Sojourns and extremes of Fourier sums and series with random coefficients

Let X(t) be the trigonometric polynomial Σ^k_j=0a_j(U_tcosjt+V_jsinjt), –∞< t<∞, where the coefficients U_t and V_t are random variables and a_j is real. Suppose that these random variables have a joint distribution which is invariant under all orthogonal transformations of R^2k–2. Then X(t) is stationary but not necessarily Gaussian. Put L_t(u) = Lebesgue measure {s: 0?s?t, X(s) > u}, and M(t) = max{X(s): 0?s?t}. Limit theorems for L_t(u) and $\text{P}\text{(M(t) > u)}$ for u→∞ are obtained under the hypothesis that the distribution of the random norm (Σ^k_j=0(U²_j+V²_j))^{1 2} belongs to the domain of attraction of the extreme value distribution exp{ e^–2}. The results are also extended to the random Fourier series (k=∞).     

A conditional dichotomy theorem for stochastic processes with independent increments

Let X(t) and Y(t) be two stochastically continuous processes with independent increments over [0, T] and Lévy spectral measures M_t and N_t, respectively, and let the “time-jump” measures M and N be defined over [0, T] × $\text{R}$?{0} by M((t₁, t₂] × A) = M_t2(A) ? M_t1(A) and N((T₁, t₂] × A) = N_t2(A) ? N_t1(A). Under the assumption that M is equivalent to N, it is shown that the measures induced on function space by X(t) and Y(t) are either equivalent or orthogonal, and necessary and sufficient conditions for equivalence are given. As a corollary a complete characterization of the set of admissible translates of such processes is obtained: a function f is an admissible translate for X(t) if and only if it is an admissible translate for the Gaussian component of X(t). In particular, if X(t) has no Gaussian component, then every nontrivial translate of X(t) is orthogonal to it.     

Singular Volterra integral equations

Existence results are presented for the singular Volterra integral equation y(t) = h(t) + ∫₀^t k(t, s) f(s, y(s)) ds, for t ∈ [0,T]. Here f may be singular at y = 0. As a consequence new results are presented for the n^th order singular initial value problem.     

A duality relation for entrance and exit laws for Markov processes

Markov processes X_t on (X, F_X) and Y_t on (Y, F_Y) are said to be dual with respect to the function f(x, y) if E_xf(X_t, y) = E_yf(x, Y_t for all x ? X, y ? Y, t ? 0. It is shown that this duality reverses the role of entrance and exit laws for the processes, and that two previously published results of the authors are dual in precisely this sense. The duality relation for the function f(x, y) = 1_{x<y} is established for one-dimensional diffusions, and several new results on entrance and exit laws for diffusions, birth-death processes, and discrete time birth-death chains are obtained.     

On a nonconvolution Volterra resolvent

Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝₀^ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝₀^tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝₀^ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f ∝₀^tr(t, s) f(s) ds maps a weighted L¹ space continuously into another weighted L¹ space, and a weighted L^∞ space into another weighted L^∞ space. Our main theorem is used to study the asymptotic behavior of two differential delay equations.     

An inversion formula for a class of integral transforms,II

In this note we give a procedure for inverting the integral transform f(x) = ∫₀^∞k(xt) φ(t) dt, where the functions f(x) and k(x) are known and φ(x) is to be found. The inversion is accomplished in two steps: by first defining a transforming function, which is an integral, followed by the application of an infinite order differential operator.     

Random evolutions in discrete and continuous time

This paper points out a connection between random evolutions and products of random matrices. This connection is useful in predicting the long-run growth rate of a single-type, continuously changing population in randomly varying environments using only observations at discrete points in time. A scalar Markov random evolution is specified by the n×n irreducible intensity matrix or infinitesimal generator Q = (q_ij) of a time-homogeneous Markov chain and by n finite real growth rates (scalars) s_i. The scalar Markov random evolution is the quantity M^C(t) = exp(Σⁿ_j=1s_jg^C_j (t)), where g^C_j(t) is the occupancy times in state j up to time t. The scalar Markov product of random matrices induced by this scalar Markov random evolution is the quantity M^D(t) = exp(Σⁿ_j=1s_jg^D_j (t)), where g^D_j(t) is the occupancy time in state j up to and including t of the discrete-time Markov chain with stochastic one-step transition matrix P = e^Q. We show that lim_t→∞(1/t)E(logM^D(t))=lim_t→∞(1/t)E(logM^C(t)) but that in general lim_t→∞(1/t)logE(M^C(t)) ≠ lim_t→∞(1/t)logE(M^D(t)). Thus the mean Malthusian parameter of population biologists is invariant with respect to the choice of continuous or discrete time, but the rate of growth of average population size is not. By contrast with a single-type population, in multitype populations whose growth is governed by non-commuting operators, the mean Malthusian parameter may be destined for a less prominent role as a measure of long-run growth.     

