The invariance principle for fractionally integrated processes with strong near-epoch dependent innovations

In this paper, we show the invariance principle for the partial sum processes of fractionally integrated processes, otherwise known as I(d + m) processes, where |d| < 1/2 and m is a nonnegative integer, with strong near-epoch dependent innovations. The results are applied to the test of unit root. The conditions given improve previous results in the literature concerning fractionally integrated processes.     

Stochastic integration for tempered fractional Brownian motion

Tempered fractional Brownian motion is obtained when the power law kernel in the moving average representation of a fractional Brownian motion is multiplied by an exponential tempering factor. This paper develops the theory of stochastic integrals for tempered fractional Brownian motion. Along the way, we develop some basic results on tempered fractional calculus.     

Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations

We consider compositions of stochastic processes that are governed by higherorder partial differential equations. The processes studied include compositions of Brownian motions, stable-like processes with Brownian time, Brownian motion whose time is an integrated telegraph process, and an iterated integrated telegraph process. The governing higher-order equations that are obtained are shown to be either of the usual parabolic type or, as in the last example, of hyperbolic type.     

Log-fractional stable processes

The first problem attacked in this paper is answering the question whether all 1/-self-similar -stable processes with stationary increments are -stable motions. The answer is yes for = 2, no for 1<2 and unknown for 0<<1. We single out the log-fractional stable processes for 1<2, different from -stable motions for ≠2. They can be regarded as the limit of fractional stable processes as the exponent in the kernel tends to 0. The paper ends with a limit theorem for partial sum processes of moving averages of iid random variables in the domain of attraction of a strictly stable law, with log-fractional stable processes as limits in law. The conditions involve de Haan's class Π of slowly varying functions.     

Finite time ruin probabilities for tempered stable insurance risk processes

We study the probability of ruin before time t$t$ for the family of tempered stable Lévy insurance risk processes, which includes the spectrally positive inverse Gaussian processes. Numerical approximations of the ruin time distribution are derived via the Laplace transform of the asymptotic ruin time distribution, for which we have an explicit expression. These are benchmarked against simulations based on importance sampling using stable processes. Theoretical consequences of the asymptotic formulae indicate that some care is needed in the choice of parameters to avoid exponential growth (in time) of the ruin probabilities in these models. This, in particular, applies to the inverse Gaussian process when the safety loading is less than one.     

Sampling distribution for a class of estimators for nonregular linear processes

Let {X_t; t = 1, 2,…} be a linear process with a location parameter θ defined by X_t ? θ = Σ₀^∞g_rZ_t?r where {Z_t; t = 0, ±1,…} is a sequence of independent and identically distributed random variables, with E∥Z₁∥^δ < ∞ for some δ > 0. If δ ? 1 we assume further than E(Z₁) = 0. Let η = δ if 0 < δ < 2, and η = 2 if δ ? 2. Then assume that Σ₀^∞∥ g_r ∥^η < ∞. Consider the class of estimators $\text{θ}\text{∧}{}_{\mathrm{n}}$ given by $\text{θ}\text{∧}{}_{\mathrm{n}}\text{= Σ}{}_1{}^{\mathrm{n}}\text{c}{}_{\mathrm{nt}}\text{X}{}_{\mathrm{t}}\text{where}\text{c}{}_{\mathrm{nt}}$ is of the form c_nt = Σ_{p = 0}^sβ_npt^p for some s ? 0. An attempt has been made to investigate the distributional properties of $\text{θ}\text{∧}{}_{\mathrm{n}}$ in large samples for various choices of β_np (0 ? p ? s), s, and the distribution of Z₁ under the constraints Σ₀^∞r^kg_r = 0, 0 ? k ? q where q in an arbitrary integer, 0 ? q ? s.     

Convergence rates for probabilities of moderate deviations for moving average processes

The present paper first shows that, without any dependent structure assumptions for a sequence of random variables, the refined results of the complete convergence for the sequence is equivalent to the corresponding complete moment convergence of the sequence. Then this paper investigates the convergence rates and refined convergence rates （or complete moment convergence） for probabilities of moderate deviations of moving average processes. The results in this paper extend and generalize some well-known results.     

Two-sided eigenvalue estimates for subordinate processes in domains

Let X={X_t,t?0} be a symmetric Markov process in a state space E and D an open set of E. Denote by X^D the subprocess of X killed upon leaving D. Let S={S_t,t?0} be a subordinator with Laplace exponent φ that is independent of X. The processes X^φ?{X_St,t?0} and are called the subordinate processes of X and X^D, respectively. Under some mild conditions, we show that, if {-μ_n,n?1} and {-λ_n,n?1} denote the eigenvalues of the generators of the subprocess of X^φ killed upon leaving D and of the process X^D respectively, then

     

Approximation theorems for random permanents and associated stochastic processes

The limit theorems for certain stochastic processes generated by permanents of random matrices of independent columns with exchangeable components are established. The results are based on the martingale decomposition of a random permanent function similar to the one known for U-statistics and on relating the components of this decomposition to some multiple stochastic integrals.Mathematics Subject Classification (2000): Primary 60F17, 62G20; Secondary 15A15, 15A52Acknowledgement A significant part of this work was completed when the first author was visiting the Center for Mathematical Sciences at the University of Wisconsin-Madison. He would like to express his gratitude to the Center and its Acting Director, Prof. Thomas G. Kurtz, for their hospitality. Thanks are also due to the first authors current host, the Institute for Mathematics and Its Applications at the University of Minnesota. Finally, both authors graciously acknowledge the comments of an anonymous referee on an earlier version of this paper.     

On r-quick limit sets for empirical and related processes based on mixing random variables

The r-quick limit points of normalized sample paths and empirical distribution functions of mixing processes are characterized. An r-quick version of Bahadur-Kiefer-type representation for sample quantiles is established, which yields the r-quick limit points of quantile processes. These results are applied to linear functions of order statistics. Some results on r-quick convergence of certain Gaussian processes are also established.     

Functional limit theorems for renewal shot noise processes with increasing response functions

We consider renewal shot noise processes with response functions which are eventually nondecreasing and regularly varying at infinity. We prove weak convergence of renewal shot noise processes, properly normalized and centered, in the space D[0,∞)$D[0,\infty )$ under the _J1$J_1$ or _M1$M_1$ topology. The limiting processes are either spectrally nonpositive stable Lévy processes, including the Brownian motion, or inverse stable subordinators (when the response function is slowly varying), or fractionally integrated stable processes or fractionally integrated inverse stable subordinators (when the index of regular variation is positive). The proof exploits fine properties of renewal processes, distributional properties of stable Lévy processes and the continuous mapping theorem.