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.     

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 → ∞.     

Coding theory in white Gaussian channel with feedback

The message m = {m(t)} is a Gaussian process that is to be transmitted through the white Gaussian channel with feedback: $\text{Y(t) = ∫}{}_0{}^{\mathrm{t}}\text{F(s, Y}{}_0{}^{\mathrm{s}}\text{, m)ds + W(t)}$. Under the average power constraint, $\text{E[F}{}^2\text{(s, Y}{}_0{}^{\mathrm{s}}\text{, m)] ≤ P}{}_0$, we construct causally the optimal coding, in the sense that the mutual information I_t(m, Y) between the message m and the channel output Y (up to t) is maximized. The optimal coding is presented by $\text{Y(t) = ∫}{}_0{}^{\mathrm{t}}\text{A(s)[m(s) ?}\text{m}\text{?}\text{(s)] ds + W(t)}$, where $\text{m}\text{?}\text{(s) = E[m(s) ¦ Y(u), 0 ≤ u ≤ s]}$ and A(s) is a positive function such that $\text{A}{}^2\text{(s) E |m(s) ?}\text{m}\text{?}\text{(s)|}{}^2\text{= P}{}_0$.     

Extinction probability for a critical general branching process

A general branching process begins with a single individual born at time t=0. At random ages during its random lifespan L it gives birth to offspring, N(t) being the number born in the age interval [0,t]. Each offspring behaves as a probabilistically independent copy of the initial individual. Let Z(t) be the population at time t, and let N=N(∞). Theorem: If a general branching process is critical, i. e E{N}=1, and if $\text{σ}{}^2\text{=E {N(N?1)}<∞, 0<a≡}\text{∫}\text{0}\text{∞}\text{tdE{N(t)}}$,and as t → ∞ both $\text{t}{}^2\text{(1?}\text{E}\text{{N(t)})→0}$ and $\text{t}{}^2\text{P}\text{[L>t]→0}$, then $\text{tP[Z(t)>0]→}\text{2a}\text{σ}{}^2$ as t→∞.     

Counting subsets of the random partition and the ‘Brownian Bridge’ process

Let Ω_m be the set of partitions, ω, of a finite m-element set; induce a uniform probability distribution on Ω_m, and define X_ms(ω) as the number of s-element subsets in ω. We alow the existence of an integer-valued function n=n^(m)(t), t?[0, 1], and centering constants b_ms, 0?s? m, such that

$\text{Z}{}^{\mathrm{(m)}}\text{(t)=}\text{∑}\text{s=0}\text{n(m)(t)}\text{(X}{}_{\mathrm{ms}}\text{?b}{}_{\mathrm{ms}}\text{)}\text{√}\text{∑}\text{s=0}\text{m}\text{b}{}_{\mathrm{ms}}$

converges to the ‘Brownian Bridge’ process in terms of its finite-dimensional distributions.     

A process of runs and its convergence to the brownian motion

Let X₁,X₂,… be i.i.d. random variables with a continuous distribution function. Let R₀=0, R_k=min{j>R_k?1, such that X_j>X_j+1}, k?1. We prove that all finite-dimensional distributions of a process $\text{W}{}^{\mathrm{(n)}}\text{(t)=}\text{(R}{}_{\mathrm{[nt]}}\text{?2[nt])}\text{2}\text{3}\text{n}\text{, t ? [0,1]}$, converge to those of the standard Brownian motion.     

Extreme value distribution for normalized sums from stationary Gaussian sequences

Let {X_i, i?0} be a sequence of independent identically distributed random variables with finite absolute third moment. Then Darling and Erdös have shown that

for -∞<t<∞ where and $\text{X}{}_{\mathrm{n}}\text{= (2 ln ln n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. The result is extended to dependent sequences but assuming that {X_i} is a standard stationary Gaussian sequence with covariance function {r_i}. When {X_i} is moderately dependent (e.g. when $\text{v(∑}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{X}{}_{\mathrm{i}}\text{) ? n}{}_{\mathrm{a}}\text{, 0 < a < 2)}$ we get

where H_a is a constant. In the strongly dependent case (e.g. when $\text{v(∑}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{X}{}_{\mathrm{i}}\text{) ? n}{}^2\text{r(n))}$ we get

for-∞<t<∞.     

Two strong limit theorems for processes with independent increments

Two related almost sure limit theorems are obtained in connection with a stochastic process {ξ(t), ?∞ < t < ∞} with independent increments. The first result deals with the existence of a simultaneous stabilizing function H(t) such that $\text{(ξ(t) ? ξ(0))}\text{H(t)}\text{→ 0}$ for almost all sample functions of the process. The second result deals with a wide-sense stationary process whose random spectral distributions is ξ. It addresses the question: Under what conditions does $\text{(2T)}{}^{\mathrm{?1}}\text{∫}{}_{\mathrm{?T}}{}^{\mathrm{T}}\text{X(t)}\text{X(t + τ)}\text{dt}$ converge as T → ∞ for all τ for almost all sample functions?     

