On the asymptotics of supremum distribution for some iterated processes

Extremes DOI 10.1007/s10687-016-0272-2 On the asymptotics of supremum distribution for some iterated processes Marek Arendarczyk1 Received: 20 April 2016 / Revised: 19 September 2016 / Accepted: 20 September 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract In this paper, we study the asymptotic behavior of supremum distribution of some classes of iterated stochastic processes {X(Y (t)) : t ∈ [0, ∞)}, where {X(t) : t ∈ R} is a centered Gaussian process and {Y (t) : t ∈ [0, ∞)} 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) as u → ∞, where T > 0, as well as limu→∞ P(sups∈[0,h(u)] X(Y (s)) > u), 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. Keywords Exact asymptotics · Supremum distribution · Iterated process · Iterated fractional brownian motion · Gaussian process AMS 2000 Subject Classifications Primary 60G15 · 60G18 · Secondary 60G70 1 Introduction Let {X(t) : t ∈ R} and {Y (t) : t ∈ [0, ∞)} be two independent stochastic processes. This contribution is devoted to the analysis of asymptotic behavior of supremum distribution of iterated process {X(Y (t)) : t ∈ [0, ∞)}.  Marek Arendarczyk 1 Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland M. Arendarczyk 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; Khoshnevisan and Lewis 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 {BH ((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 {BH ((t))} was found. Another important class of iterated processes are the so-called α-time fractional Brownian motions {BH (Y (t))}, where {Y (t)} is α-stable subordinator independent of the process {BH (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 {BH (Y (t))} was analyzed in the context of theoretical actuarial models. The process {BH (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 {BH2 (BH1 (t))}, where {BH1 (t)}, {BH2 (t)} are independent fractional Brownian motions with Hurst parameters H1 , H2 ∈ (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 X(Y (s)) > u s∈[0,T ] as u → ∞, (1) On the asymptotics of supremum distribution... 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 X(s) > u s∈[0,T ] as u → ∞, (2) where T is a non-negative random variable independent of {X(t)} with asymptotically Weibullian tail distribution, that is, P (T > u) = Cuγ exp(−βuα )(1 + o(1)) (3) 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 {BH (Y (t)) : t ∈ [0, ∞)}, where {BH (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, BH (0) = 0, and covariance function Cov(BH (t), BH (s)) =  2H  1 + |t|2H − |t − s|2H . Due to self-similarity of the process {BH (t)}, we are 2 |s| 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 {BH2 (BH1 (t)) : t ∈ [0, ∞)}, where {BH1 (t)}, {BH2 (t)} are independent fractional Brownian motions with Hurst parameters H1 , H2 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 M. Arendarczyk 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)   P sup X(s) > u = ET C 1/α Hα u2/α (u)(1 + o(1)) s∈[0,T ] as u → ∞, where Hα is the Pickands’ constant defined by the limit 1 Hα = lim E exp T →∞ T   √ α 2B (...truncated)