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.     

A note on the J1-convergence of a first-passage process

Let X₁, X₂, … be a sequence of independent and identically distributed random variables with mean zero such that the common distribution function belongs to the domain of attraction of a stable law G_α,β with 1<α<2 and β=1 or α=2. If S_n=X₁+…X_n and N(ξ)=min{k:S_k>ξ}, ξ>0, then it is shown that $\text{N(nt)}\text{B}{}^1\text{(n)}$, 0<t<1, converges weakly under the Skorohod J₁-topology to a stable subordinator of index $\text{1}\text{α}$, where B¹(n) depends on the norming constant for S_n.     

Largest and smallest maximal sets of pairwise disjoint partitions

Let $\text{X}$ be a maximal set of pairwise disjoint partitions of n into t distinct parts. Let M_t(n) (resp. m_t(n)) denote the size of the largest (resp. smallest) such maximal set $\text{X}$. Upper and lower bounds for $\text{M}{}_{\mathrm{t}}\text{(n)}\text{n}$ and $\text{m}{}_{\mathrm{t}}\text{(n)}\text{n}$ are established.     

Stability of a chain of linear differential equations

The behavior of an infinite sequence of ordinary differential equations of the form:

$\text{dXn}\text{dt}\text{=}\text{∑}\text{i=?M}\text{N}\text{L}{}_{\mathrm{i}}\text{X}{}_{\mathrm{i+n}}\text{, 0 ? n, 0 < N, M < ∞}$

,

$\text{X}{}_{\mathrm{n}}\text{(0) = C}{}_{\mathrm{n}}\text{, (1) X}{}_{\mathrm{n}}\text{≡ 0, n < 0}$

, where X_n(t) is a vector valued function of R⁺, is studied in spaces of infinite sequences of vectors. In particular, sufficient conditions for asymptotic stability of this sequence of linear equations are established and applied to the stability analysis of a string of vehicles with a simple form of automatic control.     

On a ramsey-type problem of J. A. Bondy and P. Erdös. I

Let m → (C_k, C_n) signify the truth of the following statement: Let {V(G); ≥ m; if G contains no C_k, then $\text{G}$ contains a C_n. Bondy and Erdös [1] proved that for n > 3 2n ? 1 → (C_n, C_n). They conjectured that 2n ? 1 → (C_n, C_k) for all n > 3 and all k < n and could prove it only for $\text{k < (2n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. In this paper we prove this for all n > 4 and for all k < n.     

On a problem of Chvátal and Erdös on hypergraphs containing no generalized simplex

Let X be an n-element set and $\text{T}$ a family of k-subsets of X. Let r be an integer, k > r ? 2. Suppose that $\text{T}$ does not contain r + 1 members having empty intersection such that any r of them intersect non-trivially. Chvátal and Erdös conjectured that for (r + 1) k ? rn we have |$\text{F}$|$\text{?}\text{n?1}\text{k?1}$. In this paper we first prove that This conjecture holds asymptotically (Theory 1). In Theorems 4 and 5 we prove it for r = 2, K ? 5, n > n_o(k); k ? 3r, n > n_o(k, r), respectively.     

On intersecting families of finite sets

Let X = [1, n] be a finite set of cardinality n and let $\text{F}$ be a family of k-subsets of X. Suppose that any two members of $\text{F}$ intersect in at least t elements and for some given positive constant c, every element of X is contained in less than c |$\text{F}$| members of $\text{F}$. How large |$\text{F}$| can be and which are the extremal families were problems posed by Erdös, Rothschild, and Szemerédi. In this paper we answer some of these questions for n > n₀(k, c). One of the results is the following: let $\text{t = 1,}\text{3}\text{7}\text{< c <}\text{1}\text{2}$. Then whenever $\text{F}$ is an extremal family we can find a 7-3 Steiner system $\text{B}$ such that $\text{F}$ consists exactly of those k-subsets of X which contain some member of $\text{B}$.     

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 $\text{V(}\text{C}\text{)=n}$.     

Approximation by rational modules in Sobolev and Lipschitz norms

Let X?C be compact, 0>n∈Z, and g a continuous function on X. Let R(n,g,X) be the rational module consisting of the functions on X of the type r₀ + r₁g + ··· + r_ngⁿ, where r_j is a rational function with poles off X, 0 ? j ? n. It is shown that if X is nowhere dense, g is sufficiently smooth, and $\text{}\text{?}\text{t6g(z) ≠ 0, z ∈ X}$, then the restriction to X of each function in C∈(C) is approximable in the Lip(n ? 1, X)-norm, n ? 2, by functions in R(n, g, X). Also dealt with are approximation problems in Sobolev norms by more general types of rational modules.     

