Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space

Let {X_n} be a ?-irreducible Markov chain on an arbitrary space. Sufficient conditions are given under which the chain is ergodic or recurrent. These extend known results for chains on a countable state space. In particular, it is shown that if the space is a normed topological space, then under some continuity conditions on the transition probabilities of {X_n} the conditions for ergodicity will be met if there is a compact set K and an ? > 0 such that $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 — 6X}{}_{\mathrm{n}}\text{6 ∣ X}{}_{\mathrm{n}}\text{= x} ? ??}$ whenever x lies outside K and $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 ∣ X}{}_{\mathrm{n}}\text{=x}}$ is bounded, x ∈ K; whilst the conditions for recurrence will be met if there exists a compact K with $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 ? 6X}{}_{\mathrm{n}}\text{6 ∣ X}{}_{\mathrm{n}}\text{= x} ? 0}$ for all x outside K. An application to queueing theory is given.     

Qualitative behavior of special functions on compact symmetric spaces

For a symmetric space $\text{G}\text{K}$ of compact type, the highest-weight vectors for representations of G occurring in $\text{L}{}^2\text{(G}\text{K)}$ become heavily concentrated near certain submanifolds of $\text{G}\text{K}$ as the highest weight goes to infinity. This fact is applied to obtain estimates for the spectral measures of the operators qλ = P_λqP_λ, where $\text{P}{}_{\mathrm{\lambda }}\text{: L}{}^2\text{(}\text{G}\text{K}\text{) → V}{}_{\mathrm{\lambda }}$ is an orthogonal projection onto a G-irreducible summand, and q: G/K → $\text{R}$ is a continuous function acting on $\text{L}{}^2\text{(G}\text{K)}$ by multiplication.     

Sturm—Liouville problems with several parameters

We consider the regular linear Sturm-Liouville problem (second-order linear ordinary differential equation with boundary conditions at two points x = 0 and x = 1, those conditions being separated and homogeneous) with several real parameters λ₁,…,λ_N. Solutions to this problem correspond to eigenvaluesλ = (λ₁,…,λ_N) forming sets $\text{R}$^N determined by the number of zeroes in (0, 1) of solutions. We describe properties of these sets including: boundedness, and when unbounded, asymptotic directions. Using these properties some results are given for the system of N Sturm-Liouville problems which share only the parameters λ. Sharp results are given for the system of two problems sharing two parameters. The eigensurfaces for a single problem are closely related to the cone $\text{K={λ R}{}^{\mathrm{N}}\text{:λ}{}_1\text{a}{}_1\text{(x)+…+λ}{}_{\mathrm{N}}\text{a}{}_{\mathrm{N}}\text{(x)?0 for all x in [0,1]}}$, particularly in questions of boundedness. The cone K and related objects are discussed, and a result is given which relates cones with two oscillation conditions known as “Right-Definiteness” and “Left-Definiteness.”     

The asymptotic equivalence of Bayes and maximum likelihood estimation

Let (X, $\text{A}$) be a measurable space, Θ ? $\text{R}$ an open interval and P_Ω ∥ $\text{A}$, Ω ? Θ, a family of probability measures fulfilling certain regularity conditions. Let $\text{Ω}{}_{\mathrm{n}}$ be the maximum likelihood estimate for the sample size n. Let λ be a prior distribution on Θ and let $\text{R}{}_{\mathrm{<mtext>n,</mtext><mtext>x</mtext>}}$ be the posterior distribution for the sample size n given $\text{x}\text{? X}{}^{\mathrm{n}}$. $\text{L:}\text{Θ}\text{× Θ →}\text{R}$ denotes a loss function fulfilling certain regularity conditions and T_n denotes the Bayes estimate relative to λ and L for the sample size n. It is proved that for every compact K ? Θ there exists c_K ≥ 0 such that

This theorem improves results of Bickel and Yahav [3], and Ibragimov and Has'minskii [4], as far as the speed of convergence is concerned.     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