Existence and boundedness of solutions of abstract nonlinear integrodifferential equations of nonconvolution type

Existence and boundedness theorems are given for solutions of nonlinear integrodifferential equations of type $\text{d}\text{dt}\text{u(t) + Bu(t) + ∝}{}_0{}^{\mathrm{t}}\text{a(t, s) Au(s) ds ? f(t) (t > 0)}$, (1.1) u(0) = u₀, Here A and B are nonlinear, possibly multivalued, operators on a Banach space W and a Hilbert space H, where W ? H. The function f (0, ∞) → H and the kernel a(t, s): $\text{R}$ × $\text{R}$ → $\text{R}$ are known functions. The results of this paper extend the results of Crandall, Londen, and Nohel [4] for equation (1.1). They assumed the kernel to be of the type a(t, s) = a(t ? s). We relax this assumption and obtain similar results. Examples of kernels satisfying the conditions we require are given in section 4.     

Three problems on the lengths of increasing runs

Let U₁, U₂,… be a sequence of independent, uniform (0, 1) r.v.'s and let R₁, R₂,… be the lengths of increasing runs of {U_i}, i.e., X₁=R₁=inf{i:U_i+1<U_i},…, X_n=R₁+R₂+?+R_n=inf{i:i>X_n?1,U_i+1<U_i}. The first theorem states that the sequence $\text{(}\text{3}\text{2}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(X}{}_{\mathrm{n}}\text{?2n)}$ can be approximated by a Wiener process in strong sense.Let τ(n) be the largest integer for which R₁+R₂+?+R_τ(n)?n, $\text{R}{}^1{}_{\mathrm{n}}\text{=n?(R}{}_1\text{+R}{}_2\text{+?+R}{}_{\mathrm{\tau (n)}}\text{)}$ and $\text{M}{}_{\mathrm{n}}\text{=}\text{max}\text{{R}{}_1\text{,R}{}_2\text{,…,R}{}_{\mathrm{\tau (n)}}\text{,R}{}^1{}_{\mathrm{n}}\text{}}$. Here M_n is the length of the longest increasing block. A strong theorem is given to characterize the limit behaviour of M_n.The limit distribution of the lengths of increasing runs is our third problem.     

Diffusion with interactions between two types of particles and Pressureless gas equations

We construct two d-dimensional independent diffusions $\text{X}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{=a+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{a}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{,}\text{X}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{=b+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{b}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{b}}$, with the same viscosity ν≠0 and the same drift u(x,t)=(pρ_t^a(x)v₁+(1?p)ρ_t^b(x)v₂)/(pρ_t^a(x)+(1?p)ρ_t^b(x)), where ρ_t^a,ρ_t^b are respectively the density of X_t^a and X_t^b. Here $\text{a,b,v}{}_1\text{,v}{}_2\text{∈}\text{R}{}^{\mathrm{d}}$ and p∈(0,1) are given. We show that $\text{(ρ}{}_{\mathrm{t}}\text{(x)=pρ}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{(x)+(1?p)ρ}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{(x),u(x,t):}\text{t?0,}\text{x∈}\text{R}{}^{\mathrm{d}}\text{)}$ is the unique weak solution of the following pressureless gas system

such that $\text{ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pδ}{}_{\mathrm{a}}\text{+(1?p)δ}{}_{\mathrm{b}}\text{,}\text{u(x,t)ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pv}{}_1\text{δ}{}_{\mathrm{a}}\text{+(1?p)v}{}_2\text{δ}{}_{\mathrm{b}}$ as t→0+. To cite this article: A. Dermoune, S. Filali, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

How small are the increments of a Wiener process?

Let $\text{γ}{}_{\mathrm{т}}\text{=(8(}\text{logTa}{}^{-1}{}_{\mathrm{T}}\text{+log log T)}\text{π}{}^2\text{a}{}_{\mathrm{T}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, 0<a_T?T<∞, and {W(t);0?t<∞} be a standard Wiener process. This exposition studies the almost sure behaviour of

, under varying conditions on a_T and T/a_T. The following analogue of Lévy's modulus of continuity of a Wiener Process is also given:

$\text{lim}{}_{\mathrm{h\to 0}}\text{inf}{}_{\mathrm{0?t?1}}\text{sup}{}_{\mathrm{0?s?h}}\text{(}\text{8 log h}{}^{-1}\text{π}{}^2\text{h}\text{)}\text{1}\text{2}\text{|W(t+s)?W(t)| =}{}^{\mathrm{a.s.}}\text{1.}$

and this may be viewed as the exact “modulus of non-differentiability” of a Wiener Process.     

