On the asymptotics of supremum distribution for some iterated processes

In this paper, we study the asymptotic behavior of supremum distribution of some classes of iterated stochastic processes {X(Y(t)):t∈[0,∞)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{X(Y(t)) : t \in [0, \infty )\}$\end{document}, where {X(t):t∈ℝ}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{X(t) : t \in \mathbb {R} \}$\end{document} is a centered Gaussian process and {Y(t):t∈[0,∞)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{Y(t): t \in [0, \infty )\}$\end{document} is an independent of {X(t)} stochastic process with a.s. continuous sample paths. In particular, the asymptotic behavior of ℙ(sups∈[0,T]X(Y(s))>u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {P}(\sup _{s\in [0,T]} X(Y (s)) > u)$\end{document} as u→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u \to \infty $\end{document}, where T>0, as well as limu→∞ℙ(sups∈[0,h(u)]X(Y(s))>u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lim _{u\to \infty } \mathbb {P}(\sup _{s\in [0,h(u)]} X(Y (s)) > u)$\end{document}, for some suitably chosen function h(u) are analyzed. As an illustration, we study the asymptotic behavior of the supremum distribution of iterated fractional Brownian motion process.

Originated by Burdzy (1993;1994) for the case of iterated Brownian motion, the problem of analyzing the properties of iterated processes was intensively studied in recent years. Motivation for the analysis of the process {X(Y (t))} in case of {X(t)} and {Y (t)} being independent Brownian motions was delivered by its connections to the 4th order PDE's (see, e.g., Funki 1979;Allouba and Zheng 2001;Nourdin and Peccati 2008). A vast literature is devoted to the analysis of many interesting probabilistic properties of iterated Brownian motions (see, e.g., Burdzy and Khoshnevisan 1995;Hu et al. 1995;Shi 1995;Bertoin 1996;Eisenbaum and Shi 1999;Khoshnevisan and Lewis 1999). We also refer to (Curien and Konstantopoulos 2014) where convergence of finite dimensional distributions of nth iterated Brownian motion is studied and (Turban 2004) where infinite iterations of i.i.d. random walks are analyzed.
Recent studies also focus on properties of {X(Y (t)) : t ∈ [0, ∞)} for the case of more general Gaussian processes {X(t)}. One of interesting example of such processes is fractional Laplace motion {B H ( (t)) : t ∈ [0, ∞)}, where { (t) : t ∈ [0, ∞)} is a Gamma process. Motivation for analyzing fractional Laplace motions stems from hydrodynamic models (see, e.g., Kozubowski et al. 2004). This kind of processes were described in (Kozubowski et al. 2006), see also (Arendarczyk and Dȩbicki 2011) where asymptotic behavior of exit-time distribution for the process {B H ( (t))} was found. Another important class of iterated processes are the so-called α-time fractional Brownian motions {B H (Y (t))}, where {Y (t)} is α-stable subordinator independent of the process {B H (t)} (see, e.g., Linde and Shi 2004;Nane 2006;Linde and Zipfel 2008;Aurzada and Lifshits 2009). We also refer to (Michna 1998) and (Dȩbicki et al. 2014) where the process {B H (Y (t))} was analyzed in the context of theoretical actuarial models.
The process {B H (Y (t))} in the case of {Y (t)} not being a subordinator was studied in (Aurzada and Lifshits 2009). In this case, the small deviations asymptotics was found for the so-called iterated fractional Brownian motion process {B H 2 (B H 1 (t))}, where {B H 1 (t)}, {B H 2 (t)} are independent fractional Brownian motions with Hurst parameters H 1 , H 2 ∈ (0, 1] respectively. In this paper, we focus on the analysis of asymptotic behavior of supremum distribution of the process {X(Y (t)) : t ∈ [0, ∞)} for general classes of stochastic processes {X(t)}, {Y (t)} with a.s. continuous sample paths.
Notation and organization of the paper: In Section 2, we study the asymptotic behavior of P sup s∈ [0,T ] X(Y (s)) > u as u → ∞, where T > 0 and {X(t) : t ∈ R}, {Y (t) : t ∈ [0, ∞)} are independent stochastic processes. This problem is closely related to the analysis of asymptotic behavior of the supremum distribution of the process {X(t)} over a random time interval (see, e.g., Dȩbicki et al. 2004;Arendarczyk and Dȩbicki 2011;2012;Tan and Hashorva 2013;Dȩbicki et al. 2014). We start in Section 2.1 by giving general result for the case of {X(t)} being Gaussian process with stationary increments and convex variance function (see Section 2.1, assumptions A1 -A3). In this case, under some general conditions on the process {Y (t)} (see Section 2.1, assumptions L1, L2), we show that (1) reduces to P sup where T is a non-negative random variable independent of {X(t)} with asymptotically Weibullian tail distribution, that is, as u → ∞, where α, β, C > 0, γ ∈ R (see, e.g., Arendarczyk and Dȩbicki (2011) for details). We write T ∈ W(α, β, γ , C) if T satisfies (3). Section 2.2 is devoted to the special case of the process {B H (Y (t)) : t ∈ [0, ∞)}, where {B H (t) : t ∈ R} is a fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1], that is, a centered Gaussian process with stationary increments, a.s. continuous sample paths, B H (0) = 0, and covariance function Cov(B H (t), B H (s)) = 1 2 |s| 2H + |t| 2H − |t − s| 2H . Due to self-similarity of the process {B H (t)}, we are able to provide the exact asymptotics of (1) for the whole range of Hurst parameters H ∈ (0, 1]. As an illustration, in Proposition 2.4, we work out the exact asymptotics of the supremum distribution of iterated fractional Brownian motion Brownian motions with Hurst parameters H 1 , H 2 respectively. Note that small deviation counterpart of this problem was recently studied in (Aurzada and Lifshits 2009).
In Section 2.3, the case of {X(t)} being a stationary Gaussian process is analyzed (see Section 2.3, assumptions D1, D2). In this case the exact asymptotics of (1) can be achieved under a general condition of finite average span of the process {Y (t)} (see Section 2.3, assumption S1). This problem is strongly related to the analysis of (2) in case of T being a random variable with finite mean. In this case the asymptotics of (2) has the form (see Arendarczyk and Dȩbicki (2012), Theorem 3.1, and also Pickands (1969) for the classical result of Pickands' on deterministic time interval) as u → ∞, where H α is the Pickands' constant defined by the limit and (u) := P(N > u) with N denoting the standard normal random variable.
In the second part of the paper, we study for some suitably chosen function h(u). First, in Theorem 3.1 we investigate limiting behavior of (4) for the case of {X(t)} and {Y (t)} being independent Gaussian processes with stationary increments that satisfy some general regularity conditions (see Section 3, assumptions B1 -B3). Then, in Theorem 3.2 and Proposition 3.3, the case of {X(t)} being stationary Gaussian process is studied. We analyze {X(Y (t))} for both weakly and strongly dependent stationary Gaussian processes {X(t)} (see Section 3, assumptions D1 -D3). In these settings we provide (4) in the case of {Y (t)} being a centered Gaussian process with stationary increments, as well as for self-similar process {Y (t)} that is not necessarily Gaussian.