The minimal generating space of a polynomial

Given a polynomial $\text{P(X}{}_1\text{,…,X}{}_{\mathrm{N}}\text{)∈}\text{R}\text{[}\text{X}\text{]}$, we calculate a subspace G_p of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property $\text{P∈}\text{R}\text{[}\text{G}{}_{\mathrm{p}}\text{]}$ (the algebra generated by G_p, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X₁,…,X_n) and Q(X₁,…,X_n) for which there exists an isomorphism T:〈X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as $\text{P}{}^1\text{(Y}{}_1\text{,…,Y}{}_{\mathrm{k}}\text{)}$ and $\text{Q}{}^1\text{(Y}{}_{\mathrm{<mtext>k+</mtext><mtext>1</mtext>}}\text{,…,Y}{}_{\mathrm{n}}\text{)}$ respectively, where $\text{Y}\text{=T}\text{X}$.     

On the invariants of some Zi-extensions

Let k ? k₁ ? … ? K be a Z_i-extension. The relations of $\text{λ(}\text{K}\text{k}\text{)}$ and $\text{λ(}\text{KF}\text{F}\text{)}$ is studied, where $\text{F}\text{k}$ is a cyclic l-extension. If $\text{M}\text{k}$ is another Z_i-extension of k, it is shown that for i ? 0$\text{λ(}\text{Mk}{}_{\mathrm{i}}\text{k}{}_{\mathrm{i}}\text{) = rl}{}^{\mathrm{i}}\text{+ C}$, under minimal additional hypotheses. Finally if $\text{MK}\text{k}$ has a unique totally ramified prime, and X_K is cyclic, it is shown that MK can contain at most one Z_i-extension with non-zero μ invariant.     

On some results of Strichartz and of Rallis and Schiffman

Let (i, H, E) and (j, K, F) be abstract Wiener spaces and let α be a reasonable norm on E ? F. We are interested in the following problem: is ($\text{i ? j, H}\text{}\text{?}\text{bo}{}_2\text{K, E}\text{}\text{?}\text{bo}{}_{\mathrm{\alpha F}}$) an abstract Wiener space ? The first thing we do is to prove that the setting of the problem is meaningfull: namely, i ? j is always a continuous one to one map from $\text{H}\text{}\text{?}\text{bo}{}_2\text{K}$ into $\text{E}\text{}\text{?}\text{bo}{}_{\mathrm{\alpha }}\text{F}$. Then we exhibit an example which shows that the answer cannot be positive in full generality. Finally we prove that if F=L^p(X,$\text{X}$,λ) for some σ-finite measure λ ? 0 then $\text{(i?j, H}\text{?}{}_2\text{K,L}{}^{\mathrm{p}}$(X,$\text{X}$,λ) is an abstract Wiener space. By-products are some new results on γ-radonifying operators, and new examples of Banach spaces and cross norms for which the answer is affirmative (in particular α = π the projective norm, and F=L¹(X,$\text{X}$,λ)).     

On the law of large numbers for stationary sequences

We show that if X_i is a stationary sequence for which $\text{S}{}_{\mathrm{n}}\text{B}{}_{\mathrm{n}}$ converges to a finite non zero random variable of constant sign, where S_n=X₁+X₂+?+X_n and B_n is a sequence of constants, then B_n is regularly varying with index 1. If in addition $\text{Σ}\text{P}\text{(|X}{}_1\text{|>B}{}_{\mathrm{n}}$ is finite, then $\text{E}\text{|X}{}_1\text{|}$ is finite, and if in addition to this X_i satisfies an asymptotic independence condition, $\text{E}\text{X}{}_1\text{≠ 0}$.     

