On diffusivity of a tagged particle in asymmetric zero-range dynamics

Consider a distinguished, or tagged particle in zero-range dynamics on Zd with rate g whose finite-range jump probabilities p possess a drift ∑jp(j)≠0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d?1, and at most order t in d=1 and d?3 for a wide class of rates g. Also, in d=1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g(k) increases, and g(k)/k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given.     

Annealed deviations of random walk in random scenery

Let (Zn)_n∈N be a d-dimensional random walk in random scenery, i.e., with (Sk)_k∈N0 a random walk in Zd and (Y(z))_z∈Zd an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay of for various choices of sequences n(bn) in [1,∞). Depending on n(bn) and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work [A. Asselah, F. Castell, Large deviations for Brownian motion in a random scenery, Probab. Theory Related Fields 126 (2003) 497-527] by A. Asselah and F. Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen [X. Chen, Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab. 32 (4) 2004].     

A note on random walk in random scenery

We consider a random walk in random scenery {Xn=η(S₀)+?+η(Sn),n∈N}, where a centered walk {Sn,n∈N} is independent of the scenery {η(x),x∈Zd}, consisting of symmetric i.i.d. with tail distribution P(η(x)>t)∼exp(−cαtα), with 1?α<d/2. We study the probability, when averaged over both randomness, that {Xn>ny} for y>0, and n large. In this note, we show that the large deviation estimate is of order exp(−ca(ny)), with a=α/(α+1).     

Central limit theorems for a supercritical branching process in a random environment

For a supercritical branching process (Z_n) in a stationary and ergodic environment ξ, we study the rate of convergence of the normalized population W_n=Z_n/E[Z_n|ξ] to its limit W_∞: we show a central limit theorem for W_∞−W_n with suitable normalization and derive a Berry-Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distributed. Similar results are also shown for W_n+k−W_n for each fixed k∈N^∗.     

Moments, moderate and large deviations for a branching process in a random environment

Let (Z_n) be a supercritical branching process in a random environment ξ, and W be the limit of the normalized population size Z_n/E[Z_n|ξ]. We show large and moderate deviation principles for the sequence logZ_n (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of W, and show an equivalence for all the moments of Z_n. Central limit theorems on W−W_n and logZ_n are also established.     

On the moments of some first passage times for sums of dependent random variables

Let S_n,n = 1, 2, …, denote the partial sums of integrable random variables. No assumptions about independence are made. Conditions for the finiteness of the moments of the first passage times N(c) = min {n: S_n>ca(n)}, where c ≥ 0and a(y) is a positive continuous function on [0, ∞), such that a(y) = o(y)as y → ∞, are given. With the further assumption that a(y) = yP,0 ≤ p < 1, a law of large numbers and the asymptotic behaviour of the moments when c → ∞ are obtained. The corresponding stopped sums are also studied.     

Split orders and convex polytopes in buildings

As part of his work to develop an explicit trace formula for Hecke operators on congruence subgroups of SL₂(Z), Hijikata (1974) [13] defines and characterizes the notion of a split order in M₂(k), where k is a local field. In this paper, we generalize the notion of a split order to Mn(k) for n>2 and give a natural geometric characterization in terms of the affine building for SLn(k). In particular, we show that there is a one-to-one correspondence between split orders in Mn(k) and a collection of convex polytopes in apartments of the building such that the split order is the intersection of all the maximal orders representing the vertices in the polytope. This generalizes the geometric interpretation in the n=2 case in which split orders correspond to geodesics in the tree for SL₂(k) with the split order given as the intersection of the endpoints of the geodesic.     

On globally generated vector bundles on projective spaces

A classification is given for globally generated vector bundles E of rank k on Pⁿ having first Chern class c₁(E)=2. In particular, we get that they split if k<n unless E is a twisted null-correlation bundle on P³. In view of the well-known correspondence between globally generated vector bundles and maps to Grassmannians, we obtain, as a corollary, a classification of double Veronese embeddings of Pⁿ into a Grassmannian G(k−1,N) of (k−1)-planes in P^N.     

Eigenvalue Distributions and Geometry of Banach Spaces

We consider the sums Z⁽ⁿ⁾ of i.i.d. random vectors taking values in a d-dimensional Euclidean space. It is assumed that the so-called CRAMER condition holds in a neighbourhood of the origin. We establish lower bounds for the large deviation probabilities P(Z⁽ⁿ⁾ ? A) with A belonging to a large class of sets.     

