Stationary distributions of the simplest dynamical systems in R perturbed by generalized Poisson process

Let z(t) Rⁿ be a generalized Poisson process with parameter λ and let A: Rⁿ → Rⁿ be a linear operator. The conditions of existence and limiting properties as λ → ∞ or as λ → 0 of the stationary distribution of the process x(t) Rⁿ which satisfies the equation dx(t) = Ax(t)dt + dz(t) are investigated.     

Scaling limits of Gaussian vector fields

For Gaussian vector fields {X(t) ∈ Rⁿ:t ∈ R^d} we describe the covariance functions of all scaling limits Y(t) = $\text{L}$lim_α↓0 B^?1(α) X(αt) which can occur when B(α) is a d × d matrix function with B(α) → 0. These matrix covariance functions $\text{r(t, s) = EY(t) Y}{}^1\text{(s)}$ are found to be homogeneous in the sense that for some matrix L and each α > 0, $\text{(}{}^1\text{) r(αt, αs) = α}{}^{\mathrm{L1}}\text{r(t, s) α}{}^{\mathrm{L}}$. Processes with stationary increments satisfying (¹) are further analysed and are found to be natural generalizations of Lévy's multiparameter Brownian motion.     

Point processes of exits by bivariate Gaussian processes and extremal theory for the χ2-process and its concomitants

Let ζ(t), η(t) be continuously differentiable Gaussian processes with mean zero, unit variance, and common covariance function r(t), and such that ζ(t) and η(t) are independent for all t, and consider the movements of a particle with time-varying coordinates (ζ(t), η(t)). The time and location of the exists of the particle across a circle with radius u defines a point process in R³ with its points located on the cylinder {(t, u cos θ, u sin θ); t ≥ 0, 0 ≤ θ < 2π}. It is shown that if r(t) log t → 0 as t → ∞, the time and space-normalized point process of exits converges in distribution to a Poisson process on the unit cylinder. As a consequence one obtains the asymptotic distribution of the maximum of a χ²-process, χ²(t) = ζ²(t) + η²(t), P{sup_0≤t≤Tχ²(t) ≤ u²} → e^?τ if $\text{T(}\text{?r″(0)}\text{2π}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{u ×}\text{exp}\text{(}\text{?u}{}^2\text{2}\text{) → τ}$ as T, u → ∞. Furthermore, it is shown that the points in R³ generated by the local ?-maxima of χ²(t) converges to a Poisson process in R³ with intensity measure (in cylindrical polar coordinates) (2πr²)^?1dtdθdr. As a consequence one obtains the asymptotic extremal distribution for any function g(ζ(t), η(t)) which is “almost quadratic” in the sense that $\text{g}{}^1\text{(r}\text{cos}\text{θ, r}\text{sin}\text{θ) =}\text{1}\text{2}\text{(r}{}^2\text{? g(r}\text{cos}\text{θ, r}\text{sin}\text{θ))}$ has a limit $\text{g}{}^1\text{(θ)}$ as r → ∞. Then if $\text{T(}\text{?r″(0)}\text{2π}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{u}\text{exp}\text{(}\text{?u}{}^2\text{2}\text{) → τ}$ as T, u → ∞.     

On boundary value problems for systems of nonlinear generalized ordinary differential equations

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R ⁿ → R ⁿ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → R ^n×n with bounded total variation components, and h: BV_s([a, b],R ⁿ ) → R ⁿ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t₁(x)) = B(x) · x(t ₂(x))+c ₀, where t _i: BV_s([a, b],R ⁿ ) → [a, b] (i = 1, 2) and B: BV_s([a, b], R ⁿ ) → R ⁿ are continuous operators, and c ₀ ∈ R ⁿ .     

Diffusion approximations of age-and-position dependent branching processes

By a (G, F, h) age-and-position dependent branching process we mean a process in which individuals reproduce according to an age dependent branching process with age distribution function G(t) and offspring distribution generating function F, the individuals (located in R^N) can not move and the distance of a new individual from its parent is governed by a probability density function h(r). For each positive integer n, let Z_n(t,dx) be the number of individuals in dx at time t of the (G, F_n,h_n) age-and-position dependent branching process. It is shown that under appropriate conditions on G, F_n and h_n, the finite dimensional distribution of $\text{Z}{}_{\mathrm{n}}\text{(nt, dx)}\text{n}$ converges, as n → ∞, to the corresponding law of a diffusion continuous state branching process X(t,dx) determined by a ψ-semigroup {ψ_t: t ? 0}. The ψ-semigroup {ψ_t} is the solution of a non-linear evolution equation. A semigroup convergence theorem due to Kurtz [10], which gives conditions for convergence in distribution of a sequence of non-Markovian processes to a Markov process, provides the main tools.     