Limit distributions for the maxima of stationary Gaussian processes

Let {X_n} be a stationary Gaussian sequence with E{X₀} = 0, {X²₀} = 1 and E{X₀X_n} = rⁿ_n Let c_n = (2ln n)^{$\text{built1}\text{2}$}, b_n = c_n? $\text{1}\text{2}$c^-1_n ln(4π ln n), and set M_n = max_{0 ?k?n}X_k. A classical result for independent normal random variables is that

$\text{P[c}{}_{\mathrm{n}}\text{(M}{}_{\mathrm{n}}\text{?b}{}_{\mathrm{n}}\text{)?x]→exp[-e}{}^{\mathrm{-x}}\text{] as n → ∞ for all x.}$

Berman has shown that (1) applies as well to dependent sequences provided r_nlnn = o(1). Suppose now that {r_n} is a convex correlation sequence satisfying r_n = o(1), (r_nlnn)^-1 is monotone for large n and o(1). Then

$\text{P[r}{}_{\mathrm{n}}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{(M}{}_{\mathrm{n}}\text{? (1?r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{b}{}_{\mathrm{n}}\text{)?x] → Ф(x)}$

for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {M_n}, there are others. In particular, the limit distribution is given below when r_n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r_n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).     

The construction of stationary two-dimensional Markoff fields with an application to quantum field theory

It is shown that if φ(f)  ∝_R^dφ(y) f(y) dy is a Markoff random field and X_α are multiplicative functionals of φ (with E(X_α) = 1) which converge locally in L₁, then there exists a locally Markoff random field $\text{φ}{}_1$ such that $\text{E(exp(iφ}{}_1\text{(f))) =}\text{lim}{}_{\mathrm{\alpha }}\text{E(X}{}_{\mathrm{<mtext>\alpha </mtext>}}\text{exp}\text{(iφ(φ)))}$. We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take X_α proportional to exp(?λ∝_R² : P(φ(y)) : g_α(y) dy), where: P(φ(y)) : is a regularized even degree polynomial in φ(y). It is then proved that for an appropriate choice of g_α → 1 and small λ, {X_α} does converge locally in L₁ and that the corresponding $\text{φ}{}_1$ is stationary.     

Cohérence différentielle des F-isocristaux unités

Let $\text{V}$ be a mixed characteristic complete discrete valuation ring, $\text{P}$ a smooth formal scheme over $\text{V}$, P its special fiber, X a smooth subscheme of P, T a divisor in P such that T_X=T∩X is a divisor in X and $\text{D}{}^2{}_{\mathrm{<mtext>P</mtext>}}$ the weak completion of the sheaf of differential operators on $\text{P}$. We prove that the unit-root F-isocrystals on X?T_X overconvergent along T_X are coherent over $\text{D}{}^2{}_{\mathrm{<mtext>P</mtext><mtext>,</mtext><mtext>Q</mtext>}}$. To cite this article: D. Caro, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

The Fréchet distance between multivariate normal distributions

The Fréchet distance between two multivariate normal distributions having means μ_X, μ_Y and covariance matrices Σ_X, Σ_Y is shown to be given by $\text{d}{}^2\text{= |μ}{}_{\mathrm{X}}\text{? μ}{}_{\mathrm{Y}}\text{|}{}^2\text{+}\text{tr}\text{(Σ}{}_{\mathrm{X}}\text{+ Σ}{}_{\mathrm{Y}}\text{? 2(Σ}{}_{\mathrm{X}}\text{Σ}{}_{\mathrm{Y}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$. The quantity d₀ given by $\text{d}{}_0{}^2\text{=}\text{tr}\text{(Σ}{}_{\mathrm{X}}\text{+ Σ}{}_{\mathrm{Y}}\text{? 2(Σ}{}_{\mathrm{X}}\text{Σ}{}_{\mathrm{Y}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ is a natural metric on the space of real covariance matrices of given order.     

A periodicity lemma in linear diophantine analysis

Let g = (g₁,…,g_r) ≥ 0 and h = (h₁,…,h_r) ≥ 0, g_?, h_? ∈ $\text{J}$, be two vectors of nonnegative integers and let λ ? $\text{J}$, λ ≥ 0, λ ≡ 0 mod d, where d denotes g.c.d. (g₁,…,g_r). Define

$\text{Δ(λ)=Δ(λg,h):=}\text{min}\text{∑}\text{?=1}\text{r}\text{x}{}_{\mathrm{?}}\text{h}{}_{\mathrm{?}}\text{:x}{}_{\mathrm{?}}\text{?0,x}{}_{\mathrm{?}}\text{∈J,}\text{∑}\text{?=1}\text{?}\text{x}{}_{\mathrm{?}}\text{g}{}_{\mathrm{?}}\text{=λ}$

