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.     

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

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

     

Branching brownian motion with absorption

We consider a branching diffusion {Z_t}_t?0 in which particles move during their life time according to a Brownian motion with drift -μ and variance coefficient σ², and in which each particle which enters the negative half line is instantaneously removed from the population. If particles die with probability c dt+o(dt) in [t,t+dt] and if the mean number of offspring per particle is m>1, then Z_t dies out w.p.l. if $\text{μ?μ}{}_0\text{≡{2σ}{}^2\text{c(m?1)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. If μ<μ₀, then itZ_t grows exponentially with positive probability. Our main concern here is with the critical case where μ=μ₀. Even though $\text{E}\text{{}\text{Z}{}_{\mathrm{T}}\text{}∽}\text{const.}\text{T}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>2</mtext>}}$ in this case, we find that $\text{P}\text{{}\text{Z}{}_{\mathrm{T}}\text{>0}}$ is only exp$\text{{–}\text{const.}\text{T}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{+0(}\text{log}\text{T)}{}^2\text{}}$, and conditionally on {Z_T>0} there are with high probability much fewer particles alive at time T than $\text{E}\text{{Z}{}_{\mathrm{T}}\text{|Z}{}_{\mathrm{T}}\text{0}}$.     

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 $\text{A}{}^2\text{(s) E |m(s) ?}\text{m}\text{?}\text{(s)|}{}^2\text{= P}{}_0$.     

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?     

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 → ∞.     

Cooperative birth processes with linear or sublinear intensity

Let {Z_t} be an increasing Markov process on N ⁿ and {σ(k)} the corresponding sequence of jump times. Let the increments of Z_t be i.i.d. with finite expectation and covariances, and let

$\text{E(σ(k+1)?σ(k)|Z}{}_0\text{,Z}{}_{\mathrm{\sigma (1)}}\text{,…Z}{}_{\mathrm{\sigma (k)}}\text{=}\text{h}\text{Z}{}_{\mathrm{\sigma (k)}}\text{|Z}{}_{\mathrm{\sigma (k)}}\text{|}\text{?(|Z}{}_{\mathrm{\sigma (k)}}\text{|)}{}^{\mathrm{?1}}$

where h and ? are sufficiently smooth positive functions and ?Z_t? = ∑ⁿ_j=1Z_t(j), Z_t=(Z_t(1),…,Z_t(n)). While a linear f results in asymptotically exponential growth, a suitable class of sublinear f leads to a growth asymptotically at most that of a power. Covering both cases, we obtain analoga of the strong LLN, the CLT and LIL.     

Abstract equipartition of energy theorems

Let H be a self-adjoint operator on a complex Hilbert space $\text{H}$. The solution of the abstract Schrödinger equation $\text{idu}\text{dt}\text{= Hu}$ is given by u(t) = exp(?itH)u(0). The energy E = ∥u(t)∥² is independent of t. When does the energy break up into different kinds of energy E = ∑_{j = 1}^NE_j(t) which become asymptotically equipartitioned ? (That is, $\text{E}{}_{\mathrm{j}}\text{(t) →}\text{E}\text{N}\text{as}\text{t → ± ∞}$ for all j and all data u(0).) The “classical” case is the abstract wave equation $\text{d}{}^2\text{v}\text{dt}{}^2\text{+ A}{}^2\text{v = 0}\text{with}\text{A}$ self-adjoint on $\text{H}$₁. This becomes a Schrödinger equation in a Hilbert space $\text{H}$ (essentially $\text{H}$ is two copies of $\text{H}$₁), and there are two kinds of associated energy, viz., kinetic and potential. Two kinds of results are obtained. (1) Equipartition of energy is related to the C¹-algebra approach to quantum field theory and statistical mechanics. (2) Let A₁,…, A_N be commuting self-adjoint operators with N = 2 or 4. Then the equation $\text{Π}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{N}}\text{(}\text{d}\text{dt}\text{? iA}{}_{\mathrm{j}}\text{) u(t) = 0}$ admits equipartition of energy if and only if exp(it(A_j ? A_k)) → 0 in the weak operator topology as t → ± ∞ for j ≠ k.     

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.     

