-j_d}\right) ^{a'+d/\alpha } \left| {\left( \partial ^{(p_*,\ldots ,p_*)} R_{-J}\right) (\xi )} \right| \Big \}<+\infty , \end{aligned}$$
(4.50)

and for deriving (2.34), it is enough to show that, for all \(\eta \in \Upsilon ^*\),

$$\begin{aligned} \sup _{J\in \mathbb {Z}_{(\eta )}^d} \sup _{\xi \in \mathcal {K}^d}\left\{ \prod _{l=1}^d2^{j_l/\alpha }2^{-(1-\eta _l)j_l/\alpha }2^{j_l\eta _l a_l} \left| {\left( \partial ^{(p_*,\ldots ,p_*)} R_{J}\right) (\xi )} \right| \right\} <+\infty ; \end{aligned}$$
(4.51)

recall that the sets \(\Upsilon ^*\) and \(\mathbb {Z}_{(\eta )}^d\) are defined in (2.28) and (2.29), respectively.

We now focus on the proof of (4.50). In view of (4.49) and of the fact that the \({\widehat{\Phi }}_l ^{(p_*-p_l)}\)’s, \(l=1,\ldots ,d\) are bounded functions on \(\mathcal {K}\), (4.50) can be obtained by showing that

$$\begin{aligned} \sup _{p\in \{0,1,2,\ldots , p_*\}^d} \sup _{J\in \mathbb {Z}_+^d} \sup _{\xi \in \mathcal {K}^d}\Big \{ \left( 2^{-j_1}+\cdots +2^{-j_d}\right) ^{a'+d/\alpha } 2^{-(j_1 p_1+\cdots +j_d p_d)} \left| {\left( \partial ^{p} f\right) (2^{-J}\xi )} \right| \Big \}<+\infty .\nonumber \\ \end{aligned}$$
(4.52)

Observe that, for any \(\xi \in \mathcal {K}^d\) and \(J\in \mathbb {Z}_+^d\), one has \(\left| \left| {2^{-J}\xi } \right| \right| \le 8\pi \sqrt{d}/3\). Thus, assuming that \(p\in \{0,1,2,\ldots , p_*\}^d\) is arbitrary and using (1.5), one gets that

$$\begin{aligned} \left| {\partial ^pf(2^{-J}\xi )} \right| \le c_2\left( 2^{-2j_1}\xi _1^2+\cdots +2^{-2j_d}\xi _d^2\right) ^{-\frac{a'}{2}-\frac{d}{2\alpha }-\frac{\mathrm {l}(p)}{2}}, \end{aligned}$$
(4.53)

where \(c_2\) denotes the constant \(c'\) in (1.5) which does not depend on p, J and \(\xi \). On the other hand, the fact that \(\xi \in \mathcal {K}^d\) implies that

$$\begin{aligned} \min _{1\le l \le d} \left| {\xi _l} \right| \ge 2\pi /3\ge 1. \end{aligned}$$
(4.54)

It follows from these inequalities and from the equality \(\mathrm {l}(p)=p_1+\cdots +p_d\) that

$$\begin{aligned}&\left( 2^{-2j_1}\xi _1^2+\cdots +2^{-2j_d}\xi _d^2\right) ^{-\frac{a'}{2}-\frac{d}{2\alpha }-\frac{\mathrm {l}(p)}{2}}\nonumber \\&\quad \le \left( 2^{-2j_1}+\cdots +2^{-2j_d}\right) ^{-\frac{a'}{2}-\frac{d}{2\alpha }-\frac{\mathrm {l}(p)}{2}}\nonumber \\&\quad = \left( 2^{-2j_1}+\cdots +2^{-2j_d}\right) ^{-\frac{a'}{2}-\frac{d}{2\alpha }}\prod _{l=1}^d\left( 2^{-2j_1}+\cdots +2^{-2j_d}\right) ^{-\frac{p_l}{2}}\nonumber \\&\quad \le c_3\left( 2^{-j_1}+\cdots +2^{-j_d}\right) ^{-a'-\frac{d}{\alpha }}2^{j_1 p_1+\cdots +j_d p_d}, \end{aligned}$$
(4.55)

where \(c_3>0\) is a constant only depending on d, \(a'\) and \(\alpha \). (4.52) results from (4.53) and (4.55).

We now focus on the proof of (4.51), where \(\eta \in \Upsilon ^*\) is arbitrary and fixed. In view of (4.49) and of the fact that the \({\widehat{\Phi }}_l ^{(p_*-p_l)}\)’s, \(l=1,\ldots ,d\), are bounded functions on \(\mathcal {K}\), (4.51) can be obtained by showing that

$$\begin{aligned} \sup _{p\in \{0,1,2,\ldots , p_*\}^d} \sup _{J\in \mathbb {Z}_{(\eta )}^d} \sup _{\xi \in \mathcal {K}^d}\Big \{2^{j_1p_1+\cdots +j_d p_d} \left| {\left( \partial ^{p} f\right) (2^{J}\xi )} \right| \prod _{l=1}^d2^{j_l/\alpha }2^{-(1-\eta _l)j_l/\alpha }2^{j_l\eta _l a_l} \Big \}\!<\!+\infty .\nonumber \\ \end{aligned}$$
(4.56)

Let \(p=(p_1,\ldots ,p_d)\in \{0,1,2,\ldots , p_*\}^d\), \(J=(j_1,\ldots , j_d)\in \mathbb {Z}_{(\eta )}^d\) and \(\xi =(\xi _1,\ldots ,\xi _d)\in \mathcal {K}^d\) be arbitrary. Observe that we know from the definition of \(\mathbb {Z}_{(\eta )}^d\) [see (2.29) and (2.30)] that J has at least one positive coordinate, let us say \(j_r\). Therefore, using (4.54), one gets that \(\left| \left| {2^J\xi } \right| \right| \ge \left| {2^{j_r}\xi _r} \right| \ge 2\pi /3\). Then, it follows from (1.6) that

$$\begin{aligned} \left| {\partial ^pf(2^J\xi )} \right| \le c_4 \prod _{l=1}^d\left( 1+2^{j_l} \left| {\xi _l} \right| \right) ^{-a_l-\frac{1}{\alpha }-p_l}, \end{aligned}$$
(4.57)

where \(c_4\) denotes the constant c in (1.6) which does not depend on p, J and \(\xi \). We now provide a convenient upper bound for the right-hand side in (4.57). To this end, we notice that \(\{1,\ldots ,d\}=\mathbb {L}_+\cup \mathbb {L}_-\), where the disjoint sets \(\mathbb {L}_+\) and \(\mathbb {L}_-\) are defined by \(\mathbb {L}_+=\left\{ l\in \{1,\ldots ,d\}: \eta _l=1\right\} \) and \(\mathbb {L}_-=\left\{ l\in \{1,\ldots ,d\}: \eta _l=0\right\} \). Then, using (4.54) and the fact that \(-j_l\ge 0\) when \(l\in \mathbb {L}_-\), one obtains that

$$\begin{aligned} \prod _{l\in \mathbb {L}_+}\left( 1+2^{j_l} \left| {\xi _l} \right| \right) ^{-a_l-\frac{1}{\alpha }-p_l}\le & {} \prod _{l\in \mathbb {L}_+}2^{-j_l \left( a_l+\frac{1}{\alpha }+p_l\right) }\nonumber \\\le & {} 2^{-(j_1p_1+\cdots +j_d p_d)}\prod _{l=1}^d2^{-j_l\eta _l \left( a_l+\frac{1}{\alpha }\right) }. \end{aligned}$$
(4.58)