Ultimate boundedness results for a certain system of third-order non-linear differential equations

We present in this paper ultimate boundedness results for a third-order system of non-linear differential equations of the form $\text{·X + A}\text{X}\text{?}\text{+ B/.X + H(X) = P(t, X, /.X,}\text{X}\text{?}\text{)}$. where A, B are constant symmetric n × n matrices and X, H(X), P(t, X, X, X) are real n-vectors with H:R_n→R_n and P: RXR_nXR_nXR_nXR_n → XR_ncontinuous in their respective arguments. Our results give an n-dimensional analogue of an earlier result of Ezeilo in [1] and extend other earlier results for the case in which we do not necessarily require that H(X) be differentiable.     

On a property related to convergence in probability and some applications to branching processes

It is shown that any real-valued sequence of random variables {X_n} converging in probability to a non-degenerate, not necessarily a.s. finite limit X possesses the following property: for any c with P(X? (c ? δ, c + δ)) > 0 for all δ > 0, there exists a sequence {c_n} with lim_n→∞ c_n = c such that for any ε > 0, lim_n→∞ P(Xδ (c ? ε, c + ε) |X_n = c_n) = 1. This property is applied to various types of branching processes where $\text{X}{}_{\mathrm{n}}\text{=}\text{Z}{}_{\mathrm{n}}\text{C}{}_{\mathrm{n}}$ or $\text{X}{}_{\mathrm{n}}\text{=}\text{U(Z}{}_{\mathrm{n}}\text{)}\text{C}{}_{\mathrm{n}}\text{{C}{}_{\mathrm{n}}\text{}}$ being a sequence of constants or random variables and U a slowly varying function. If {Z_n} is a supercritical branching process in varying or random environment, X is shown to have a continuous and strictly increasing distribution function on (0, ∞). Characterizations of the tail of the liniting distribution of the finite mean and the infinite mean supercritical Galton-Watson processes are also obtained.     

Poisson limits for a hard-core clustering model

Let X_1n,…,X_>nn denote the locations of n points in a bounded, γ-dimensional, Euclidean region D_n which has positive γ-dimensional Lebesgue measure μ(D_n). Let {Y_n(r): r > 0} be the interpoint distance process for these points where Y_n(r) is the number of pairs of points(X_in, X_in) which with i < j have Euclidean distance 6X_in ? X_>in6 < r. In this article we study the limiting distribution of Y_n(r) when n → ∞ and μ(D_n) → ∞, and the joint density of X_1n,…,X_nnis of the form

where r₀ is a positive constant and C_n is a normalizing constant. These joint densities modify the Strauss [11] clustering model densities by introducing a hard-core component (no two points can have 6X_in ? X_in6 < r₀) found in the Matérn [4] models. In our main result we show that the interpoint distance process converges to a non-homogeneous Poisson process for r values in a bounded interval 0 < r₀ < r < r₀₀ provided sparseness conditions discussed by Saunders and Funk [9] hold. The sparseness conditions which require $\text{μ(D}{}_{\mathrm{n}}\text{)}\text{n}{}^2$ converges to a positive constant and the boundary of D_n is negligible are essentially equivalent to requiring that although the number of points n is large the region is large enough so that the points are sparse in this region. That is, it is rare for a point to have another point close to it. These results extend results for v ? 0 given by Saunders and Funk [9] where it is shown that without the hard core component such results do not hold for v > 0. Statistical applications are discussed.     

Générateurs et semi-groupes dans les espaces de Banach uniformement lisses

Let C be a closed convex subset of a uniformly smooth Banach space. Let S(t) : C → C be a semigroup of type ω. Then the generator A₀ of S(t) has a dense domain in C. Moreover there is is an operator A such that: (i) A₀ ? A and $\text{D(A)}\text{= C, (ii) A + gwI}$ accretive, (iii) R(I + λA) ? C for λ > 0 and ωλ < 1, (iv) $\text{S(t)x =}\text{lim}{}_{\mathrm{n\; \to \; \infty }}\text{(I + (}\text{t}\text{n}\text{)A)}{}^{\mathrm{?n}}\text{x}$ for every x?C.     

Sperner families over a subset

Let |X| = n > 0, |Y| = k > 0, and Y ? X. A family A of subsets of X is a Sperner family of X over Y if A₁A₂ for every pair of distinct members of A and every member of A has a nonempty intersection with Y. The maximum cardinality f(n, k) of such a family is determined in this paper. $\text{f(n,k)=(}\text{n}\text{[n}\text{2]}\text{)?(}\text{?k}\text{[n}\text{2]}\text{)}$.     