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 $\text{(}\text{1}\text{2}\text{Δ}\text{+ q)φ = 0}\text{in}\text{D, φ = ?}\text{on}\text{?D}$.     

Boundedness of solutions of a system of integro-differential equations

Let b: [?1, 0] →$\text{R}$ be a nondecreasing, strictly convex C²-function with b(? 1) = 0, and let g: $\text{R}$ⁿ → $\text{R}$ⁿ be a locally Lipschitzian mapping, which is the gradient of a function G: $\text{R}$ⁿ → $\text{R}$. Consider the following vector-valued integro-differential equation of the Levin-Nohel type

$\text{x}\text{?}\text{(t)=?∝}{}_{\mathrm{?1}}{}^0\text{b(θ)g(x(t + θ))dθ}$

. (E) This equation is used in applications to model various viscoelastic phenomena. By LaSalle's invariance principle, every bounded solution x(t) goes to a connected set of zeros of g, as time t goes to infinity. It is the purpose of this paper to give several geometric criteria assuring the boundedness of solutions of (E) or some of its components.     

Uniqueness of the blow-up boundary solution of logistic equations with absorbtion

Let $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. Assume f∈C¹[0,∞) is a non-negative function such that f(u)/u is increasing on (0,∞). Let a be a real number and let b?0, $\text{b}\text{/≡0}$ be a continuous function such that b≡0 on $\text{?Ω}$. We study the logistic equation Δu+au=b(x)f(u) in $\text{Ω}$. The special feature of this work is the uniqueness of positive solutions blowing-up on $\text{?Ω}$, in a general setting that arises in probability theory. To cite this article: F.-C. C??rstea, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 447–452.     

A test for the independence of two Gaussian processes

A bivariate Gaussian process with mean 0 and covariance

$\text{Σ(s, t, p)=}\text{Σ}{}_{11}\text{(s, t)}\text{ρΣ}{}_{12}\text{(s, t)}\text{ρΣ}{}_{21}\text{(s, t)}\text{Σ}{}_{22}\text{(s, t)}$

is observed in some region Ω of R′, where {Σ_ij(s,t)} are given functions and p an unknown parameter. A test of H₀: p = 0, locally equivalent to the likelihood ratio test, is given for the case when Ω consists of p points. An unbiased estimate of p is given. The case where Ω has positive (but finite) Lebesgue measure is treated by spreading the p points evenly over Ω and letting p → ∞. Two distinct cases arise, depending on whether Δ_2,p, the sum of squares of the canonical correlations associated with Σ(s, t, 1) on $\text{Ω}{}^2$, remains bounded. In the case of primary interest as p → ∞, Δ_2,p → ∞, in which case $\text{p}\text{?}$ converges to p and the power of the one-sided and two-sided tests of H₀ tends to 1. (For example, this case occurs when Σ_ij(s, t) ≡ Σ₁₁(s, t).)     

The inverse balayage problem for Markov chains,part II

This paper continues the study of the inverse balayage problem for Markov chains. Let X be a Markov chain with state space $\text{A ? B}{}_2$, let v be a probability measure on B₂ and let M(v) consist of probability measures μ on A whose X-balayage onto B₂ is v. The faces of the compact, convex set M(v) are characterized. For fixed μ?M(v) the set M(μ,v) of the measures ? of the form $\text{?(·) = P}{}^{\mathrm{\mu }}\text{{X(S) ? ·}}$, where S is a randomized stopping time, is analyzed in detail. In particular, its extreme points and edge are explicitly identified. A naturally defined reversed chain X, for which v is an inverse balayage of μ, is introduced and the relation between X and X^ is studied. The question of which $\text{? ? M(μ, v)}$ admit a natural stopping time $\text{S}{}_{\mathrm{?}}$ of X (not involving an independent randomization) such that $\text{?(·) = P}{}^{\mathrm{\mu }}\text{{X(S}{}_{\mathrm{?}}\text{) ? ·}}$, is shown to have rather different answers in discrete and continuous time. Illustrative examples are presented.     

On a problem of P. Erdös

Let n be a positive integer and let A = {a₁,…, a_s}, B = {b₁,…, b_t} be two sets of positive integers such that the product set consists of st distinct numbers. Then, for a certain positive constant c, $\text{st ≤ c}\text{n}{}^2\text{log}\text{n}$, establishing a conjecture made by P. Erdös.