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

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

     

A functional central limit theorem for ϱ-mixing sequences

In this note a functional central limit theorem for ?-mixing sequences of I. A. Ibragimov (Theory Probab. Appl.20 (1975), 135–141) is generalized to nonstationary sequences (X_n)_n ∈ $\text{N}$, satisfying some assumptions on the variances and the moment condition $\text{E |X}{}_{\mathrm{n}}\text{|}{}^{\mathrm{2\; +\; b}}\text{= O(n}{}^{\mathrm{<mtext>b</mtext><mtext>2</mtext><mtext>??</mtext>}}\text{)}$ for some b > 0, ? > 0.     

On the Smale theorem in weak shape theory

We modify various lemmas from Dydak's paper on the Vietoris-Smale theorem to obtain more general results. We consider closed subsets X and Y of paracompact spaces M, N respectively, and a map F:M→N such that F∥X:X→Y is closed and surjective and X=F^-1(Y) to obtain the following result.(Theorem). If $\text{F}{}^{-1}\text{(y)ϵAC}{}^{\mathrm{n}}{}_{\mathrm{M}}\text{(}\text{K}\text{)}$ for each y ϵ Y and N is LCⁿ⁺¹, then the morphism in-pro-$\text{π}{}_{\mathrm{k}}\text{[F]:π}{}_{\mathrm{k}}\text{U}{}^{\mathrm{<mtext>K</mtext>}}{}_{\mathrm{M}}\text{(X,x)→π}{}_{\mathrm{k}}\text{U}{}^{\mathrm{<mtext>K</mtext>}}{}_{\mathrm{N}}\text{(Y,y)}$ induced by the morphism $\text{[F]:U}{}^{\mathrm{<mtext>K</mtext>}}{}_{\mathrm{M}}\text{(X,x)→U}{}^{\mathrm{<mtext>K</mtext>}}{}_{\mathrm{N}}\text{(Y,y)}$ is an isomorphism of in-pro-Grp for k⩽n and an epimorphism for k=n+1.     

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 $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$.     

Zeta matrices of elliptic curves

Let $\text{O =}\text{lim}{}_{\mathrm{n}}\text{Z/p}{}^{\mathrm{n}}\text{Z}$, let $\text{A}\text{= O[g}{}_2\text{, g}{}_3\text{]}{}_{\mathrm{\Delta }}$, where g₂ and g₃ are coefficients of the elliptic curve: Y² = 4X³ ? g₂X ? g₃ over a finite field and Δ = g₂³ ? 27g₃² and let $\text{B}\text{=}\text{A}\text{[X, Y]}\text{(Y}{}^2\text{? 4X}{}^3\text{+ g}{}_2\text{X + g}{}_3\text{)}$. Then the p-adic cohomology theory will be applied to compute explicitly the zeta matrices of the elliptic curves, induced by the pth power map on the free $\text{A}{}^2\text{?}{}_{\mathrm{Z}}\text{Q}$-module $\text{H}{}^1\text{(X,}\text{A}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$. Main results are; Theorem 1.1: X²dY and YdX are basis elements for $\text{H}{}^1\text{(}\text{X}\text{, Γ}{}_{\mathrm{<mtext>A</mtext>}}{}^1\text{(}\text{X}\text{)}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$; Theorem 1.2: YdX, X²dY, Y^?1dX, Y^?2dX and XY^?2dX are basis elements for $\text{H}{}^1\text{(}\text{X}\text{? (Y = 0), Γ}{}_{\mathrm{<mtext>A</mtext>}}{}^1\text{(}\text{X}\text{)}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$, where $\text{X}$ is a lifting of X, and all the necessary recursive formulas for this explicit computation are given.     