On the other hand, one clearly has that

$$\begin{aligned} \prod _{l\in \mathbb {L}_-}\left( 1+2^{j_l} \left| {\xi _l} \right| \right) ^{-a_l-\frac{1}{\alpha }-p_l}\le 1, \end{aligned}$$
(4.59)

with the convention that \(\prod _{l\in \mathbb {L}_-}\cdots =1\), when \(\mathbb {L}_-\) is the empty set. Next, combining (4.58) and (4.59), it follows that:

$$\begin{aligned} \prod _{l=1}^d\left( 1+2^{j_l} \left| {\xi _l} \right| \right) ^{-a_l-\frac{1}{\alpha }-p_l}\le & {} 2^{-(j_1p_1+\cdots +j_d p_d)}\prod _{l=1}^d 2^{-\eta _l j_l/\alpha }\, 2^{-j_l\eta _l a_l}\nonumber \\= & {} 2^{-(j_1p_1+\cdots +j_d p_d)}\prod _{l=1}^d 2^{- j_l/\alpha }\,2^{(1-\eta _l) j_l/\alpha }\, 2^{-j_l\eta _l a_l}. \end{aligned}$$
(4.60)

Finally (4.56) results from (4.57) and (4.60). \(\square \)

Proof of Lemma 2.9

One denotes by \(\lfloor v\rfloor \) the integer part of v, and one sets \(w(v)=v-\lfloor v\rfloor \). Then, using the triangle inequality and the inequality \( \left| {\lfloor v\rfloor } \right| \le \left| {v} \right| +1\), one obtains that

$$\begin{aligned} \sum _{k\in \mathbb {Z}}\frac{\sqrt{\log (3+\theta + \left| {k} \right| )}}{\left( 2+ \left| {v-k} \right| \right) ^{p_*}}= & {} \sum _{k\in \mathbb {Z}}\frac{\sqrt{\log (3+\theta + \left| {k+\lfloor v\rfloor } \right| )}}{\left( 2+ \left| {v-\lfloor v\rfloor -k} \right| \right) ^{p_*}}\nonumber \\\le & {} \sum _{k\in \mathbb {Z}}\frac{\sqrt{\log (3+\theta + \left| {k} \right| +1+ \left| {v} \right| )}}{\left( 2+ \left| {w(v)-k} \right| \right) ^{p_*}}. \end{aligned}$$
(4.61)

Next, let c be the constant defined as:

$$\begin{aligned} c:=2\sup _{w\in [0,1]}\Bigg \{\sum _{k\in \mathbb {Z}}\frac{\sqrt{\log {(4+ \left| {k} \right| )}}}{\left( 2+ \left| {w-k} \right| \right) ^{p_*}}\Bigg \}. \end{aligned}$$
(4.62)

Observe that (1.4) and the inequality \(2+ \left| {w-k} \right| \ge 1+ \left| {k} \right| \), for all \((k,w)\in \mathbb {Z}\times [0,1]\), imply that c is finite. Also, observe that it follows from (2.38), the fact that \(w(v)\in [0,1]\), and (4.62) that

$$\begin{aligned} \sum _{k\in \mathbb {Z}}\frac{\sqrt{\log (3+\theta + \left| {k} \right| +1+ \left| {v} \right| )}}{\left( 2+ \left| {w(v)-k} \right| \right) ^{p_*}}\le & {} 2\sum _{k\in \mathbb {Z}}\frac{\sqrt{\log (4+ \left| {k} \right| )}\sqrt{\log (3+\theta + \left| {v} \right| )}}{\left( 2+ \left| {w(v)-k} \right| \right) ^{p_*}}\nonumber \\\le & {} c\sqrt{\log (3+\theta + \left| {v} \right| )}. \end{aligned}$$
(4.63)

Finally combining (4.61) and (4.63), one gets (2.40). \(\square \)

Appendix 2: Proofs of Lemma 2.5 and Proposition 2.2

Proof of Lemma 2.5

Assume that the real numbers \(a'\in (0,1)\), \(\alpha \in (0,2]\), and \(\delta >0\) are arbitrary and fixed. Also assume that the positive integer d and \(r\in \{1,\ldots ,d\}\) are arbitrary and fixed. Then, for any fixed \(r'\in \{1,\ldots ,d\}\), let \(\Gamma _{r'}\) be the set defined as

$$\begin{aligned} \Gamma _{r'}:=\left\{ J=(j_1,\ldots ,j_d)\in \mathbb {Z}_+^d: j_{r'}=\min \{j_1,\ldots ,j_d\}\right\} , \end{aligned}$$

and let \(S_{r,r'}\) be the positive quantity defined as

$$\begin{aligned} S_{r,r'}= & {} \sum _{J\in \Gamma _{r'}}2^{-j_r(1-a')}\left( 2^{-j_1}+\cdots +2^{-j_d}\right) ^{-d/\alpha }\\&\quad \times \prod _{l=1}^d2^{-j_l/\alpha }\sqrt{\log {\left( 3+j_l\right) }}(1+j_l)^{1/\alpha +\delta }. \end{aligned}$$

The fact that \(\mathbb {Z}_+^d=\bigcup _{r'=1}^d \Gamma _{r'}\) implies that

$$\begin{aligned}&\sum _{J\in \mathbb {Z}_+^d}2^{-j_r(1-a')}\left( 2^{-j_1}+\cdots +2^{-j_d}\right) ^{-d/\alpha }\prod _{l=1}^d2^{-j_l/\alpha }\sqrt{\log {\left( 3+j_l\right) }}(1+j_l)^{1/\alpha +\delta }\\&\quad \le \sum _{r'=1}^dS_{r,r'}. \end{aligned}$$

On the other hand, standard computations, relying on the definitions of \(\Gamma _{r'}\) and \(S_{r,r'}\), allow to obtain, for each \(r'\in \{1,\ldots ,d\}\), that

$$\begin{aligned} S_{r,r'}\le & {} \sum _{n=0}^{+\infty } 2^{-n(1+1/\alpha -a'-d/\alpha )}\sqrt{\log {\left( 3+n\right) }}(1+n)^{1/\alpha +\delta }\\&\times \bigg (\sum _{m=n}^{+\infty }2^{-m/\alpha }\sqrt{\log {\left( 3+m\right) }}(1+m)^{1/\alpha +\delta }\bigg )^{d-1}. \end{aligned}$$