Short timescale case
In this section, we study the asymptotic behavior of where T > 0, for the case of {X(t) : t ∈ R} being a centered Gaussian process with a.s. continuous sample paths. We focus on two important classes of Gaussian processes. First, processes {X(t)} with stationary increments are studied. Then, we analyze the case of stationary processes {X(t)}.

The stationary increments case
Let {X(t) : t ∈ R} be a centered Gaussian process with stationary increments, a.s. continuous sample paths, X(0) = 0 a.s., and variance function σ 2 X (t) := Var(X(t)) that satisfies the following assumptions To provide general result for (5) we assume that {Y (t) : t ∈ [0, ∞)} is a stochastic process with a.s. continuous sample paths, which is independent of {X(t)} and its extremal distributions belong to the Weibullian class of random variables, that is, Remark 2.1 Note that assumptions L1, L2 cover, e.g., a class of general Gaussian processes.
In the following theorem we present structural form of the asymptotics. The explicit asymptotic expansion is presented in Corollary 2.2.
The proof of Theorem 2.1 is presented in Section 4.1.
If the variance function of {X(t)} is regular enough, then the straightforward application of Corollary 3.2 in (Arendarczyk and Dȩbicki 2011) enables us to give the exact form of the asymptotics. (1)) as u → ∞.