Uncertainty principle and Lp–Lq-sufficient pairs on noncompact real symmetric spaces

We consider a real semi-simple Lie group G with finite center and a maximal compact sub-group K of G. Let $\text{G=K}\text{exp}\text{(}\text{a}{}_{\mathrm{+}}\text{)K}$ be a Cartan decomposition of G. For x∈G denote ∥x∥ the norm of the $\text{a}{}_{\mathrm{+}}$-component of x in the Cartan decomposition of G. Let $\text{a>0,}\text{b>0}$ and 1?p,q?∞. In this Note we give necessary and sufficient conditions on $\text{a,}\text{b}$ such that for all K-bi-invariant measurable function f on G, if e^a∥x∥2f∈L^p(G) and then f=0 almost everywhere. To cite this article: S. Ben Farah, K. Mokni, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the probability that given polynomials have a specified highest common factor

The polynomial functions f₁, f₂,…, f_m are found to have highest common factor h for a set of values of the variables x₁, x₂,…,x_m whose asymptotic density is

$\text{1}\text{h}{}^{\mathrm{n}}\text{∑}\text{d∣h}\text{μ(d)Π}{}^{\mathrm{m}}{}_{\mathrm{l\; =\; 1}}\text{?(f}{}_1\text{, dh)}\text{d}{}^{\mathrm{m}}\text{Π}\text{p∣h}\text{1?}\text{Π}{}^{\mathrm{m}}{}_{\mathrm{l\; =\; 1}}\text{?(f}{}_1\text{, p)}\text{p}{}^{\mathrm{m}}$

For the special case f₁(x) = f₂(x) = … = f_m(x) = x and h = 1 the above formula reduces to $\text{Π}{}_{\mathrm{?}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{m}}\text{) =}\text{1}\text{ζ}\text{(m)}$, the density if m-tuples with highest common factor 1. Necessary and sufficient conditions on the polynomials f₁, f₂,…, f_m for the asymptotic density to be zero are found. In particular it is shown that either the polynomials may never have highest common factor h or else h is the highest common factor infinitely often and in fact with positive density.     

Some special vapnik-chervonenkis classes

For a class $\text{C}$ of subsets of a set X, let V($\text{C}$) be the smallest n such that no n-element set F?X has all its subsets of the form A ∩ F, A ∈ $\text{C}$. The condition V($\text{C}$) <+∞ has probabilistic implications. If any two-element subset A of X satisfies both A ∩ C = Ø and A ? D for some C, D∈$\text{C}$, then $\text{V(}\text{C}\text{)=2}$ if and only if $\text{C}$ is linearly ordered by inclusion. If $\text{C}$ is of the form $\text{C}\text{={∩}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{C}{}_{\mathrm{i}}\text{:C}{}_{\mathrm{i}}\text{∈}\text{C}{}_{\mathrm{i}}$, i=1,2,…,n}, where each $\text{C}{}_{\mathrm{i}}$ is linearly ordered by inclusion, then $\text{V(}\text{C}\text{)?n+1}$. If H is an (n-1)-dimensional affine hyperplane in an n-dimensional vector space of real functions on X, and $\text{C}$ is the collection of all sets {x: f(x)>0} for f in H, then $\text{V(}\text{C}\text{)=n}$.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

Higher analogues of Stickelberger's theorem

Let l be an odd prime number, F denote any totally real number field and E/F be an Abelian CM extension of F of conductor f. In this paper we prove that for every n odd and almost all prime numbers l we have where S_n(E/F,l) is the Stickelberger ideal (Ann. of Math. 135 (1992) 325–360; J. Coates, p-adic L-functions and Iwasawa's theory, in: Algebraic Number Fields by A. Fröhlich, Academic Press, London, 1977). In addition if we assume the Quillen–Lichtenbaum conjecture then To cite this article: G. Banaszak, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

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<∞.     

A transfer spectral sequence for fixed point free involutions with an application to stunted real projective spaces

We show that if X is a finite CW-complex admitting a fixed point free involution then there is a singly graded spectral sequence with $\text{E}{}^1{}_1\text{? H}{}^1\text{(X;}\text{Z}{}_2\text{)}$ and $\text{E}{}^1\text{∞ = 0}$. As an application we prove that for any n > 0 there is a natural number k(n) such that if n > k(n) and X is a homotopy $\text{R}\text{P}{}^{\mathrm{n+k}}\text{R}\text{P}{}^{\mathrm{n}}$, then X will not admit a fixed point free involution.