Thus, in order to derive (2.25), it is enough to show that

$$\begin{aligned}&\sum _{n=0}^{+\infty } 2^{-n(1+1/\alpha -a'-d/\alpha )}\sqrt{\log {\left( 3+n\right) }}(1+n)^{1/\alpha +\delta }\\&\quad \times \bigg (\sum _{m=n}^{+\infty }2^{-m/\alpha }\sqrt{\log {\left( 3+m\right) }}(1+m)^{1/\alpha +\delta }\bigg )^{d-1}<+\infty . \end{aligned}$$

This can easily be obtained by making use of the inequality

$$\begin{aligned} \sum _{m=n}^{+\infty }2^{-m/\alpha }\sqrt{\log {\left( 3+m\right) }}(1+m)^{1/\alpha +\delta }\le c 2^{-n/\alpha }\sqrt{\log {\left( 3+n\right) }}(1+n)^{1/\alpha +\delta }, \end{aligned}$$
(4.64)

which holds for any non-negative integer n and for some finite constant c only depending on \(\alpha \) and \(\delta \). The proof of (4.64) has been omitted since it is not difficult. \(\square \)

The proof of Proposition 2.2 is devided into the following two steps which will be obtained separately.

Step 1 We show that, for every fixed \(t\in \mathbb {R}^d\), there exists \(\widetilde{F}(t,\cdot )\) in \({L}^{ \alpha } \left( \mathbb {R}^d \right) \) such that, for any increasing sequence \((\mathcal {D}_n)_{n\in \mathbb {N}}\) of finite subsets of \(\mathbb {Z}^d\times \mathbb {Z}^d\) which satisfies \(\bigcup _{n\in \mathbb {N}}\mathcal {D}_n=\mathbb {Z}^d\times \mathbb {Z}^d\), one has

$$\begin{aligned} \lim _{n\rightarrow +\infty }\Delta _{\alpha }\left( \sum _{(J,K)\in \mathcal {D}_n}\big (\Psi _{\alpha ,J}(2^Jt-K)-\Psi _{\alpha ,J}(-K)\big )\overline{\widehat{\psi }_{\alpha ,J,K}(\cdot )},\widetilde{F}(t,\cdot )\right) =0. \end{aligned}$$
(4.65)

Step 2 We show that, for all \(t\in \mathbb {R}^d\) and almost all \(\xi \in \mathbb {R}^d\), \( F(t,\xi )=\widetilde{F}(t,\xi ). \)

Proof of Proposition 2.2 (Step 1)

In view of Lemma 2.4 and (2.31), it is enough to show that, for all fixed \(t\in \mathbb {R}^d\) and \(\eta \in \Upsilon \), one has

$$\begin{aligned} \sum _{(J,K)\in \mathbb {Z}_{(\eta )}^d\times \mathbb {Z}^d}\Delta _{\alpha }\left( \big (\Psi _{\alpha ,J}(2^Jt-K)-\Psi _{\alpha ,J}(-K)\big )\overline{\widehat{\psi }_{\alpha ,J,K}(\cdot )},0\right) <+\infty . \end{aligned}$$
(4.66)

We will study the following four cases:

$$\begin{aligned}&\alpha \in (0,1) \text { and } \eta =0, \quad \alpha \in [1,2] \text { and }\eta =0,\quad \alpha \in (0,1)\text { and } \eta \ne 0, \\&\quad \alpha \in [1,2] \text { and } \eta \ne 0. \end{aligned}$$

Case 1: \({\alpha \in (0,1)}\) and \({\eta =0}\). Notice that, in this case, one has \(J\in \mathbb {Z}^d_{(0)}\), so it can be rewritten as \(J=-J'\), where \(J'\) belongs to \(\mathbb {Z}_+^d\). In the sequel, \(J'\) is denoted by J. Then (2.13), (2.14) and the change of variable \(\eta =2^{-J}\xi \) imply that, for all \(K\in \mathbb {Z}^d\), one has

$$\begin{aligned}&\Delta _{\alpha }\left( \big (\Psi _{\alpha ,-J}(2^{-J}t-K)-\Psi _{\alpha ,{-J}}(-K)\big )\overline{\widehat{\psi }_{\alpha ,{-J},K}(\cdot )},0\right) \nonumber \\&\quad =c_1 \left| {\Psi _{\alpha ,{-J}}(2^{-J}t-K)-\Psi _{\alpha ,{-J}}(-K)} \right| ^{\alpha }, \end{aligned}$$
(4.67)

where the constant \(c_1:=\left( \int _{\mathbb {R}}|\widehat{\psi ^1}(\eta )|^\alpha \,\mathrm d\xi \right) ^d\) is finite. Next, let \(T:=\max _{1\le l\le d}{ \left| {t_l} \right| }\). Using the mean value theorem and the triangle inequality, we get that

$$\begin{aligned} \left| {\Psi _{\alpha ,-J}(2^{-J}t-K)-\Psi _{\alpha ,-J}(-K)} \right| \le T\sum _{r=1}^d2^{-j_r}\sup _{s\in [-T,T]^d}\bigg |\frac{\partial \Psi _{\alpha ,-J}}{\partial x_r}\big (2^{-J}s-K\big )\bigg |, \end{aligned}$$
(4.68)

Moreover, combining (2.33) with the inequality,

$$\begin{aligned} 1+T+ \left| {2^{-j_l}s_l-k_l} \right| \ge 1+ \left| {k_l} \right| ,\quad \hbox { for all } l\in \{1,\ldots ,d\} \hbox { and } s_l\in [-T,T], \end{aligned}$$

we obtain, for every \(r\in \{1,\ldots ,d\}\), that

$$\begin{aligned}&2^{-j_r}\sup _{s\in [-T,T]^d}\bigg |\frac{\partial \Psi _{\alpha ,-J}}{\partial x_r}\big (2^{-J}s-K\big )\bigg |\nonumber \\&\quad \le c_2\frac{2^{-j_r(1-a')}\left( 2^{-j_1}+\cdots +2^{-j_d}\right) ^{-d/\alpha }\prod _{l=1}^d2^{-j_l/\alpha }}{\prod _{l=1}^d\big (1+ \left| {k_l} \right| \big )^{p_*}}, \end{aligned}$$
(4.69)

where \(c_2\) is a constant not depending on (JK). On the other hand, (1.4) implies that

$$\begin{aligned} \sum _{K\in \mathbb {Z}^d}\,\prod _{l=1}^d \big (1+ \left| {k_l} \right| \big )^{-\alpha p_*}<+\infty . \end{aligned}$$
(4.70)

Finally, using (4.67)–(4.70), and the same arguments as in the proof of (2.25), we get (4.66).

Case 2: \({\alpha \in [1,2]}\) and \({\eta =0}\). The proof follows the same lines as in the case 1, except that one has to use (2.12) instead of (2.13).