The case of fBm
Let {B H (t) : t ∈ R} be a fractional Brownian motion with Hurst parameter H ∈ (0, 1]. In this section, we analyze the asymptotic behavior of P sup where T > 0 and {Y (t) : t ∈ [0, ∞)} is an independent of {B H (t)} stochastic process with a.s. continuous sample paths that satisfies assumptions L1, L2. Due to self-similarity of the process {B H (t)}, we are able to provide the exact asymptotics of (6) for the whole range of Hurst parameters H ∈ (0, 1], which includes cases of both convex and concave variance functions.

Proposition 2.3 Let {B H (t) : t ∈ R} be a fractional Brownian motion with Hurst
stochastic process with a.s. continuous sample paths that satisfies L1, L2. If: The proof of Proposition 2.3 is presented in Section 4.2.
We now apply Proposition 2.3 to calculate the exact asymptotics for the special case of iterated fractional Brownian motion process {B H 2 (B H 1 (t))}.
Moreover, due to Lemma 4.2 in (Arendarczyk and Dȩbicki 2011) (see also Piterbarg (1996) Additionally, by stationarity of the increments of fBm Now, in order to complete the proof it suffices to apply Proposition 2.3.

The stationary case
In this section, we analyze the asymptotic behavior of (5) for the case of {X(t) : t ∈ [0, ∞)} being a centered stationary Gaussian process with a.s. continuous sample paths and covariance function r(t) := Cov(X(s), X(s +t)). We impose the following assumptions on r(t) (see, e.g., Piterbarg 1996): In this case, we are able to give the exact form of the asymptotics for general class of stochastic processes Proof Due to stationarity of the process {X(t)}, we have Now, in order to complete the proof it suffices to apply Theorem 3.1 in (Arendarczyk and Dȩbicki 2012).
Remark 2.2 Equivalently, Proposition 2.5 states that P sup

Long timescale case
In this section, we investigate for a suitably chosen function h(u).
In order to formulate the results, it is convenient to introduce the notation for the generalized inverse of the function σ (t).
We start with the observation that (7) can be straightforwardly obtained for any independent, self-similar processes {X(t)} and {Y (t)} with a.s. continuous sample paths.
In order to formulate the result, it is convenient to introduce the notation where B α X /2 (t) , B α Y /2 (t) are independent fractional Brownian motions with Hurst parameters α X /2 and α Y /2 respectively.

Theorem 3.1 Let {X(t) : t ∈ R} and {Y (t) : t ∈ [0, ∞)} be independent, centered Gaussian processes with stationary increments that satisfy B1 -B3 with parameters
The proof of Theorem 3.1 is presented in Section 4.3.
We study (7) for both weakly and strongly dependent stationary Gaussian processes, i.e., for r = 0 and r > 0 respectively. We refer to (Tan et al. 2012;Tan and Hashorva 2013) for recent results on asymptotic behavior of supremas of strongly dependent Gaussian processes.
In this settings, in Theorem 3.2, we provide (7) in the case of {Y (t) : t ∈ [0, ∞)} being a centered Gaussian process with stationary increments and variance function σ 2 Y (t) that satisfies conditions B1 -B3. Moreover, in Proposition 3.3, we analyze (7) for self-similar process {Y (t)} that is not necessarily Gaussian.

Theorem 3.2 Let {X(t) : t ∈ R} be a centered stationary Gaussian process with covariance function that satisfies D1 -D3 and {Y (t) : t ∈ [0, ∞)} be an independent of {X(t)} Gaussian process with a.s. continuous sample paths, stationary increments, and variance function σ 2 Y (t) that satisfies B1 -B3 with parameters α Y , β Y . Then, for
The proof of Theorem 3.2 is given in Section 4.4.

Proposition 3.3 Let {X(t) : t ∈ R} be a stationary Gaussian process with covariance function that satisfies D1 -D3 and {Y (t) : t ∈ [0, ∞)} be a self-similar stochastic process with parameter λ Y , independent of the process {X(t)}. Then, for
Proof Due to stationarity of the process {X(t)} and self-similarity of the process where (8)

Proofs
In this section, we present detailed proofs of Theorem 2.1, Proposition 2.3, Theorem 3.1 and Theorem 3.2.