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.     

On oscillation of solutions of a nonautonomous quasilinear second-order equation

Sufficient conditions are obtained for the initial values of nontrivial oscillating (for t=ω) solutions of the nonautonomous quasilinear equation $$y'' \pm \lambda (t)y = F(t,y,y'),$$ wheret ∈ Δ=[a, ω[,-∞ <a < ω ≤+ ∞, λ(t) > 0, λ(t) ∈ C _Δ ⁽¹⁾ , |F((t,x,y))|≤L(t)(|x|+|y|)^1+α, L(t) ≥-0, α ∈ [0,+∞[, F: Δ × R² →R,F ∈C _Δ×R ²,R is the set of real numbers, and R² is the two-dimensional real Euclidean space.     

Necessary and sufficient conditions for quadratic stabilizability of an uncertain system

Consider an uncertain system (Σ) described by the equationx(t)=A(r(t))x(t)+B(s(t))u(t), wherex(t) ∈R ⁿ is the state,u(t) ∈R ^m is the control,r(t) ∈ ? ?R ^p represents the model parameter uncertainty, ands(t) ∈L ?R ^l represents the input connection parameter uncertainty. The matrix functionsA(·),B(·) are assumed to be continuous and the restraint sets ?,L are assumed to be compact. Within this framework, a notion of quadratic stabilizability is defined. It is important to note that this type of stabilization is robust in the following sense: The Lyapunov function and the control are constructed using only the bounds ?,L. Much of the previous literature has concentrated on a fundamental question: Under what conditions onA(·),B(·), ?,L can quadratic stabilizability be assured? In dealing with this question, previous authors have shown that, if (Σ) satisfies certain matching conditions, then quadratic stabilizability is indeed assured (e.g., Refs. 1–2). Given the fact that matching is only a sufficient condition for quadratic stabilizability, the objective here is to characterize the class of systems for which quadratic stabilizability can be guaranteed.     

Multiobjective programming and penalty functions

This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R ^m, with each component equal to unity;f:R ⁿ →R ^m, represents them objective functions {f ⁱ} defined onX \( \subseteq \) R ⁿ; λ ∈R ¹, λ>0;P:R ⁿ →R ¹ X \( \subseteq \) Z \( \subseteq \) R ⁿ,P(x)≦0, ∨x ∈R ⁿ,P(x) = 0 ?x ∈X. The paper studies properties of {E (Z, λ _r )} for a sequence of positive {λ _r } converging to 0 in relationship toE(X), whereE(Z, λ _r ) is the efficient set ofZ with respect tog(·, λ_r) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.     

Approximation and estimation of some compound distributions

The distribution of the total amount claimed up to time t can often be written in the form of a compound distribution G_t(x) = Σp_n(t)F⁽ⁿ⁾(x) where p_n(t) is the probability of exactly n claims while F is the distribution of a single claim. In the actuarial literature one often finds approximations of G_t(x) when the time t is large. It seems more natural to take t fixed and to look for approximations for x large. This paper contains a number of such results for a Poisson process and for a Pascal process. Different hypotheses on the tail behaviour of F(t) yield different expressions to estimate 1 - G_t(x). The results obtained should prove to have wider applicability than suggested by the insurance context. Within it, however, applications to premium calculation principles are immediate.     

A remark on Schrödinger operators

We study the almost everythere convergence to the initial dataf(x)=u(x, 0) of the solutionu(x, t) of the two-dimensional linear Schrödinger equation Δu=i? _t u. The main result is thatu(x, t) →f(x) almost everywhere fort → 0 iff ∈H ^p (R²), wherep may be chosen <1/2. To get this result (improving on Vega’s work, see [6]), we devise a strategy to capture certain cancellations, which we believe has other applications in related problems.     

On the Laplace transforms of retarded,Lorentz-invariant functions

Let ø(t) (t ∈ $\text{R}$ⁿ) be a retarded, Lorentz-invariant function which satisfies, in addition, condition (c). We call “R” the family of such functions. Let f(z) be the Laplace transform of ø(t) ∈ $\text{R}$. We prove (Theorem 1) that f(z) can be expressed as a K-transform (formula (I, 2; 1)). We apply this formula to evaluate several Laplace transforms. We show that it affords simple proofs of important known results. Formula (I, 2; 1) is an effective complement to L. Schwartz' method of evaluating Fourier transforms via Laplace transforms (“Théorie des distributions,” p. 264, Hermann, Paris, 1966). We think this is the most useful application of our formula.     

On the Riemann zeta-function—Mean value theorems and the distribution of |S(T)|

Asymptotic formulae for the mean values of |S(t)|^λ, where λ is any nonnegative number are proved,. These are then used with λ ∈ $\text{N}$, to obtain information of the distribution of |S(t)|.     

