Stationary queues with one autonomous server

Given a stationary stochastic continuous demand of service σ(θ_tω) dt with ∫ σ(ω)P(dω) < 1, we construct real stationary point processes $\text{(T}{}_{\mathrm{n}}\text{, n ∈}\text{Z}\text{)[T}{}_{\mathrm{n}}\text{< T}{}_{\mathrm{n}}\text{+1, lim}{}_{\mathrm{\pm \infty }}\text{T}{}_{\mathrm{n}}\text{= ±∞]}$ such that

for a given constant D \2>0. These point processes correspond to a service discipline for which a single server services during the time intervals [T_n, T_n+1[ the demand of service accumulated during the proceding intervals [T_n?1, T_n[ and take a rest of fixed duration D.     

Exposants de Lyapounov pour opérateurs de Schrödinger discrètes quasi-périodiquesLyapounov exponents for discrete,quasi-periodic Schrödinger operators

We consider quasi-periodic Schrödinger operators H on $\text{Z}$ of the form H=H_λ,x,ω=λv(x+nω)δ_n,n′+Δ where v is a non-constant real analytic function on the d-torus $\text{T}{}^{\mathrm{d}}\text{(d?1)}$ and Δ denotes the discrete lattice Laplacian on $\text{Z}$. Denote by L_ω(E) the Lyapounov exponent, considered as function of the energy E and the rotation vector $\text{ω∈}\text{T}{}^{\mathrm{d}}$. It is shown that for |λ|>λ₀(v), there is the uniform minoration $\text{L}{}_{\mathrm{\omega }}\text{(E)>}\text{1}\text{2}\text{log}\text{|λ|}$ for all E and ω. For all λ and ω, L_ω(E) is a continuous function of E. Moreover, L_ω(E) is jointly continuous in (ω,E), at any point $\text{(ω}{}_0\text{,E}{}_0\text{)∈}\text{T}{}^{\mathrm{d}}\text{×}\text{R}$ such that k·ω₀≠0 for all $\text{k∈}\text{Z}{}^{\mathrm{d}}\text{?{0}}$. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 529–531.     

Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

An elementary proof of the uniqueness of invariant product measures for some infinite dimensional processesUne preuve élémentaire de l'unicité des mesures invariantes produits pour certains processus en dimension infinie

Consider an infinite dimensional diffusion process with state space , where T is the circle, and defined by an infinitesimal generator L which acts on local functions f as . Suppose that the coefficients a_i and b_i are smooth, bounded, of finite range, have uniformly bounded second order partial derivatives, that a_i are uniformly bounded from below by some strictly positive constant, and that a_i is a function only of η_i. Suppose that there is a product measure ν which is invariant. Then if ν is the Lebesgue measure or if d=1,2, it is the unique invariant measure. Furthermore, if ν is translation invariant, it is the unique invariant, translation invariant measure. The proofs are elementary. Similar results can be proved in the context of an interacting particle system with state space , with uniformly positive bounded flip rates which are finite range. To cite this article: A.F. Ram??rez, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 139–144     

A Feynman-Kac gauge for solvability of the Schrödinger equation

Let {X_t, t ≥ 0} be Brownian motion in $\text{R}$^d (d ≥ 1). Let D be a bounded domain in $\text{R}$^d with C² boundary, ?D, and let q be a continuous (if d = 1), Hölder continuous (if d ≥ 2) function in D?. If the Feynman-Kac “gauge” E_x{exp(∝₀^τ_Dq(X_t)dt)1_A(X_τD)}, where τ_D is the first exit time from D, is finite for some non-empty open set A on ?D and some x?D, then for any $\text{? ? C}{}^0\text{(?D}$), is the unique solution in $\text{C}{}^2\text{(D) ∩ C}{}^0\text{(}\text{D}\text{?}\text{)}$ of the Schrödinger boundary value problem $\text{(}\text{1}\text{2}\text{Δ}\text{+ q)φ = 0}\text{in}\text{D, φ = ?}\text{on}\text{?D}$.     