Case 3: \({\alpha \in (0,1)}\) and \({\eta \ne 0}\). It follows from (2.13), the triangle inequality and the sub-additivity on \([0,+\infty )\) of the function \(z\mapsto z^{\alpha }\), that, for all \((J,K)\in \mathbb {Z}_{(\eta )}^d\times \mathbb {Z}^d\), one has

$$\begin{aligned}&\Delta _{\alpha }\left( \big (\Psi _{\alpha ,J}(2^{J}t-K)-\Psi _{\alpha ,{J}}(-K)\big )\overline{\widehat{\psi }_{\alpha ,{J},K}(\cdot )},0\right) \\&\quad = c_1 \left| {\Psi _{\alpha ,{J}}(2^{J}t-K)-\Psi _{\alpha ,{J}}(-K)} \right| ^{\alpha }\\&\quad \le c_1 \left| {\Psi _{\alpha ,{J}}(2^{J}t-K)} \right| ^{\alpha }+ \left| {\Psi _{\alpha ,{J}}(-K)} \right| ^{\alpha }\\&\quad \le c_3 \prod _{l=1}^d 2^{(1-\eta _l)j_l}2^{-j_l\eta _l a_l \alpha }\left( \frac{1}{\big (2+ \left| {2^{j_l}t_l-k_l} \right| \big )^{\alpha p_*}}+\frac{1}{\big (2+ \left| {k_l} \right| \big )^{\alpha p_*}}\right) . \end{aligned}$$

Notice that \(c_3\) is a constant not depending on (JK). Also notice that the last inequality is obtained by using (2.34) in the case where \(T=1\). Next, this inequality, (2.29), (2.30), and (1.4) yield that

$$\begin{aligned}&\sum _{(J,K)\in \mathbb {Z}_{(\eta )}^d\times \mathbb {Z}^d}\Delta _{\alpha }\left( \big (\Psi _{\alpha ,J}(2^{J}t-K)-\Psi _{\alpha ,{J}}(-K)\big )\overline{\widehat{\psi }_{\alpha ,{J},K}(\cdot )},0\right) \\&\quad \le c_3 \sum _{J\in \mathbb {Z}_{(\eta )}^d}\prod _{l=1}^d 2^{(1-\eta _l)j_l}2^{-j_l\eta _l a_l \alpha }\left( \sum _{k_l\in \mathbb {Z}}\frac{1}{\big (2+ \left| {2^{j_l}t_l-k_l} \right| \big )^{\alpha p_*}}\right. \\&\quad \left. +\sum _{k_l\in \mathbb {Z}}\frac{1}{\big (2+ \left| {k_l} \right| \big )^{\alpha p_*}}\right) \\&\quad =c_3\sum _{J\in \mathbb {Z}_{(\eta )}^d}\prod _{l=1}^d 2^{(1-\eta _l)j_l}2^{-j_l\eta _l a_l \alpha }\left( \sum _{k_l\in \mathbb {Z}}\frac{1}{\big (2+ \left| {2^{j_l}t_l-\lfloor 2^{j_l}t_l\rfloor -k_l} \right| \big )^{\alpha p_*}}\right. \\&\quad \left. +\sum _{k_l\in \mathbb {Z}}\frac{1}{\big (2+ \left| {k_l} \right| \big )^{\alpha p_*}}\right) \\&\quad \le 2^d c_3\prod _{l=1}^d\left\{ \left( \sum _{j_l\in \mathbb {Z}_{\eta _l}} 2^{(1-\eta _l)j_l}2^{-j_l\eta _l a_l\alpha }\right) \left( \sum _{k_l\in \mathbb {Z}}\frac{1}{\big (1+ \left| {k_l} \right| \big )^{\alpha p_*}}\right) \right\} <+\infty , \end{aligned}$$

which show that (4.66) holds.

Case 4: \({\alpha \in [1,2]}\) and \({\eta \ne 0}\). The proof follows the same lines as in the case 3, except that one has to use (2.12) instead of (2.13). \(\square \)

Proof of Proposition 2.2 (Step 2)

For any fixed \(m\in \mathbb {N}\), we denote by \(\Theta _m\) the closed subset of \(\mathbb {R}^d\) defined as

$$\begin{aligned} \Theta _m:=\Big \{\xi =(\xi _1,\ldots ,\xi _d)\in \mathbb {R}^d:\min \big \{|\xi _1|,\ldots ,|\xi _d|\big \}\ge 2^{-m+1}\pi /3\Big \}. \end{aligned}$$
(4.71)

In view of (2.20) and Definition 1.1, it can easily be seen that, for any fixed \(t\in \mathbb {R}^d\), the function \(F(t,\cdot )\mathbbm {1}_{\Theta _m}(\cdot ):\xi \mapsto F(t,\xi )\mathbbm {1}_{\Theta _m}(\xi )\) belongs to the Hilbert space \({L}^{ 2 } \left( \mathbb {R}^d \right) .\) Therefore, using the fact that \(\{\psi _{J,K}: (J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\}\) is an orthonormal basis of this space, similarly to (2.5), one gets that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\int _{\mathbb {R}^d}\bigg |F(t,\xi )\mathbbm {1}_{\Theta _m}(\xi )-\sum _{(J,K)\in \mathcal {D}_n}w_{J,K}(t)\overline{\widehat{\psi }_{{J},K}(\xi )}\bigg |^2\,\mathrm d\xi =0, \end{aligned}$$
(4.72)

where

$$\begin{aligned} \begin{aligned} w_{J,K}(t):=\int _{\mathbb {R}^d}F(t,\xi ){1}_{\Theta _m}(\xi )\widehat{\psi }_{J,K}(\xi )\,\mathrm {d}\xi =\int _{\Theta _m}\left( e^{it\cdot \xi }-1\right) f(\xi )\widehat{\psi }_{J,K}(\xi )\,\mathrm {d}\xi , \end{aligned} \end{aligned}$$
(4.73)

and \((\mathcal {D}_n)_{n\in \mathbb {N}}\) is an arbitrary increasing sequence of finite subsets of \(\mathbb {Z}^d\times \mathbb {Z}^d\) such that \(\bigcup _{n\in \mathbb {N}}\mathcal {D}_n=\mathbb {Z}^d\times \mathbb {Z}^d\). Next, we denote by \(\mathcal {C}_m\) the compact subset of \(\Theta _m\) defined as

$$\begin{aligned} \mathcal {C}_m:= & {} \Big \{\xi =(\xi _1,\ldots ,\xi _d)\in \mathbb {R}^d: 2^{m+3}\pi /3\ge \max \big \{|\xi _1|,\ldots ,|\xi _d|\big \}\nonumber \\\ge & {} \min \big \{|\xi _1|,\ldots ,|\xi _d|\big \} \ge 2^{-m+3}\pi /3\Big \}. \end{aligned}$$
(4.74)