In view of inclusion -exclusion principle
where Observe that by definition of the process {X(t)}, The case (i) is a consequence of the fact that, by (10) and Theorem 3.1 in (Arendarczyk and Dȩbicki 2011), P(K > u) = o(P(M > u)) implies P 1 (u) = o(P 2 (u)). Thus, as u → ∞, which in view of Theorem 3.1 in (Arendarczyk and Dȩbicki 2011), completes the proof for the case (i). A similar reasoning implies that for the case (ii), we have as u → ∞, which in view of Theorem 3.1 in (Arendarczyk and Dȩbicki 2011), completes the proof for the case (ii). In order to prove (iii), without loss of generality, we assume that as u → ∞. Due to (9) combined with (10) and Theorem 3.1 in (Arendarczyk and Dȩbicki 2011), it suffices to show that P 3 (u) is negligible. We distinguish the case K ≤ M and the case K > M and obtain as u → ∞, where (12) is due to the assumption (11). To find an upper bound of (12) it is convenient to make the following decomposition where Let ε > 0. We analyze each of the integrals I 1 , I 2 , I 3 separately. Integral I 1 : (14) is due to (16) in (Arendarczyk and Dȩbicki 2011) (see also the proof of Lemma 6.3 in Arendarczyk and Dçbicki 2011).
Integral I 3 : as u → ∞. The above, combined with the observation that for each η > 0 and sufficiently large u, leads to the conclusion that I 1 and I 3 are negligible. Integral I 2 : Observe that, due to A1, σ 2 Var(X(s) + X(t)) = 2σ 2 X (|s|) + 2σ 2 for (s, t) ∈ [−w, 0]×[0, w]. Thus, according to the Borell inequality (see, e.g., Adler (1990), Theorem 2.1), combined with (15), I 2 is bounded by To find the upper bound of E sup t∈[0,w] X(t), we use metric entropy method (see, e.g., Lifshits (2012), Chapter 10). At the beginning, for any T ⊆ R define the semimetric The metric entropy H d (T, ) is defined as log N d (T, ), where N d (T, ) denotes the minimal number of points in an -net in T with respect to the semimetric d.

Proof of Proposition 2.3
The idea of the proof is analogous to the proof of Theorem 2.1, thus we present only main steps of the argumentation. In view of inclusion -exclusion principle P sup Since the arguments for the cases P(K > u) = o(P(M > u)) as u → ∞, and P(M > u) = o(P(K > u)) as u → ∞ are similar to those in the proof of Theorem 2.1, then we focus on the case P(K > u) = C 2 C 1 P(M > u)(1 + o(1)) as u → ∞.
Without loss of generality, we assume that as u → ∞. Due to (22) combined with (23) and Theorem 4.1 in (Arendarczyk and Dȩbicki 2011), it suffices to show that P 3 (u) is negligible. In an analogous way to (12), we obtain the following upper bound Let ε > 0. We investigate the asymptotic behavior of each of the integrals. Integral I 3 : We have, as u → ∞, Observe that, due to Theorem 4.1 in (Arendarczyk and Dȩbicki 2011) (27), we obtain, for sufficiently large u, the following upper bound, which in view of Theorem 4.1 in (Arendarczyk and Dȩbicki 2011), implies that lim sup u→∞ I 2 P (X(M) > u) ≤ 2ε.
In order to complete the proof it suffices to pass with ε → 0.
By continuity of the sample paths of the processes {X(t)} and {Y (t)}, To find an upper bound of (30) we consider the following decomposition We analyze each of the integrals I 1 , I 2 , I 3 separately.
In order to complete the proof it suffices to pass with A ∞ → ∞, and ε → 0.

Proof of Theorem 3.2
In further analysis we use the following notation  (h(u)) .
Let ε > 0 and 0 < A 0 < A ∞ < ∞. Note that due to Lemma 5.2 in (Dȩbicki et al. 2004) where ⇒ denotes convergence in distribution. It is convenient to consider the following decomposition = I 1 + I 2 + I 3 .
We analyze each of the integrals I 1 , I 2 , I 3 separately.
Integral I 1 : Due to Lemma 3.3 in (Tan and Hashorva 2013), for sufficiently large u, as u → ∞.