Levy's stochastic area formula in higher dimensions

Let (W_t) = (W¹_t,W²_t,…,W^d_t), d ? 2, be a d-dimensional standard Brownian motion and let A(t) be a bounded measurable function from $\text{R}$⁺ into the space of d × d skew-symmetric matrices and x(t) such a function into $\text{R}$^d. A class of stochastic processes (L_t^A,x), a particular example of which is Levy's “stochastic area” $\text{L}{}_{\mathrm{t}}\text{=}\text{1}\text{2}\text{∝}{}_0{}^{\mathrm{?t}}\text{(W}{}^1{}_{\mathrm{s}}\text{,dW}{}^2{}_{\mathrm{s}}\text{? W}{}^2{}_{\mathrm{s}}\text{,dW}{}^1{}_{\mathrm{s}}\text{)}$, is dealt with.The joint characteristic function of W_t and L₁^A,x is calculated and based on this result a formula for fundamental solutions for the hypoelliptic operators which generate the diffusions (W_t,L_t^A,x) is given.     

A decomposition theorem for an analytic function

Let f(z), an analytic function with radius of convergence R (0 < R < ∞) be represented by the gap series ∑_{k = 0}^∞c_kz^λ_k. Set and define the growth constants ?, λ, T, t by

$\text{?}\text{λ}\text{=}\text{lim sup}\text{r→R}\text{inf}\text{{}\text{log}\text{[Rr /(R?r)]}{}^{\mathrm{?1}}\text{log}{}^{\mathrm{+}}\text{log}{}^{\mathrm{+}}\text{M(r)}}$

, and if 0 < ? < ∞,

$\text{T}\text{t}\text{=}\text{lim sup}\text{r→R}\text{inf}\text{{[Rr /(R?r)]}{}^{\mathrm{??}}\text{log}{}^{\mathrm{+}}\text{M(r)}}$

. Then, assuming 0 < t < T < ∞, we obtain a decomposition theorem for f(z).     

On Dynkin's Markov property of random fields associated with symmetric processes

Let p(t, x, y) be a symmetric transition density with respect to a σ-finite measure m on (E, $\text{E}$), g(x,y)=∫p(t,x,y)dt, and $\text{M}\text{={σ-finite measures μ?0:∫g(x,y)μ(}\text{d}\text{x)μ(}\text{d}\text{y)<∞}}$. There exists a Gaussian random field $\text{Φ={?}{}_{\mathrm{\mu }}\text{:μ?}\text{M}\text{}}$ with mean 0 and covariance $\text{E}\text{?}{}_{\mathrm{\mu }}\text{?}{}_{\mathrm{\nu }}\text{=∫g(x,y)μ(}\text{d}\text{x)ν(}\text{d}\text{y)}$. Letting $\text{F}\text{(B)=σ{?}{}_{\mathrm{\mu }}\text{:μ(B}{}^{\mathrm{c}}\text{)=0}}$ we consider necessary and sufficient conditions for the Markov property (MP) on sets B, C: $\text{F}$(B), $\text{F}$(C) c.i. given $\text{F}$(B ∩ C). Of crucial importance is the following, proved by Dynkin: $\text{E}\text{{?}{}_{\mathrm{\mu }}\text{∣}\text{F}\text{(B)}=?}{}_{\mathrm{\mu B}}$, where μ_B is the hitting distribution of the process corresponding to p, m with initial law μ. Another important fact is that ?_μ=?_ν iff μ, ν have the same potential. Putting these together with an additional transience assumption, we present a potential theoretic proof of the following necessary and sufficient condition for (MP) on sets B, C: For every x?E, T_B∩C=T_B+T_C∮ θ_TB=T_C+T_B∮θ_TC a.s. P^x where, for D ? $\text{E}$, T_D is the hitting time of D for the process associated with p, m. This implies a necessary condition proved by Dynkin in a recent preprint for the case where B∪C=E and B, C are finely closed.     

Strong approximation theorems for density dependent Markov chains

A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form

$\text{X}{}_{\mathrm{N}}\text{(t)=x}{}_0\text{+∑}\text{1}\text{N}\text{lY}{}_1\text{N ∫}{}^{\mathrm{t}}{}_0\text{f}{}_1\text{(X}{}_{\mathrm{N}}\text{(s))ds}$

where $\text{l∈}\text{Z}{}^{\mathrm{t}}$, the Y₁ are independent Poisson processes, and N is a parameter with a natural interpretation (e.g. total population size or volume of a reacting solution).The corresponding deterministic model, satisfies