Let us show that, for all \((J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\) and \(\xi \in \mathcal {C}_m\), one has

$$\begin{aligned} w_{J,K}(t)\overline{\widehat{\psi }_{{J},K}(\xi )}=\big (\Psi _J(2^Jt-K)-\Psi _J(-K)\big )\overline{\widehat{\psi }_{J,K}(\xi )}, \end{aligned}$$
(4.75)

where the function \(\Psi _J\) is as in (2.7). To this end, we will study the following two cases: \(\min \{j_1,\ldots , j_d\}< -m\) and \(\min \{j_1,\ldots , j_d\}\ge -m\), where the integers \(j_1, \ldots , j_d\) are the coordinates of J, that is, \(J=(j_1,\ldots , j_d)\). In the first case \(\min \{j_1,\ldots , j_d\}< -m\), using (2.4) and (4.74), one gets that \(\widehat{\psi }_{J,K}(\xi )=0\), for each \(\xi \in \mathcal {C}_m\); therefore (4.75) holds. In the second case \(\min \{j_1,\ldots , j_d\}\ge -m\), it follows from (2.4) and (4.71) that \(\mathrm {supp}\,\widehat{\psi }_{J,K}\subset \Theta _m\). Thus, (4.73), (2.3), the change of variable \((\eta _1,\ldots ,\eta _d)=(2^{-j_1}\xi _1,\ldots , 2^{-j_d}\xi _d)\) and (2.7) imply that \( w_{J,K}(t)=\Psi _J(2^Jt-K)-\Psi _J(-K). \) Therefore, (4.75) is satisfied.

Next, using (4.75), (2.19), (4.72) and the inclusion \(\mathcal {C}_m\subset \Theta _m\), one gets that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\int _{\mathcal {C}_m}\bigg |F(t,\xi )-\sum _{(J,K)\in \mathcal {D}_n}\left( \Psi _{\alpha ,J}\left( 2^Jt-K\right) \!-\!\Psi _{\alpha ,J}\left( -K\right) \right) \overline{\widehat{\psi }_{\alpha ,{J},K}(\xi )}\bigg |^2\,\mathrm d\xi \!=\!0. \end{aligned}$$

Then the Hölder inequality, combined with the fact that \(\mathcal {C}_m\) has a finite Lebesgue measure, implies that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\int _{\mathcal {C}_m}\Big |F(t,\xi )-\!\sum _{(J,K)\in \mathcal {D}_n}\left( \Psi _{\alpha ,J}\left( 2^Jt-K\right) -\Psi _{\alpha ,J}\left( -K\right) \right) \overline{\widehat{\psi }_{\alpha ,{J},K}(\xi )}\Big |^\alpha \,\mathrm d\xi \!=\!0. \end{aligned}$$
(4.76)

On the other hand, (4.65) entails that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\int _{\mathcal {C}_m} \left| {\widetilde{F}(t,\xi )-\sum _{(J,K)\in \mathcal {D}_n}\big (\Psi _{\alpha ,J}(2^Jt-K)-\Psi _{\alpha ,J}(-K)\big )\overline{\widehat{\psi }_{\alpha ,J,K}(\xi )}} \right| ^\alpha \,\mathrm d\xi =0. \end{aligned}$$
(4.77)

Finally, it follows from (4.76) and (4.77) that, for all \(m\in \mathbb {N}\) and for almost all \(\xi \in \mathcal {C}_m\), one has \(\widetilde{F}(t,\xi )={F}(t,\xi )\); this amounts to saying that \(\widetilde{F}(t,\xi )={F}(t,\xi )\), for almost all \(\xi \in \mathbb {R}^d\), since \(\bigcup _{m\in \mathbb {N}}\mathcal {C}_m=(\mathbb {R}\setminus \{0\})^d\). \(\square \)

Appendix 3: Proof of Lemma 2.7

In order to show that Lemma 2.7 holds, we need two preliminary results. The following proposition provides, when \(\alpha \in (0,2)\), a LePage series representation of the complex-valued \(\alpha \)-stable process

$$\begin{aligned} \bigg \{\int _{\mathbb {R}^d}\overline{\widehat{\psi }_{\alpha ,J,K}(\xi )}\,\mathrm d\widetilde{M}_{\alpha }(\xi ): (J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\bigg \}. \end{aligned}$$

Its proof has been omitted since it is rather similar to that of Theorem 4.2 in [14].

Proposition 4.3

We assume that the stability parameter \(\alpha \) belongs to the open interval (0, 2), and we set

$$\begin{aligned} a(\alpha ):=\left( \int _0^{+\infty }x^{-\alpha }\sin (x)\,\mathrm {d} x\right) ^{-1/\alpha }. \end{aligned}$$
(4.78)

Let \(\{\kappa ^m: m\in \mathbb {N}\}\), \(\{\Gamma _m: m\in \mathbb {N}\}\) and \(\{g_m: m\in \mathbb {N}\}\) be three arbitrary mutually independent sequences of random variables, defined on the same probability space \((\Omega ,\mathcal {G},\mathbb {P})\), having the following properties.

  • The \(\kappa ^m\)’s, \(m\in \mathbb {N}\), are \(\mathbb {R}^d\)-valued, independent, identically distributed and absolutely continuous, with a probability density function, denoted by \(\phi \), such that the measure \(\phi (\xi )\mathrm d\xi \) is equivalent to the Lebesgue measure \(\mathrm d\xi \) on \(\mathbb {R}^d\).

  • The \(\Gamma _m\)’s, \(m\in \mathbb {N}\), are Poisson arrival times with unit rate; that is, for all \(m\in \mathbb {N}^*\), one has

    $$\begin{aligned} \Gamma _m=\sum _{n=1}^m \nu _n, \end{aligned}$$
    (4.79)

    where \((\nu _n)_{n\in \mathbb {N}}\) denotes a sequence of independent exponential random variables with the same parameter equal to 1.

  • The \(g_m\)’s, \(m\in \mathbb {N}\), are complex-valued, independent, identically distributed, rotationally invariantFootnote 8 and satisfy \(\mathbb {E}[ \left| {\mathcal {R}e(g_m)} \right| ^\alpha ]=1\).

On the other hand, for every fixed \((J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\), let \(\widehat{\psi }_{\alpha ,J,K}\) be the function defined in (2.14).

Then, the random series of complex numbers

$$\begin{aligned} \sum _{m=1}^{+\infty }g_m\Gamma _m^{-1/\alpha }\phi (\kappa ^m)^{-1/\alpha }\overline{\widehat{\psi }_{\alpha ,J,K}(\kappa ^m)} \end{aligned}$$

is almost surely convergent. Moreover, the stochastic processes

$$\begin{aligned}&\left\{ a(\alpha )\sum _{m=1}^{+\infty }g_m\Gamma _m^{-1/\alpha }\phi (\kappa ^m)^{-1/\alpha }\overline{\widehat{\psi }_{\alpha ,J,K}(\kappa ^m)}:\,\, (J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\right\} \quad \text {and}\\&\quad \Bigg \{\int _{\mathbb {R}^d}\overline{\widehat{\psi }_{\alpha ,J,K}(\xi )}\,\mathrm d\widetilde{M}_{\alpha }(\xi ): (J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\Bigg \} \end{aligned}$$

have the same distribution. These two processes are identified throughout our article.

