Nonuniqueness ing-functions

We give an example of two distinct stationary processes {X _n} and {X′ _n} on {0, 1} for whichP[X₀=1|X₋₁=a₋₁,X₋₂=a₋₂, …]=P[X′₀=1|X′₋₁=a₋₁,X′₋₂=a₋₂, …] for all {a _i},i=−1, −2, …, even though these probabilities are bounded away from 0 and 1, and are continuous in {a _i}. Supported in part by NSF Grant DMS 89-01545. Supported in part by the US Army Research Office.     

On the range of random walk

Let {S _n , n=0, 1, 2, …} be a random walk (S _n being thenth partial sum of a sequence of independent, identically distributed, random variables) with values inE _d , thed-dimensional integer lattice. Letf _n =Prob {S ₁ ≠ 0, …,S _n −1 ≠ 0,S _n =0 |S ₀=0}. The random walk is said to be transient if and strongly transient if . LetR _n =cardinality of the set {S ₀,S ₁, …,S _n }. It is shown that for a strongly transient random walk with p<1, the distribution of [R _n −np]/σ √n converges to the normal distribution with mean 0 and variance 1 asn tends to infinity, where σ is an appropriate positive constant. The other main result concerns the “capacity” of {S ₀, …,S _n }. For a finite setA inE _d , let C(A=Σ _x∈A ) Prob {S _n ∉A, n≧1 |S ₀=x} be the capacity ofA. A strong law forC{S ₀, …,S _n } is proved for a transient random walk, and some related questions are also considered. This research was partially supported by the National Science Foundation.     

A limit theorem for the moments of sums of independent random variables

Forn≧1, letS _n=ΣX _n,i (1≦i≦r _n <∞), where the summands ofS _n are independent random variables having medians bounded in absolute value by a finite number which is independent ofn. Letf be a nonnegative function on (− ∞, ∞) which vanishes and is continuous at the origin, and which satisfies, for some for allt≧1 and all values ofx. Theorem.For centering constants c _n,let S _n − c _n converge in distribution to a random variable S. (A)In order that Ef(S_n − c_n) converge to a limit L, it is necessary and sufficient that there exist a common limit (B)If L exists, then L<∞ if and only if R<∞, and when L is finite, L=Ef(S)+R. Applications are given to infinite series of independent random variables, and to normed sums of independent, identically distributed random variables.     

Multidimensional analog of the Hardy condition for power series

In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f ⁻¹(x) = $ \sum\limits_{n = 0}^\infty {b_n x^n } $ \sum\limits_{n = 0}^\infty {b_n x^n } b _n x ⁿ be such that b ₀ > 0 and b _n ≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x ₁, x ₂, …, x _m ) with positive coefficients in order that the series f ⁻¹(x ₁, x ₂, …, x _m ) = $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } satisfies the property b _{0, …, 0} > 0, $ bi_1 ,i_2 , \ldots ,i_m $ bi_1 ,i_2 , \ldots ,i_m ≤ 0, i ₁² + i ₂² + … + i _m ² > 0, which is similar to the one-dimensional case.     

The <Emphasis Type="Italic">f</Emphasis>-Vector of the Descent Polytope

For a positive integer n and a subset S⊆[n−1], the descent polytope DP  _S is the set of points (x ₁,…,x _n ) in the n-dimensional unit cube [0,1] ⁿ such that x _i ≥x _i+1 if i∈S and x _i ≤x _i+1 otherwise. First, we express the f-vector as a sum over all subsets of [n−1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5,…}∩[n−1]. We derive a generating function for F _S (t), written as a formal power series in two non-commuting variables with coefficients in ℤ[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.     

A general Hobby-Rice Theorem and cake cutting

LetX be a Borel subset of a separable Banach spaceE. Letμ be a non-atomic,σ-finite, Borel measure onX. LetG ⊆L ₁ (X, Σ,μ) bem-dimensional. Theorem:There is an l ∈ E* and real numbers −∞=x ₀<x ₁<x ₂<…<x _n<x _n+1=∞with n≤m, such that for all g ∈ G,      

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

The Johnson–Lindenstrauss Lemma Almost Characterizes Hilbert Space, But Not Quite

Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer n and any x ₁,…,x _n ∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that ‖x _i −x _j ‖≤‖L(x _i )−L(x _j )‖≤O(1)⋅‖x _i −x _j ‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion 2^2O(log*n)2^{2^{O(\log^{*}n)}} . On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-dimensional subspace E _n ⊆Y whose Euclidean distortion is at least 2^Ω(α(n)), where α is the inverse Ackermann function.     