It is shown in this paper that Λ(λ) is periodic in λ with constant jump. If i? {1,…,r} is such that

$\text{det}\text{g}{}_{\mathrm{i}}\text{h}{}_{\mathrm{i}}\text{g}{}_{\mathrm{?}}\text{h}{}_{\mathrm{?}}\text{? (?1,…r)}$

then

$\text{Δ(λ)+g}{}_{\mathrm{i}}\text{Δ(λ)+h}{}_{\mathrm{i}}$

holds true for all sufficiently large λ, λ ≡ 0 mod d.     

Corresponding residue systems in nonnormal extensions of prime degree

Let K₁ and K₂ be number fields and $\text{F = K}{}_1\text{? K}{}_2$. Suppose $\text{K}{}_1\text{F}$ and $\text{K}{}_2\text{F}$ are of prime degree p but are not necessarily normal. Let N₁ and N₂ be the normal closures of K₁ and K₂ over F, respectively, L = K₁K₂, N = N₁N₂, and $\text{B}$ be a prime divisor of N which divides p and is totally ramified in $\text{K}{}_1\text{F}$ and $\text{K}{}_2\text{F}$. Let ${}_{\mathrm{<mtext>N</mtext><mtext>L</mtext>}}$ be the ramification index of $\text{B}$ in $\text{N}\text{L}$, $\text{t}{}_{\mathrm{<mtext>L</mtext><mtext>F</mtext>}}$ be the total ramification number of $\text{B}$ in $\text{L}\text{F}$, and . Then $\text{M}$(K₁, K₂) is exactly divisible by $\text{B}$^M, where .     

On moment conditions for normed sums of independent variables and martingale differences

Let X₁, X₂,… be a sequence of i.i.d. random variables and S_n their partial sums. Necessary and sufficient conditions are given for $\text{{n}{}^{\mathrm{<mtext>?1</mtext><mtext>q</mtext>}}\text{S}{}_{\mathrm{n}}\text{}}{}^{\mathrm{\propto }}{}_1$ to have uniformly bounded pth moments, 0<p<q?2.Some of the results are generalized to martingle differences.     

On the simulation of shot noise and some other random variables

It is shown that the random voltage V_t resulting from pulses with independent random amplitude Y_i Poisson arrivals, and exponential decay, can be asymptotically represented, in the stationary case, by the following random variable; namely a sum of products of random variables:

$\text{W=}\text{∑}\text{i=1}\text{∞}\text{U}{}_{\mathrm{i}}\text{Y}{}_{\mathrm{i}}\text{,}$

where

$\text{U}{}_{\mathrm{i}}\text{=}\text{∏}\text{j=1}\text{i}\text{X}{}_{\mathrm{j}}{}^{\mathrm{\beta /\lambda }}\text{.}$

Here X_j are independent uniform random variables, β>0 is the decay parameter, λ>0 is the rate of the Poisson process.     

On Berry-Esséen rates,a law of the iterated logarithm and an invariance principle for the proportion of the sample below the sample mean

Let F_n(x) be the empirical distribution function based on n independent random variables X₁,…,X_n from a common distribution function F(x), and let $\text{X}\text{= Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{X}{}_{\mathrm{i}}\text{n}$ be the sample mean. We derive the rate of convergence of $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$ to normality (for the regular as well as nonregular cases), a law of iterated logarithm, and an invariance principle for $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$.     

L-spaces and S-spaces in P(ω)

We show that CH implies that $\text{P}\text{(ω)}$, when equipped with the Vietoris topology, has a subspace which is an L-space and a subspace which is an S-space. This is an immediate consequence of the following purely combinatorial result: CH implies the existence of an ω₁-sequence 〈x_α: α < ω₁〉 in $\text{P}\text{(ω)}$ such that (1) if α<β<ω₁, then $\text{X}{}_{\mathrm{\beta }}\text{?}{}^1\text{X}{}_{\mathrm{\alpha }}$; (2) if I ?ω₁ is unaccountable, then there are distinct α, β ∈ I with X_β ?X_α.     

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.