Lemma 4.4

There exists a positive constant c such that for any sequence of complex-valued centredFootnote 9 Gaussian random variables \(\left\{ G_{J,K}:(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\right\} \), defined on \((\Omega ,\mathcal {G},\mathbb {P})\), one has

$$\begin{aligned}&\mathbb {E}\left\{ \sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\left( \frac{ \left| {G_{J,K}} \right| }{\sqrt{\log {\left( 3+\sum _{l=1}^d\big ( \left| {j_l} \right| + \left| {k_l} \right| \big )\right) }}}\right) \right\} \nonumber \\&\quad \le c\sqrt{\sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\mathbb {E}\left[ \left| {G_{J,K}} \right| ^2\right] }, \end{aligned}$$
(4.80)

where the \(j_l\)’s and \(k_l\)’s, respectively, denote the coordinates of J and K.

Proof

We set,

$$\begin{aligned} \Sigma (G):= & {} \sqrt{\sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\mathbb {E}\left[ \left| {G_{J,K}} \right| ^2\right] } \text { and, for all } (J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d, b_{J,K}\nonumber \\:= & {} \sqrt{\log {\left( 3+\sum _{l=1}^d\big ( \left| {j_l} \right| + \left| {k_l} \right| \big )\right) }}. \end{aligned}$$
(4.81)

Clearly, the lemma holds when \(\Sigma (G)=0\) and also when \(\Sigma (G)=+\infty \). Thus, in the sequel, we assume that \(0<\Sigma (G)<+\infty \). Using the fact that the expectation of an arbitrary non-negative random variable Z can be expressed as \(\mathbb {E}[Z]=\int _{0}^{+\infty }\mathbb {P}(Z>x)\,\mathrm dx,\) we get that

$$\begin{aligned}&\mathbb {E}\left[ \sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\left( \frac{ \left| {G_{J,K}} \right| }{\Sigma (G)b_{J,K}}\right) \right] \nonumber \\&\quad =\int _{0}^{+\infty }\mathbb {P}\left( \sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\left( \frac{ \left| {G_{J,K}} \right| }{\Sigma (G)b_{J,K}}\right)>x\right) \,\mathrm dx\nonumber \\&\quad \le 2^{d+1}+\int _{2^{d+1}}^{+\infty }\mathbb {P}\left( \sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\left( \frac{ \left| {G_{J,K}} \right| }{\Sigma (G)b_{J,K}}\right)>x\right) \,\mathrm dx\nonumber \\&\quad \le 2^{d+1}+\sum _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\int _{2^{d+1}}^{+\infty }\mathbb {P}\left( \frac{ \left| {G_{J,K}} \right| }{\Sigma (G)b_{J,K}}>x\right) \,\mathrm dx, \end{aligned}$$
(4.82)

where the last inequality follows from the equality

$$\begin{aligned} \bigg \{\omega \!\in \!\Omega : \sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\left( \frac{ \left| {G_{J,K}(\omega )} \right| }{\Sigma (G)b_{J,K}}\right)>x\bigg \}{=}\bigcup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\bigg \{\omega \!\in \!\Omega : \frac{ \left| {G_{J,K}(\omega )} \right| }{\Sigma (G)b_{J,K}}\!>\!x\bigg \}. \end{aligned}$$

Next, denoting by \(\mathcal {R}e(G_{J,K})\) and \(\mathcal {I}m(G_{J,K})\) the real and the imaginary parts of \(G_{J,K}\), then, in view of the equality \(|G_{J,K}|=\sqrt{ \left| {\mathcal {R}e(G_{J,K})} \right| ^2+ \left| {\mathcal {I}m(G_{J,K})} \right| ^2}\), for all \(x\ge 2^{d+1}\), one has

$$\begin{aligned} \mathbb {P}\left( \frac{ \left| {G_{J,K}} \right| }{\Sigma (G)b_{J,K}}\!>\!x\right) \le \mathbb {P}\left( \frac{ \left| {\mathcal {R}e(G_{J,K})} \right| }{\Sigma (G)b_{J,K}}>2^{-1/2}x\right) +\mathbb {P}\left( \frac{ \left| {\mathcal {I}m(G_{J,K})} \right| }{\Sigma (G)b_{J,K}}>2^{-1/2}x\right) . \end{aligned}$$
(4.83)

Now, we are going to show that

$$\begin{aligned} \mathbb {P}\left( \frac{ \left| {\mathcal {R}e(G_{J,K})} \right| }{\Sigma (G)b_{J,K}}>2^{-1/2}x\right) \le \exp \left( -2^{-2}\,b_{J,K}^2\,x^2\right) ; \end{aligned}$$
(4.84)

similarly, it can be shown that

$$\begin{aligned} \mathbb {P}\left( \frac{ \left| {\mathcal {I}m(G_{J,K})} \right| }{\Sigma (G)b_{J,K}}>2^{-1/2}x\right) \le \exp \left( -2^{-2}\,b_{J,K}^2\,x^2\right) . \end{aligned}$$
(4.85)

We set

$$\begin{aligned} \sigma (G_{J,K}):=\sqrt{\mathbb {E}\left[ \left| {\mathcal {R}e(G_{J,K})} \right| ^2\right] }; \end{aligned}$$

observe that, in view of the first equality in (4.81), one has

$$\begin{aligned} \Sigma (G)\ge \sigma (G_{J,K}). \end{aligned}$$
(4.86)

It is clear that (4.84) holds when \(\sigma (G_{J,K})=0\), since \(\mathcal {R}e(G_{J,K})\) is then vanishing almost surely. So, in the sequel we assume that \(\sigma (G_{J,K})>0\). Hence \(\mathcal {R}e(G_{J,K})/\sigma (G_{J,K})\) is a well-defined real-valued standard Gaussian random variable. Therefore, using (4.86) and the fact that \(2^{-1/2}b_{J,K}x\ge 2^d\,\sqrt{2\log 3}\ge 1\), we get that

$$\begin{aligned} \mathbb {P}\left( \frac{ \left| {\mathcal {R}e(G_{J,K})} \right| }{\Sigma (G)b_{J,K}}>2^{-1/2}x\right)\le & {} \mathbb {P}\left( \frac{ \left| {\mathcal {R}e(G_{J,K})} \right| }{\sigma (G_{J,K})b_{J,K}}>2^{-1/2}x\right) \\\le & {} \int _{2^{-1/2}b_{J,K}x}^{+\infty }e^{-y^2/2}\,\mathrm dy\le \int _{2^{-1/2}b_{J,K}x}^{+\infty } ye^{-y^2/2}\,\mathrm dy\\= & {} \exp \left( -2^{-2}\,b_{J,K}^2\,x^2\right) , \end{aligned}$$

which shows that (4.84) holds.

