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 $\{X(Y(t)) : t \in [0, \infty)\}$, where $\{X(t) : t \in \mathbb{R} \}$ is a centered Gaussian process and $\{Y(t): t \in [0, \infty)\}$ is an independent of $\{X(t)\}$ stochastic process with a.s. continuous sample paths. In particular, the asymptotic behavior of $\mathbb{P}\left(\sup_{s \in [0,T]} X(Y(s))>u\right)$ as $u \to \infty$, where $T>0$, as well as $\lim_{u\to\infty} \mathbb{P}\left(\sup_{s \in [0, h(u)]} X(Y(s))>u\right)$, 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 [7,8] 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., [14,2,25]). A vast literature is devoted to the analysis of many interesting probabilistic properties of iterated Brownian motions (see, e.g., [9,15,28,6,16,13,17]). We also refer to [10] where convergence of finite dimensional distributions of nth iterated Brownian motion is studied and [31] 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 (Γ(t)) : t ∈ [0, ∞)}, where {Γ(t) : t ∈ [0, ∞)} is a Gamma process. Motivation for analyzing fractional Laplace motions stems from hydrodynamic models (see, e.g., [18]). This kind of processes were described in [19], see also [3] where asymptotic behavior of exit-time distribution for the process {B H (Γ(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., [21,24,22,5]). We also refer to [23] and [11] 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 [5]. 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 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., [12,3,4,30,11] 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., [3] 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 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 . 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 fractional Brownian motions with Hurst parameters H 1 , H 2 respectively. Note that small deviation counterpart of this problem was recently studied in [5]. 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 [4], Theorem 3.1, and also [26] for the classical result of Pickands' on deterministic time interval) P sup as u → ∞, where H α is the Pickands' constant defined by the limit and Ψ(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 P sup 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.  (ii) P(M > u) = o(P(K > u)) as u → ∞, then P sup 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 [3] enables us to give the exact form of the asymptotics.

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.
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 Proof. Due to self-similarity of fBm Moreover, due to Lemma 4.2 in [3] (see also [27], Theorem D3) 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., [27]): 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 P sup Now, in order to complete the proof it suffices to apply Theorem 3.1 in [4].

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.
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 [29,30] 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.
The proof of Theorem 3.2 is given in Section 4.4.

s) and N is a normal random variable independent of T .
Proof. Due to stationarity of the process {X(t)} and self-similarity of the process {Y (t)}, we have P sup where (8) follows by the reasoning as in the proof of Theorem 3.2.

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

Proof of Theorem 2.1
In view of inclusion -exclusion principle P sup 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 [3], 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 [3], 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 [3], 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 [3], 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 Let ε > 0. We analyze each of the integrals I 1 , I 2 , I 3 separately. Integral I 1 : ≤ exp −u as u → ∞, where (14) is due to (16) in [3] (see also the proof of Lemma 6.3 in [3]).
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 Thus, according to the Borell inequality (see, e.g., [1], Theorem 2.1), combined with (15), I 2 is bounded by Moreover, To find the upper bound of E sup t∈[0,w] X(t), we use metric entropy method (see, e.g., [20], Chapter 10). At the beginning, for any T ⊆ R define the semimetric The metric entropy H d (T, ǫ) is defined as log N d (T, ǫ), where N d (T, ǫ) denotes the minimal number of points in an ǫ-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 Moreover observe that 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) Without loss of generality, we assume that as u → ∞. Due to (22) combined with (23) and Theorem 4.1 in [3], 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 1 : Due to self-similarity of {B H (t)} combined with Lemma 4.2 in [3], we have, as u → ∞, (1)).

Proof of Theorem 3.1
In further analysis we use the following notation Moreover, we denote Let ε > 0 and 0 < A ∞ < ∞. We start with the observation that lim u→∞ h(u) = ∞, which also implies that lim u→∞ σ Y (h(u)) = ∞. Hence, due to Lemma 5.2 in [12] ( and where ⇒ denotes convergence in distribution. 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. Integral I 1 : Due to (28), for sufficiently large u, Integral I 3 : Due to (28), for sufficiently large u, Integral I 2 : For u sufficiently large, where (31) is due to (29) and the fact that lim u→∞ u σ X (σ Y (h(u))) = 1, and (32) is due to (28), and the observation that P sup t∈[v,w] B α X /2 (t) > 1 is bounded and continuous function with respect to (v, w). Thus, for each ε > 0, A ∞ > A 0 > 0, B α X /2 (t) > 1 .
In order to complete the proof it suffices to pass with A 0 → 0, A ∞ → ∞, and ε → 0.

Proof of Theorem 3.2
In further analysis we use the following notation Let ε > 0 and 0 < A 0 < A ∞ < ∞. Note that due to Lemma 5.2 in [12] T u ⇒ T as u → ∞, where ⇒ denotes convergence in distribution. It is convenient to consider the following decomposition X(s) > u dF Tu (t) = 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 [30], for sufficiently large u, as u → ∞. Integral I 3 : Due to (33), for sufficiently large u, Integral I 2 : X(s) > u dF Tu (t) where (34) is by Lemma 3.3 in [30] and (35) is due to (33), and the observation that In order to complete the proof it suffices to pass with A 0 → 0, A ∞ → ∞, and ε → 0.