Asymptotic behavior of unbounded solutions of linear Volterra integral equations

The asymptotic behavior as t → ∞ of solutions of ∝₀^tu(t ? s) dA(s) = f(t) is studied when f(t) satisfies a “o” estimate as $\text{t ” ∞}$, and A belongs to a weighted space and its Laplace-Stieltjes transform has finitely many zeros in its closed half-plane of convergence. Results for systems of integral equations as well as for integrodifferential systems are also given.     

Distribution and characteristic functions for correlated complex Wishart matrices

Let A(t) be a complex Wishart process defined in terms of the M×N complex Gaussian matrix X(t) by A(t)=X(t)X(t)^H. The covariance matrix of the columns of X(t) is Σ. If X(t), the underlying Gaussian process, is a correlated process over time, then we have dependence between samples of the Wishart process. In this paper, we study the joint statistics of the Wishart process at two points in time, t₁, t₂, where t₁<t₂. In particular, we derive the following results: the joint density of the elements of A(t₁), A(t₂), the joint density of the eigenvalues of Σ^-1A(t₁),Σ^-1A(t₂), the characteristic function of the elements of A(t₁), A(t₂), the characteristic function of the eigenvalues of Σ^-1A(t₁),Σ^-1A(t₂). In addition, we give the characteristic functions of the eigenvalues of a central and non-central complex Wishart, and some applications of the results in statistics, engineering and information theory are outlined.     

Decay of solutions of some nonlinear parabolic inequalities associated with time-dependent convexes

A simple result concerning integral inequalities enables us to give an alternative proof of Waltman's theorem: lim_{t → ∞} ∝^t₀a(s) ds = ∞ implies oscillation of the second order nonlinear equation y″(t) + a(t) f(y(t)) = 0; to prove an analog of Wintner's theorem that relates the nonoscillation of the second order nonlinear equations to the existence of solutions of some integral equations, assuming that lim_{t → ∞} ∝^t₀a(s) ds exists; and to give an alternative proof and to extend a result of Butler. An often used condition on the coefficient a(t) is given a more familiar equivalent form and an oscillation criterion involving this condition is established.     

Weighted pseudo almost periodic mild solutions of semilinear evolution equations with nonlocal conditions

In this paper, applying the theory of semigroups of operators to evolution families and Banach fixed point theorem, we prove the existence and uniqueness of the weighted pseudo almost periodic mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) with nonlocal conditions x(0)=x₀+g(x) in Banach space X under some suitable hypotheses.     

Maximum and minimum of one-dimensional diffusions

Let M_t be the maximum of a recurrent one-dimensional diffusion up till time t. Under appropriate conditions, there exists a distribution function F such that |P(M_t?x) ? F^t(x)|→0as t and x go to infinity. This reduces the asymptotic behavior of the maximum to that of the maximum of independent and identically distributed random variables with distribution function F. A new proof of this fact is given which is based on a time change of the Ornstein-Uhlenbeck process. Using this technique, the asymptotic independence of the maximum and minimum is also established. Moreover, this method allows one to construct stationary processes in which the limiting behavior of M_t is essentially unaffected by the stationary distribution. That is, there may be no relationship between the distribution F above and the marginal distribution of the process.     

Semi-stable Markov processes in rn

A Markov process in Rⁿ{x_t} with transition function P_t is called semi-stable of order α>0 if for every a>0, P_t(x, E) = P_at(a^ax, a^aE). Let ?_t(ω)=∫^t₀|x_s(ω)|^-1/α ds, T(t) be its inverse and {y_t}={x_T(t)}.Theorem 1: {Y_t} is a multiplicative invariant process; i.e., it has transition function q_t satisfying q_t(x,E)=q_t(ax,aE) for all a > 0.Theorem 2: If {x_t} is Feller, right continuous and uniformly stochastic continuous on a neighborhood of the origin, then {y_t} is Feller.     