Next putting together (4.83)–(4.85) and the inequalities \(2^{-2}\,b_{J,K}^2\,x\ge 2^{d-1}\log 3\ge 1\), we obtain that

$$\begin{aligned} \int _{2^{d+1}}^{+\infty }\mathbb {P}\left( \frac{ \left| {G_{J,K}} \right| }{\Sigma (G)b_{J,K}}>x\right) \,\mathrm dx\le 2\int _{2^{d+1}}^{+\infty }2^{-2}\,b_{J,K}^2\,x\exp \left( -2^{-2}\,b_{J,K}^2\,x^2\right) \,\mathrm \nonumber \\ dx=\exp \left( -2^{2d}\,b_{J,K}^2\right) .\nonumber \\ \end{aligned}$$
(4.87)

Finally, in view of (4.81), (4.82) and (4.87), it turns out that in order to obtain (4.80) it is enough to show that

$$\begin{aligned} \sum _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\bigg (3+\sum _{l=1}^d\big ( \left| {j_l} \right| + \left| {k_l} \right| \big )\bigg )^{-4^{d}}<+\infty . \end{aligned}$$

This can be shown by noticing that \(4^d\ge 4d\) and that

$$\begin{aligned} \bigg (3+\sum _{l=1}^d\big ( \left| {j_l} \right| + \left| {k_l} \right| \big )\bigg )^{-4^{d}}\le & {} \bigg (3+\sum _{l=1}^d\big ( \left| {j_l} \right| + \left| {k_l} \right| \big )\bigg )^{-4d} \\= & {} \prod _{m=1}^d \bigg (3+\sum _{l=1}^d\big ( \left| {j_l} \right| + \left| {k_l} \right| \big )\bigg )^{-4} \\\le & {} \prod _{m=1}^d \bigg (3+\big ( \left| {j_m} \right| + \left| {k_m} \right| \big )\bigg )^{-4}\\\le & {} \prod _{m=1}^d \big (3+ \left| {j_m} \right| \big )^{-2} \big (3+ \left| {k_m} \right| \big )^{-2}. \end{aligned}$$

\(\square \)

We are now in the position to prove Lemma 2.7.

Proof of Lemma 2.7

First we recall that the third result provided by Lemma 2.7 (in other words the inequality (2.37) which holds in the Gaussian case \(\alpha =2\)) is rather classical. We will skip its proof; it can be found in e.g. [4]. In all the sequel, we assume that \(\alpha \in (0,2)\). Notice that, in view of (2.17), for all \((J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\), one clearly has

$$\begin{aligned} |\varepsilon _{\alpha ,J,K}|\le \Big |\int _{\mathbb {R}^d}\overline{\widehat{\psi }_{\alpha ,J,K}(\xi )}\,\mathrm d\widetilde{M}_{\alpha }(\xi )\Big |. \end{aligned}$$
(4.88)

Thus, in order to get (2.35) and (2.36), it is enough to show that these two inequalities are satisfied when \(\varepsilon _{\alpha ,J,K}\) in them is replaced by \(\int _{\mathbb {R}^d}\overline{\widehat{\psi }_{\alpha ,J,K}(\xi )}\,\mathrm d\widetilde{M}_{\alpha }(\xi )\). The advantage of this strategy is that we know from Proposition 4.3 that, for each \((J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\),

$$\begin{aligned} \int _{\mathbb {R}^d}\overline{\widehat{\psi }_{\alpha ,J,K}(\xi )}\,\mathrm d\widetilde{M}_{\alpha }(\xi )=a(\alpha )\sum _{m=1}^{+\infty } g_m\Gamma _m^{-1/\alpha }\phi (\kappa ^m)^{-1/\alpha }\overline{\widehat{\psi }_{\alpha ,J,K}(\kappa ^m)}; \end{aligned}$$
(4.89)

moreover, we can and will assume that the \(g_m\)’s, \(m\in \mathbb {N}\), are complex-valued centred Gaussian random variables and that the function \(\phi \) is such that for all \(\xi =(\xi _1,\ldots ,\xi _d)\in \big (\mathbb {R}\setminus \{0\}\big )^d\), one has

$$\begin{aligned} \phi (\xi ):=\left( \frac{\epsilon }{4}\right) ^d\,\prod _{l=1}^d \left| {\xi _l} \right| ^{-1}\left( 1+ \left| {\log { \left| {\xi _l} \right| }} \right| \right) ^{-1-\epsilon }, \end{aligned}$$

where \(\epsilon \) is an arbitrary fixed positive real number. Therefore, using (2.14) and (2.15), we obtain, for every \(m\in \mathbb {N}\) and \((J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\), that

$$\begin{aligned}&\left| {\phi (\kappa ^m)^{-1/\alpha }\overline{\widehat{\psi }_{\alpha ,J,K}(\kappa ^m)}} \right| \nonumber \\&\quad \le \left( \frac{\epsilon }{4}\right) ^{-d/\alpha }\prod _{l=1}^d \left| {2^{-j_l}\kappa _l^m} \right| ^{1/\alpha }\left( 1+ \left| {j_l} \right| + \left| {\log { \left| {2^{-j_l}\kappa _l^m} \right| }} \right| \right) ^{(1+\epsilon )/\alpha } \left| {\widehat{\psi ^1}(2^{-j_l}\kappa _l^m)} \right| \nonumber \\&\quad \le c_1 \prod _{l=1}^d\left( 1+ \left| {j_l} \right| \right) ^{(1+\epsilon )/\alpha }, \end{aligned}$$
(4.90)

where \(c_1\) is a deterministic constant not depending on (JK) and m. On the other hand, in view of the Gaussianity assumption on the \(g_m\)’s, \(m\in \mathbb {N}\), it can be derived from the Borel–Cantelli Lemma that, almost surely, for all \(m\in \mathbb {N}\), one has

$$\begin{aligned} |g_m|\le C_2\sqrt{\log {\left( 3+m\right) }}, \end{aligned}$$
(4.91)

where \(C_2\) is a finite random variable not depending on (JK) and m. Also, observe that, in view of (4.79), it results from the strong law of large number that almost surely, for any \(m\in \mathbb {N}\), the Poisson arrival time \(\Gamma _m\) satisfies

$$\begin{aligned} C_3 m\le \Gamma _m\le C_4 m, \end{aligned}$$
(4.92)

where \(C_3\) and \(C_4\) are two positive finite random variables not depending on (JK) and m. Next, we suppose for a while that \(\alpha \in (0,1)\), then the random variable

$$\begin{aligned} C_5:=a(\alpha )c_1C_2C_3^{-1/\alpha }\sum _{m=1}^{+\infty }m^{-1/\alpha }\sqrt{\log {\left( 3+m\right) }} \end{aligned}$$

is almost surely finite; moreover, it follows from the triangle inequality and from (4.89) to (4.92) that, almost surely, for all \((J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\), one has

$$\begin{aligned} \Big |\int _{\mathbb {R}^d}\overline{\widehat{\psi }_{\alpha ,J,K}(\xi )}\,\mathrm d\widetilde{M}_{\alpha }(\xi )\Big |\le & {} a(\alpha )\sum _{m=1}^{+\infty } \left| {g_m} \right| \Gamma _m^{-1/\alpha }\phi (\kappa ^m)^{-1/\alpha } \left| {\overline{\widehat{\psi }_{\alpha ,J,K}(\kappa ^m)}} \right| \\\le & {} C_5\prod _{l=1}^d\left( 1+ \left| {j_l} \right| \right) ^{(1+\epsilon )/\alpha }. \end{aligned}$$

These inequalities combined with (4.88) show that (2.35) holds.