On the sequence of Bayesian estimates of normal random walk process

Let (μ_t)^∞_t=0 be a k-variate (k?1) normal random walk process with successive increments being independently distributed as normal N(δ, R), and μ₀ being distributed as normal N(0, V₀). Let X_t have normal distribution N(μ_t, Σ) when μ_t is given, t = 1, 2,….Then the conditional distribution of μ_t given X₁, X₂,…, X_t is shown to be normal N(U_t, V_t) where U_t's and V_t's satisfy some recursive relations. It is found that there exists a positive definite matrix V and a constant θ, 0 < θ < 1, such that, for all t?1,

$\text{|R}\text{1}\text{2}\text{(V}{}^{\mathrm{?1}}{}_{\mathrm{t}}\text{?V}{}^{\mathrm{?1}}\text{R}\text{1}\text{2}\text{|<θ}{}^{\mathrm{t}}\text{|R}\text{1}\text{2}\text{(V}{}^{\mathrm{?1}}{}_0\text{?V}{}^{\mathrm{?1}}\text{)R}\text{1}\text{2}\text{|}$

where the norm |·| means that |A| is the largest eigenvalue of a positive definite matrix A. Thus, V_t approaches to V as t approaches to infinity. Under the quadratic loss, the Bayesian estimate of μ_t is U_t and the process {U_t}^∞_t=₀, U₀=0, is proved to have independent successive increments with normal N(θ, V_t?V_t+1+R) distribution. In particular, when V₀ =V then V_t = V for all t and {U_t}^∞_t=0 is the same as {μ_t}^∞_t=0 except that U₀ = 0 and μ₀ is random.     

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).     

On Guerra's broken replica-symmetry bound

Consider a random Hamiltonian $\text{H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{)}$ for $\text{σ}\text{→}\text{∈Σ}{}_{\mathrm{N}}\text{={0,1}}{}^{\mathrm{<mtext>N</mtext>}}\text{.}$ We assume that the family $\text{(H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{))}$ is jointly Gaussian centered and that for $\text{σ}\text{→}{}^1\text{,}\text{σ}\text{→}{}^2\text{∈Σ}{}_{\mathrm{N}}\text{,}$$\text{N}{}^{\mathrm{?1}}\text{EH}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}{}^1\text{)H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}{}^2\text{)}$ =ξ(N^?1∑_i?Nσ¹_iσ²_i) for a certain function ξ on $\text{R}$. F. Guerra proved the remarkable fact that the free energy of the system with Hamiltonian $\text{H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{)+h∑}{}_{\mathrm{i?N}}\text{σ}{}_{\mathrm{i}}$ is bounded below by the free energy of the Parisi solution provided that ξ is convex on $\text{R}$. We prove that this fact remains (asymptotically) true when the function ξ is only assumed to be convex on $\text{R}{}^{\mathrm{+}}$. This covers in particular the case of the p-spin interaction model for any p. To cite this article: M. Talagrand, 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.     

Sublinear perturbations of the differential equation y(n) = 0 and of the analogous difference equation

According to a result of A. Ghizzetti, for any solution y(t) of the differential equation where $\text{y}{}^{\mathrm{(n)}}\text{(t)+}\text{∑}\text{i=0}\text{n?1}\text{g}{}_{\mathrm{i}}\text{(t) y}{}^{\mathrm{i}}\text{(t)=0}\text{(t ? 1)}$, $\text{∝}{}_1{}^{\mathrm{\infty }}\text{¦g}{}_{\mathrm{i}}\text{(x)¦x}{}^{\mathrm{n?I?1}}\text{dx < ∞}$ (0 ?i ? n ?1, either y(t) = 0 for t ? 1 or there is an integer r with 0 ? r ? n ? 1 such that $\text{lim}\text{t → ∞}\text{y(t)}\text{t}{}^{\mathrm{r}}$ exists and ≠0. Related results are obtained for difference and differential inequalities. A special case of the former has interesting applications in the study of orthogonal polynomials.     

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.     

On exponential bases for the Sobolev spaces over an interval

We suppose that K is a countable index set and that $\text{Λ = {λ}{}_{\mathrm{k}}\text{¦ k ? K}}$ is a sequence of distinct complex numbers such that forms a Riesz (strong) basis for L²[a, b], a < b. Let Σ = {σ₁, σ₂,…, σ_m} consist of m complex numbers not in Λ. Then, with p(λ) = Π_{k = 1}^m (λ ? σ_k), forms a Riesz (strong) bas Sobolev space H^m[a, b]. If we take σ₁, σ₂,…, σ_m to be complex numbers already in Λ, then, defining p(λ) as before, forms a Riesz (strong) basis for the space H^?m[a, b]. We also discuss the extension of these results to “generalized exponentials” tⁿe^λ_kt.