$\text{X(t)=x}{}_0\text{+ ∫}{}^{\mathrm{t}}{}_0\text{∑ lf}{}_1\text{(X(s))ds}$

Under very general conditions lim_N→∞X_N(t)=X(t) a.s. The process X_N(t) is compared to the diffusion processes given by

$\text{Z}{}_{\mathrm{N}}\text{(t)=x}{}_0\text{+∑}\text{1}\text{N}\text{lB}{}_1\text{N∫}{}^{\mathrm{t}}{}_0\text{ft(Z}{}_{\mathrm{N}}\text{(s))ds}$

and

$\text{V(t)=∑ l∫}{}^{\mathrm{t}}{}_0\text{f}{}_1\text{(X(s))}\text{d}\text{W}\text{?}{}_1\text{+∫}{}^{\mathrm{t}}{}_0\text{?F(X(s))·V(s)}\text{ds.}$

Under conditions satisfied by most of the applied probability models, it is shown that X_N,Z_N and V can be constructed on the same sample space in such a way that

$\text{X}{}_{\mathrm{N}}\text{(t)=Z}{}_{\mathrm{N}}\text{(t)+O}\text{logN}\text{N}$

and

$\text{N}\text{(X}{}_{\mathrm{N}}\text{(t)?X(t))=V(t)+O}\text{log N}\text{N}$

     

Laws of iterated logarithm of multiparameter wiener processes

Let {$\text{X(}\text{t}\text{) :}\text{t}\text{∈ R}{}_{\mathrm{+}}{}^{\mathrm{N}}$} denote the N-parameter Wiener process on $\text{R}{}_{\mathrm{+}}{}^{\mathrm{N}}\text{= [0, ∞)}{}^{\mathrm{n}}$. For multiple sequences of certain independent random variables the authors find lower bounds for the distributions of maximum of partial sums of these random variables, and as a consequence a useful upper bound for the yet unknown function , c ≥ 0, is obtained where D_N = Π_{k = 1}^N [0, T_k]. The latter bound is used to give three different varieties of N-parameter generalization of the classical law of iterated logarithm for the standard Brownian motion process.     

A remark on the tail probability of a distribution

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let S_n = Σ_j=1ⁿX_j and $\text{E{N}{}_{\mathrm{\infty }}\text{(r, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{r?2}}\text{P{|S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>r</mtext><mtext>t</mtext>}}\text{}}$. In this paper, we prove that (1) lim_?→0+?^α(r?1)E{N_∞(r, t, ?)} = K(r, t) if E(X₁) = 0, Var(X₁) = 1, and E(| X₁ |^t) < ∞, where 2 ≤ t < 2r ≤ 2t, $\text{K(r, t) =}\text{{2}{}^{\mathrm{<mtext>\alpha (r?1)</mtext><mtext>2</mtext>}}\text{Γ(}\text{(1 + α(r ? 1))}\text{2}\text{)}}\text{{(r ? 1) Γ(}\text{1}\text{2}\text{)}}$, and $\text{α =}\text{2t}\text{(2r ? t)}$; (2) $\text{lim}{}_{\mathrm{?\to 0+}}\text{G(t, ?)}\text{H(t, ?)}\text{= 0}$ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(|X₁|^t) < ∞, where G(t, ?) = E{N_∞(t, t, ?)} = Σ_n=1^∞n^t?2P{| S_n | > ?n} → ∞ as ? → 0+ and $\text{H(t, ?) = E{N}{}_{\mathrm{\infty }}\text{(t, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{t?2}}\text{P{| S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>2</mtext><mtext>t</mtext>}}\text{} → ∞}\text{as}\text{? → 0+}$, i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(| X₁ |^t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.     

Delayed random walks

A delayed random walk $\text{{S}{}^1{}_{\mathrm{n}}\text{, n ≥ 0}}$ is defined here as a partial sum process of independent random variables in which the first N summands (N optional) are distributed F¹,…,F_N, respectively, while all remaining summands are distributed F₀, where {F_k, _k ≥ 0} is a sequence of proper distribution functions on the real line. Delayed random walks arise naturally in the study of certain generalized single server queues. This paper examines optional times of the process such as $\text{π = inf {n: n ≥ 1 and S}{}^1{}_{\mathrm{n}}\text{≥ 0}}$. Conditions insuring the finiteness of E {π} and E {π²} are obtained, generating functions calculated, and illustrative examples given. The bivariate functions $\text{E{r}{}^{\mathrm{\pi }}\text{explsqbitS}{}^1{}_{\mathrm{\pi }}\text{rsqb}}$ and $\text{E {}\text{∑}\text{n=0}\text{π?1}\text{explsqbitS}{}^1{}_{\mathrm{n}}\text{rsqb}}$ are studied for the case where N ≡ 1.     