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).     

Lower large deviations for supercritical branching processes in random environment

Branching processes in random environment (Z _n : n ≥ 0) are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical regime, the process survives with a positive probability and grows exponentially on the non-extinction event. We focus on rare events when the process takes positive values but lower than expected. More precisely, we are interested in the lower large deviations of Z, which means the asymptotic behavior of the probability {1 ≤ Z _n ≤ exp(nθ)} as n → ∞. We provide an expression for the rate of decrease of this probability under some moment assumptions, which yields the rate function. With this result we generalize the lower large deviation theorem of Bansaye and Berestycki (2009) by considering processes where ?(Z ₁ = 0 | Z ₀ = 1) > 0 and also much weaker moment assumptions.     

Limit theory for random coefficient first-order autoregressive process under martingale difference error sequence

Phillips and Magdalinos (2007) [1] gave the asymptotic theory for autoregressive time series with a root of the form ρ_n=1+c/k_n, where k_n is a deterministic sequence. In this paper, an extension to the more general case where the coefficients of an AR(1) model is a random variable and the error sequence is a sequence of martingale differences is discussed. A conditional least squares estimator of the autoregressive coefficient is derived and shown to be asymptotically normal. This extends the result of Phillips and Magdalinos (2007) [1] for stationary and near-stationary cases.     

Submodule categories of wild representation type

Let Λ be a commutative local uniserial ring with radical factor field k. We consider the category S(Λ) of embeddings of all possible submodules of finitely generated Λ-modules. In case Λ=Z/〈pⁿ〉, where p is a prime, the problem of classifying the objects in S(Λ), up to isomorphism, has been posed by Garrett Birkhoff in 1934. In this paper we assume that Λ has Loewy length at least seven. We show that S(Λ) is controlled k-wild with a single control object I∈S(Λ). It follows that each finite dimensional k-algebra can be realized as a quotient End(X)/End(X)_I of the endomorphism ring of some object X∈S(Λ) modulo the ideal End(X)_I of all maps which factor through a finite direct sum of copies of I.     

Nonsmoothable group actions on elliptic surfaces

Let G be a cyclic group of order 3, 5 or 7, and X=E(n) be the relatively minimal elliptic surface with rational base. In this paper, we prove that under certain conditions on n, there exists a locally linear G-action on X which is nonsmoothable with respect to infinitely many smooth structures on X. This extends the main result of [X. Liu, N. Nakamura, Pseudofree Z/3-actions on K3 surfaces, Proc. Amer. Math. Soc. 135 (3) (2007) 903-910].     

Hamiltonicity, independence number, and pancyclicity

A graph on n vertices is called pancyclic if it contains a cycle of length ? for all 3≤?≤n. In 1972, Erd?s proved that if G is a Hamiltonian graph on n>4k⁴ vertices with independence number k, then G is pancyclic. He then suggested that n=Ω(k²) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such that n>ck^7/3 suffices.     

On Ochoa's special matrices in matrix classes

It is well known that the ideal classes of an order Z[μ], generated over Z by the integral algebraic number μ, are in a bijective correspondence with certain matrix classes, that is, classes of unimodularly equivalent matrices with rational integer coefficients. If the degree of μ is ?3, we construct explicitly a particularly simple ideal matrix for an ideal which is a product of different prime ideals of degree 1. We obtain the following special n×n matrix (c_ij) in the matrix class corresponding to the ideal class of our ideal: c_i+1,_i=1(i=1,…,n?2); c_ij=0(?i?n, 1?j?n? 2, and i≠j+1); c_nj=0(j)=2,…,n?1). The remaining coefficients are given as explicit polynomials in an integer z which depends on the ideal. It is shown that the matrix class of every regular ideal class of Z[μ] contains a special matrix of this kind.     

Superlarge deviations for sums of random variables with arithmetical super-exponential distributions

Local limit theorems are obtained for superlarge deviations of sums S(n) = ξ(1) + ... + ξ(n) of independent identically distributed random variables having an arithmetical distribution with the right-hand tail decreasing faster that that of a Gaussian law. The distribution of ξ has the form ?(ξ = k) = \(e^{ - k^\beta L(k)} \), where β > 2, k ∈ ? (? is the set of all integers), and L(t) is a slowly varying function as t → ∞ which satisfies some regularity conditions. These theorems describing an asymptotic behavior of the probabilities ?(S(n) = k) as k/n → ∞, complement the results on superlarge deviations in [4, 5].