A finiteness criterion and asymptotics for codimensions of generalized identities

Let A be an associative algebra over a field of characteristic zero. Then either all codimensions gc _n (A) of its generalized polynomial identities are infinite or A is the sum of ideals I and J such that dim _F I < ∞ and J is nilpotent. In the latter case, there exist numbers n ₀ ∈ ?, C ∈ ?₊, and t ∈ ?₊ for which gc _n (A) < +∞ if n ≥ n ₀ and gc _n (A) ～ Cn ^t d ⁿ as n → ∞, where d = PIexp(A) ∈ ?₊. Thus, in the latter case, conjectures of Amitsur and Regev on generalized codimensions hold.     

Idempotence preserving maps on spaces of triangular matrices

SupposeF is an arbitrary field. Let |F| be the number of the elements ofF. LetT _n (F) be the space of allnxn upper-triangular matrices overF. A map Ψ: T _N (F) → T _N (F) is said to preserve idempotence ifA - λ B is idempotent if and only if Ψ(A) - λΨ(B) is idempotent for anyA, B ∈ T _n (F) and λ ∈ F. It is shown that: when the characteristic ofF is not 2, |F|>3 and n ≥ 3, Ψ:T _n (F) → T _n (F) is a map preserving idempotence if and only if there exists an invertible matrixP τ T _n (F) such that either ?(A) = PAP ^?1 for everyA ∈ T _n (F) or Ψ(A) = PJA ^t JP ^?1 for everyA ∈ T _n (F), whereJ = ∑ _n=1 ⁿ E _i,n+1?i and E_ij is the matrix with 1 in the (i,j)th entry and 0 elsewhere.     

A note on the conditional moments of a multivariate normal distribution confined to a convex set

Let Y be an N(μ, Σ) random variable on R^m, 1 ≤ m ≤ ∞, where Σ is positive definite. Let C be a nonempty convex set in R^m with closure $\text{C}$. Let (·,-·) be the Eculidean inner product on R^m, and let μ_c be the conditional expected value of Y given Y ∈ C. For v ∈ R^m and s ≥ 0, let β_s(v) be the expected value of |(v, Y) ? (v, μ)|^s and let γ_s(v) be the conditional expected value of |(v, Y) ? (v, μ_c)|^s given Y ∈ C. For s ≥ 1, γ_s(v) < β_s(v) if and only if $\text{C}\text{+ Σ v ≠}\text{C}$, and γ_s(v) < β_s(v) for all v ≠ 0 if and only if $\text{C}\text{+ v ≠}\text{C}$ for any v ∈ R^m such that v ≠ 0.     

Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)^1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0⁺. Then we find, as t → ∞, (P{Z(t)> 0})^αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u^α)^?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.     

The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

We derive sufficient conditions for ∝ λ (dx)6Pⁿ(x, ·) - π6 to be of order o(ψ(n)^-1), where Pⁿ (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, π is the invariant probability measure, λ an initial distribution and ψ belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order ψ, which in a countable state space is equivalent to $\text{Σ ψ(n)}\text{P}{}_{\mathrm{i}}\text{{τ}{}_{\mathrm{i}}\text{?n} <∞}$ for some i, where τ_i is the hitting time of the tate i. We also show that for a general Markov chain to be ergodic of order ψ it suffices that a corresponding condition is satisfied by a small set.We apply these results to non-singular renewal measures on $\text{R}$ providing a probabilisite method to estimate the right tail of the renewal measure when the increment distribution F satisfies ∝ tF(dt) 0; > 0 and ∝ ψ(t)(1- F(t))dt< ∞.     

Spherically symmetric solutions of a reaction-diffusion equation

Two theorems are proved for the spherically symmetric solutions of the “bistable” reaction-diffusion equation $\text{u}{}_{\mathrm{t}}\text{= Δ}{}_{\mathrm{x}}\text{u + ?(u)}$, where ? is cubic-like and x ∈ Rⁿ. The first theorem says that, for a suitable class of initial data, there are only two types of asymptotic behavior, u(x, t) tends to an equilibrium solution as t → + ∞ or u(x, t) → 1 uniformly on compact sets. The second theorem says that in the latter case, if the solution is followed out along any ray, it approaches, in shape, the one-dimensional travelling wave.     

When are intermediate processes of the same stochastic order?

Let Z_m⁽ⁿ⁾ represent the mth largest order statistic in a random sample of size n. Here we study the process Z_[mt]⁽ⁿ⁾, t > 0, where m(n) is an intermediate sequence such that m → ∞, m/n → 0 as n → ∞.