Some comments on the hazard gradient

For random variables T₁,…,T_n, the gradient of $\text{R(t) = ?log}\text{P}\text{{T}{}_1\text{> t}{}_1\text{,…,T}{}_{\mathrm{n}}\text{> t}{}_{\mathrm{n}}\text{}}$ is called the hazard gradient. Some properties of this multivariate version of the hazard rate are demonstrated, and some examples are given to show the usefulness of the hazard gradient in characterizing distributions or families of distributions.     

About some problems of spectral synthesis

For a given pair $\text{(A,b)∈}\text{R}{}^{\mathrm{n\times n}}\text{×}\text{R}{}^{\mathrm{n\times 1}}$ such that A is cyclic and b is a cyclic generator (with respect to A) of $\text{R}{}^{\mathrm{n\times 1}}$, it is shown that for every nonnegative integer m we can find a nonnegative integer t and a sequence $\text{{f}{}_{\mathrm{j\}}}{}^{\mathrm{t}}{}_{\mathrm{j=0}}\text{,f}{}_{\mathrm{j}}\text{∈}\text{R}{}^{\mathrm{1\times n}}$,so that a the zeros of the rational function det P(z), where $\text{P(z) = zI ? A ? ∑}{}^{\mathrm{t}}{}_{\mathrm{j=0}}\text{z}{}^{\mathrm{-(m+j)}}\text{b?}$_f, lie in the open unit disc in the complex plane. The result is directly applicable to a stabilizability problem for linear systems with a time delay in control action.     

Minimum variance estimators for misclassification probabilities in discriminant analysis

Let α(n₁, n₂) be the probability of classifying an observation from population Π₁ into population Π₂ using Fisher's linear discriminant function based on samples of size n₁ and n₂. A standard estimator of α, denoted by T₁, is the proportion of observations in the first sample misclassified by the discriminant function. A modification of T₁, denoted by T₂, is obtained by eliminating the observation being classified from the calculation of the discriminant function. The UMVU estimators, $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$, of ET₁ = τ₁(n₁, n₂) and ET₂ = τ₂(n₁, n₂) = α(n₁ ? 1, n₂) are derived for the case when the populations have multivariate normal distributions with common dispersion matrix. It is shown that $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$ are nonincreasing functions of D², the Mahalanobis sample distance. This result is used to derive the sampling distributions and moments of $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$. It is also shown that α is a decreasing function of Δ² = (μ₁ ? μ₂)′Σ^?1(μ₁ ? μ₂). Hence, by truncating $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$ (or any estimator) at the value of α for Σ = 0, new estimators are obtained which, for all samples, are as close or closer to α.     

The approximation of flows

Let U, V be two strongly continuous one-parameter groups of bounded operators on a Banach space $\text{X}$ with corresponding infinitesimal generators S, T. We prove the following: ∥U_t, ? V_t ∥ = O(t), t → 0, if and only if U = V; ∥U_t ? V_t∥ = O(t^α), t → 0; with 0 ? α ? 1, if and only if $\text{S = Ω(T + P)Ω}{}^{\mathrm{?1}}$, where Ω, P, are bounded operators on $\text{X}$ such that $\text{∥U}{}_{\mathrm{t}}\text{Ω ? ΩU}{}_{\mathrm{t}}\text{∥ = O(t}{}^{\mathrm{\alpha }}\text{), ∥U}{}_{\mathrm{t}}\text{P ? PU}{}_{\mathrm{t}}\text{∥ = ?O(t}{}^{\mathrm{\alpha }}\text{), t → 0; ∥U}{}_{\mathrm{t}}\text{? V}{}_{\mathrm{t}}\text{∥ = O(t)}$ if and only if $\text{S}{}^1\text{? T}{}^1$ has a bounded extension to $\text{X}$¹. Further results of this nature are inferred for semigroups, reflexive spaces, Hilbert spaces, and von Neumann algebras.     