Fluctuations of the empirical quantiles of independent Brownian motions

We consider iid Brownian motions, B_j(t), where B_j(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by B_j:n(t), and we consider a sequence Q_n(t)=B_j(n):n(t), where j(n)/n→α∈(0,1). This sequence converges in probability to q(t), the α-quantile of the law of B_j(t). We first show convergence in law in C[0,∞) of F_n=n^1/2(Q_n−q). We then investigate properties of the limit process F, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=1/4.     

Hitting time of a moving boundary for a diffusion

We obtain asymptotic estimates for the quantity r = log P[T_f[rang]t] as t → ∞ where T_f = inf\s{s : |X(s)|[rang]f(s)\s} and X is a real diffusion in natural scale with generator a(x) d²(·)/dx² and the ‘boundary’ f(s) is an increasing function. We impose regular variation on a and f and the result is expressed as r = ∫^t₀ λ₁ (f(s) ds(1 + o(1)) where λ₁(f) is the smallest eigenvalue for the process killed at ±f.     

Continuity of l 2 two-parameter ornstein-uhlenbeck processes

Let {Y(t);t=(t ₁,t₂)≥0}={X_k(t₁,t₂);t₁≥0,t₂≥0} _k=1 ^∞ , be a sequence of two-parameter Ornstein-Uhlenbeck processes (OUP₂) with coefficient a _k>0,ß_k>0.. A Fernique type inequality is established and the sufficient condition for a. s. l ² continuity of Y(?) is studied by means of the inequality.     

On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn

Let B₁, B₂, ... be a sequence of independent, identically distributed random variables, letX₀ be a random variable that is independent ofB_n forn?1, let ρ be a constant such that 0<ρ<1 and letX₁,X₂, ... be another sequence of random variables that are defined recursively by the relationshipsX_n=ρX_n-1+B_n. It can be shown that the sequence of random variablesX₁,X₂, ... converges in law to a random variableX if and only ifE[log⁺¦B₁¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX₁,X₂,...stands toB₁,B₂,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log⁺¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.     

Loewner chains and parametric representation in several complex variables

Let B be the unit ball of with respect to an arbitrary norm. We study certain properties of Loewner chains and their transition mappings on the unit ball B. We show that any Loewner chain f(z,t) and the transition mapping v(z,s,t) associated to f(z,t) satisfy locally Lipschitz conditions in t locally uniformly with respect to z∈B. Moreover, we prove that a mapping f∈H(B) has parametric representation if and only if there exists a Loewner chain f(z,t) such that the family {e^−tf(z,t)}_t?0 is a normal family on B and f(z)=f(z,0) for z∈B. Also we show that univalent solutions f(z,t) of the generalized Loewner differential equation in higher dimensions are unique when {e^−tf(z,t)}_t?0 is a normal family on B. Finally we show that the set S⁰(B) of mappings which have parametric representation on B is compact.     

Uniform asymptotic stability for perturbed neutral delay differential equations

One-dimensional perturbed neutral delay differential equations of the form (x(t)−P(t,x(t−τ)))′=f(t,x_t)+g(t,x_t) are considered assuming that f satisfies −v(t)M(φ)?f(t,φ)?v(t)M(−φ), where M(φ)=max{0,max_s∈[−r,0]φ(s)}. A typical result is the following: if ‖g(t,φ)‖?w(t)‖φ‖ and , then the zero solution is uniformly asymptotically stable providing that the zero solution of the corresponding equation without perturbation (x(t)−P(t,x(t−τ)))′=f(t,x_t) is uniformly asymptotically stable. Some known results associated with this equation are extended and improved.     

Regularity and long time behavior of the Boltzmann equation for granular gases

This paper investigates regularity of solutions of the Boltzmann equation with dissipative collisions in a thermal bath. In the case of pseudo-Maxwellian approximation, we prove that for any initial datum f₀(ξ) in the set of probability density with zero bulk velocity and finite temperature, the unique solution of the equation satisfies f(ξ,t)∈H^∞(R³) for all t>0. Furthermore, for any t₀>0 and s?0 the Hs norm of f(ξ,t) is bounded for t?t₀. As a consequence, the exponential convergence to the unique steady state is also established under the same initial condition.