Ergodicity of a class of nonlinear time series models in random environment domain

In this paper, we study the problem of a variety of p, onlinear time series model Xn＋ 1= TZn＋1（X（n）, … ,X（n - Zn＋l）, en＋1（Zn＋1）） in which {Zn} is a Markov chain with finite state space, and for every state i of the Markov chain, {en（i）} is a sequence of independent and identically distributed random variables. Also, the limit behavior of the sequence {Xn} defined by the above model is investigated. Some new novel results on the underlying models are presented.     

Fourier and Hermite series estimates of regression functions

Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.     

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.     

Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

For x = (x ₁, x ₂, …, x _n ) ∈ (0, 1 ] ⁿ and r ∈ { 1, 2, … , n}, a symmetric function F _n (x, r) is defined by the relation

Limit theorems for sums of dependent random variables

Summary In Lai and Stout [7] the upper half of the law of the iterated logarithm (LIL) is established for sums of strongly dependent stationary Gaussian random variables. Herein, the upper half of the LIL is established for strongly dependent random variables {X _i} which are however not necessarily Gaussian. Applications are made to multiplicative random variables and to f(Z _i ) where the Z _i are strongly dependent Gaussian. A maximal inequality and a Marcinkiewicz-Zygmund type strong law are established for sums of strongly dependent random variables X _i satisfying a moment condition of the form E¦S _a,n ¦^pg(n), where , generalizing the work of Serfling [13, 14].Research supported by the National Science Foundation under grant NSF-MCS-78-09179Research supported by the National Science Foundation under grant NSF-MCS-78-04014     

Asymptotic properties of solutions of semilinear second-order elliptic equations in cylindrical domains

The equations under consideration have the following structure:

Some path properties of generalized Lévy sheet

Let X(t) be an N parameter generalized Lévy sheet taking values in ℝ^d with a lower index α, ℜ = {(s, t] = ∏ _i=1 ^N (s _i, t _i], s _i < t _i}, E(x, Q) = {t ∈ Q: X(t) = x}, Q ∈ ℜ be the level set of X at x and X(Q) = {x: ∃t ∈ Q such that X(t) = x} be the image of X on Q. In this paper, the problems of the existence and increment size of the local times for X(t) are studied. In addition, the Hausdorff dimension of E(x, Q) and the upper bound of a uniform dimension for X(Q) are also established.     

On measure concentration for separately Lipschitz functions in product spaces

Let M _n = X ₁ + ⋯ + X _n be a martingale with bounded differences X _m = M _m − M _m −1 such that ℙ{a _m − σ _m ≤ X _m ≤ a _m + σ _m } = 1 with nonrandom nonnegative σ _m and σ(X ₁, …, X _m −1)-measurable random variables a _m . Write σ ² = σ ₁ ² + ⋯ + σ _n ² . Let I(x) = 1 − Φ(x), where Φ is the standard normal distribution function. We prove the inequalities

The density of rational points on non-singular hypersurfaces

LetF(x) =F[x₁,…,x_n]∈ℤ[x₁,…,x_n] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ ⁿ ;F(x)=0, |x|⩽X}, where . It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X ^n−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process. Dedicated to the memory of Professor K G Ramanathan     

On normal approximation of discounted and strongly mixing random variables

We estimate the difference for bounded functions h: ℝ → ℝ satisfying the Lipschitz condition, where Z _v = B _v ⁻¹ ∑ _i=0 ^∞ v ⁱ X _i and with discount factor ν such that 0 < ν < 1. Here {X _n , n ≥ 0} is a sequence of strongly mixing random variables with , and N is a standard normal random variable. In a particular case, the obtained upper bounds are of order O((1 − ν)^1/2). Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 399–409, July–September, 2007. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-15/07.     

Limiting distribution of sums of nonnegative stationary random variables

Summary Let {X _n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS _n=X_n,1+…+X _n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f _n(x)∼ defined on a stationary sequence {X _j∼, whereX _n.f=f_n(X_j). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.     

Record values and the exponential distribution

A sequence {X _n,n≧1} of independent and identically distributed random variables with continuous cumulative distribution functionF(x) is considered.X _j is a record value of this sequence ifX _j>max (X ₁, …,X _j−1). Let {X _L(n) n≧0} be the sequence of such record values. Some properties ofX _L(n) andX _L(n)−X_L(n−1) are studied when {X _n,n≧1} has the exponential distribution. Characterizations of the exponential distribution are given in terms of the sequence {X _L(n),n≧0} The work was partly completed when the author was at the Department of Statistics, University of Brasilia, Brazil.