Barriers for uniformly elliptic equations and the exterior cone condition

We consider the mixed boundary value problem $\text{Au = f}\text{in}\text{Ω, B}{}_0\text{u = g}{}_0\text{in}\text{Γ}{}^{\mathrm{?}}\text{, B}{}_1\text{u = g}{}_1\text{in}\text{Γ}{}^{\mathrm{+}}$, where Ω is a bounded open subset of $\text{R}$ⁿ whose boundary Γ is divided into disjoint open subsets Γ⁺ and Γ^? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on $\text{Ω}$ and B_j, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on $\text{Γ}{}^{\mathrm{+}}$. The coefficients of the operators and Γ⁺, Γ^? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset $\text{T}$ of the reals such that, if $\text{D}{}^{\mathrm{s}}\text{= {u ? H}{}^{\mathrm{s}}\text{(Ω): Au = 0}}$ then for $\text{s ? =}\text{1}\text{2}\text{(}\text{mod}\text{1), (B}{}_0\text{,B}{}_1\text{): D}{}^{\mathrm{s}}\text{→ H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{?}}\text{) × H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>3</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{+}}\text{)}$ is a Fredholm operator if and only if s ∈$\text{T}$ . Moreover, $\text{T}$ = ?_xew$\text{T}$_x, where the sets $\text{T}$_x are determined algebraically by the coefficients of the operators at x. If n = 2, $\text{T}$_x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, $\text{T}$_x is either an open interval of length 1 or is empty; and finally, if n ? 4, $\text{T}$_x is an open interval of length 1.     

Asymptotic behavior of a stochastic growth process associated with a system of interacting branching random walksComportement asymptotique d'un processus stochastique de croissance associé à un système de marches aléatoires en interaction

We study a continuous time growth process on $\text{Z}{}^{\mathrm{d}}$ (d?1) associated to the following interacting particle system: initially there is only one simple symmetric continuous time random walk of total jump rate one located at the origin; then, whenever a random walk visits a site still unvisited by any other random walk, it creates a new independent random walk starting from that site. Let us call P_d the law of such a process and S⁰_d(t) the set of sites, visited by all walks by time t. We prove that there exists a bounded, non-empty, convex set $\text{C}{}_{\mathrm{d}}\text{?}\text{R}{}^{\mathrm{d}}$, such that for every ε>0, P_d-a.s. eventually in t, the set S_d⁰(t) is within an ε neighborhood of the set [C_dt], where for $\text{A?}\text{R}{}^{\mathrm{d}}$ we define $\text{[A]:=A∩}\text{Z}{}^{\mathrm{d}}$. Moreover, for d large enough, the set C_d is not a ball under the Euclidean norm. We also show that the empirical density of particles within S_d⁰(t) converges weakly to a product Poisson measure of parameter one. To cite this article: A.F. Ram??rez, V. Sidoravicius, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 821–826.     

The one-dimensional diffusion process and unitary representations of R1

Given a cocycle a(t) of a unitary group {U¹}, ?∞ < t < ∞, on a Hilbert space $\text{H}$, such that a(t) is of bounded variation on [O, T] for every T > O, a(t) is decomposed as a(t) = f;^t₀U^sxds + β(t) for a unique x ? $\text{H}$, β(t) yielding a vector measure singular with respect to Lebesgue measure. The variance is defined as $\text{σ}{}^2\text{({rmU}{}^{\mathrm{t}}\text{}, a(t)) =}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{)∥∝}{}^{\mathrm{t}}{}_0\text{U}{}^{\mathrm{s}}\text{x ds∥}{}^2$ if existing. For a stationary diffusion process on $\text{R}$¹, with Ω₁, the space of paths which are natural extensions backwards in time, of paths confined to one nonsingular interval J of positive recurrent type, an information function I(ω) is defined on $\text{Ω}{}_1$, based on the paths restricted to the time interval [0, 1]. It is shown that $\text{I(Ω)}$ is continuous and bounded on $\text{Ω}{}_1$. The shift τ_t, defines a unitary representation {U^t}. Assuming , dm being the stationary measure defined by the transition probabilities and the invariant measure on J, $\text{I(Ω)}$ has a C^∞ spectral density function f;. It is then shown that σ²({U^t}, I) = f;(O).