From now on, we assume that \(\alpha \in [1,2)\) and our goal is to derive (2.36); notice that the previous strategy has to be modified since \(C_5\) is no longer finite. Let \(\mathcal {F}_{\Gamma ,\kappa }\) be the sub-\(\sigma \)-field of \(\mathcal {G}\) generated by the two sequences of random variables \( \left\{ \Gamma _m:m\in \mathbb {N}\right\} \) and \( \left\{ \kappa ^m:m\in \mathbb {N}\right\} \). We denote by \( \mathbb {E}_{\Gamma ,\kappa }[\,\cdot \,] \) the conditional expectation operator with respect to \(\mathcal {F}_{\Gamma ,\kappa }\); recall that \(\mathbb {E}(\,\cdot \,)\) denotes the classical expectation operator. We know from (4.89) that conditionally to \(\mathcal {F}_{\Gamma ,\kappa }\), for any arbitrary \( (J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d, \) the random variable

$$\begin{aligned} G_{J,K}:=\Big (\prod _{l=1}^d\left( 1+ \left| {j_l} \right| \right) ^{-(1+\epsilon )/\alpha }\Big ) \int _{\mathbb {R}^d}\overline{\widehat{\psi }_{\alpha ,J,K}(\xi )}\,\mathrm d\widetilde{M}_{\alpha }(\xi ) \end{aligned}$$
(4.93)

has a centred Gaussian distribution over \(\mathbb {C}\). Then, assuming that \(c_6\) denotes the constant c in (4.80), one can derive from Lemma 4.4 that the following inequality holds almost surely:

$$\begin{aligned}&\mathbb {E}_{\Gamma ,\kappa }\left[ \sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\Bigg (\frac{ \left| {G_{J,K}} \right| }{\sqrt{\log {\left( 3+\sum _{l=1}^d\big ( \left| {j_l} \right| + \left| {k_l} \right| \big )\right) }}}\Bigg )\right] \nonumber \\&\quad \le c_6\sqrt{\sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\mathbb {E}_{\Gamma ,\kappa }\left[ \left| {G_{J,K}} \right| ^2\right] }. \end{aligned}$$
(4.94)

Next, using the fact that \(\mathbb {E}(\,\cdot \,)=\mathbb {E}\big (\mathbb {E}_{\Gamma ,\kappa }[\,\cdot \,]\big )\), Cauchy–Schwarz inequality and (4.94), one obtains that

$$\begin{aligned}&\mathbb {E}\left( \sqrt{\sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\Bigg (\frac{|G_{J,K}|}{\sqrt{\log {\left( 3+\sum _{l=1}^d \left| {j_l} \right| + \left| {k_l} \right| \right) }}}\Bigg )}\right) \nonumber \\&\quad =\mathbb {E}\left( \mathbb {E}_{\Gamma ,\kappa }\left[ \sqrt{\sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\Bigg (\frac{|G_{J,K}|}{\sqrt{\log {\left( 3+\sum _{l=1}^d \left| {j_l} \right| + \left| {k_l} \right| \right) }}}\Bigg )}\right] \right) \nonumber \\&\quad \le \mathbb {E}\left( \sqrt{\mathbb {E}_{\Gamma ,\kappa }\Bigg [\sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\Bigg ( \frac{|G_{J,K}|}{\sqrt{\log {\left( 3+\sum _{l=1}^d \left| {j_l} \right| + \left| {k_l} \right| \right) }}}\Bigg )\Bigg ]}\right) \nonumber \\&\quad \le \sqrt{c_6}\,\mathbb {E}\left( \bigg (\sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\mathbb {E}_{\Gamma ,\kappa }\Big [ \left| {G_{J,K}} \right| ^2\Big ]\bigg )^{1/4}\right) .\nonumber \\ \end{aligned}$$
(4.95)

On the other hand, (4.89) and (4.93) imply that one has, almost surely, for any arbitrary \((J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d\),

$$\begin{aligned} \mathbb {E}_{\Gamma ,\kappa }\left[ \left| {G_{J,K}} \right| ^2\right] \!=\!c_7\Big (\prod _{l=1}^d\left( 1+ \left| {j_l} \right| \right) ^{-2(1+\epsilon )/\alpha }\Big )\sum _{m=1}^{+\infty }\Gamma _m^{-2/\alpha }\phi (\kappa ^m)^{-2/\alpha } \left| {\widehat{\psi }_{\alpha ,J,K}(\kappa ^m)} \right| ^2, \end{aligned}$$

where the deterministic constant \(c_7:=a(\alpha )^2\,\mathbb {E}\big (|g_1|^2\big )\) does not depend on (JK). Then, using (4.90), one gets, almost surely, that

$$\begin{aligned} \sup _{(J,K)\in \mathbb {Z}^d\times \mathbb {Z}^d}\mathbb {E}_{\Gamma ,\kappa }\left[ \left| {G_{J,K}} \right| ^2\right] \le c_8\sum _{m=1}^{+\infty }\Gamma _m^{-2/\alpha }, \end{aligned}$$
(4.96)

where the deterministic constant \(c_8:=c_1^2c_7\). Finally, in view of (4.88), (4.93), (4.95) and (4.96), it turns out that (2.36) can be obtained by showing that

$$\begin{aligned} \mathbb {E}\left( \bigg (\sum _{m=1}^{+\infty }\Gamma _{m}^{-2/\alpha }\bigg )^{1/4}\right) <+\infty . \end{aligned}$$
(4.97)

We know from Remark 4 on page 29 in [21] that the positive random variable \(\sum _{m=1}^{+\infty }\Gamma _{m}^{-2/\alpha }\) has a stable distribution with a stability parameter equal to \(\alpha /2\). Thus combining the fact that \(\alpha /2>1/4\) with the Property 1.2.16 on page 18 in [21], one gets (4.97). \(\square \)

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Ayache, A., Boutard, G. Stationary Increments Harmonizable Stable Fields: Upper Estimates on Path Behaviour. J Theor Probab 30, 1369–1423 (2017). https://doi.org/10.1007/s10959-016-0698-0

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