On the minimum number of disjoint pairs in a family of finite sets

The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ? m ? 2ⁿ. Then there is a family $\text{F}$, |$\text{F}$| = m, such that F ∈ $\text{F}$, G ? X, | G | > | F | implies G ∈ $\text{F}$ and $\text{F}$ minimizes the number of pairs (F₁, F₂), F₁, F₂ ∈ $\text{F}$ F₁ ∩ F₂ = ? over all families consisting of m subsets of X.     

A note on the functional law of iterated logarithm for maxima of Gaussian sequences

Let {X_n, n ≥ 1} be a real-valued stationary Gaussian sequence with mean zero and variance one. Let M_n = max{X_t, i ≤ n} and H_n(t) = (M_[nt] ? b_n)a_n^?1 be the maximum resp. the properly normalised maximum process, where $\text{c}{}_{\mathrm{n}}\text{= (2}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, $\text{a}{}_{\mathrm{n}}\text{=}\text{(}\text{log log}\text{n)}\text{c}{}_{\mathrm{n}}$ and $\text{b}{}_{\mathrm{n}}\text{= c}{}_{\mathrm{n}}\text{?}\text{1}\text{2}\text{(}\text{log}\text{(4π}\text{log}\text{n))}\text{c}{}_{\mathrm{n}}$. We characterize the almost sure limit functions of (H_n)_n≥3 in the set of non-negative, non-decreasing, right-continuous, real-valued functions on (0, ∞), if r(n) (log n)^3?Δ = O(1) for all Δ > 0 or if r(n) (log n)^2?Δ = O(1) for all Δ > 0 and r(n) convex and fulfills another regularity condition, where r(n) is the correlation function of the Gaussian sequence.     

Construction of translation planes from t-spread sets

A t-spread set [1] is a set $\text{C}$ of (t + 1) × (t + 1) matrices over GF(q) such that ∥C∥ = q^t+1, 0 ? C, I?C, and det(X ? Y) ≠ 0 if X and Y are distinct elements of $\text{C}$. The amount of computation involved in constructing t-spread sets is considerable, and the following construction technique reduces somewhat this computation. Construction: Let $\text{G}$ be a subgroup of GL(t + 1, q), (the non-singular (t + 1) × (t + 1) matrices over GF(q)), such that ∥G∥|a^t+1, and det (G ? H) ≠ 0 if G and H are distinct elements of $\text{G}$. Let A₁, A₂, …, A_n?GL(t + 1, q) such that det(A_i ? G) ≠ 0 for i = 1, …, n and all G?G, and det(A_i ? A_jG) ≠ 0 for i > j and all G?G. Let $\text{C = \&{0\&} ∪ G ∪ A}{}_1\text{G ∪ … ∪ A}{}_{\mathrm{n}}\text{G}$, and ∥C∥ = q^t+1. Then $\text{C}$ is a t-spread set. A t-spread set can be used to define a left V ? W system over V(t + 1, q) as follows: x + y is the vector sum; let e?V(t + 1, q), then xoy = yM(x) where M(x) is the unique element of $\text{C}$ with x = eM(x). Theorem: Let$\text{C}$be a t-spread set and F the associatedV ? Wsystem; the left nucleus = {y | CM(y) = C}, and the middle nucleus = }y | M(y)C = C}. Theorem: For$\text{C}$constructed as aboveG ? {M(x) | x?N_λ}. This construction technique has been applied to construct a V ? W system of order 25 with ∥N_λ∥ = 6, and ∥N_μ∥ = 4. This system coordinatizes a new projective plane.     

Central limit theorems for C(S)-valued random variables

Let C(S) be the space of real-valued continuous functions on a compact metric space S. Let {X_n, n ? 1} be a sequence of independent identically distributed C(S)-valued random variables with mean zero and sup_t?sE[X₁²(t)] = 1. We show that the measures induced by ($\text{X}{}_1\text{+ ··· + X}{}_{\mathrm{n}}\text{) n}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$ converge weakly to a Gaussian measure on C(S) under different conditions on X₁, one of which consolidates and extends results of Strassen and Dudley, Giné, and Dudley. Our method of proof is different from the methods employed by these authors.     

On some covering designs

Let n ? k ? t be positive integers, and let Ω be a set of n elements. Let C(n, k, t) denote the number of k-tuples of Ω in a minimal system of k-tuples such that every t-tuple is contained in at least one k-tuple of the system. C(n, k, t) has been determined in all cases for which $\text{C(n, k, t) ?}\text{3(t + 1)}\text{2}$ [W. H. Mills, Ars Combinatoria8 (1979), 199–315]. C(n, k, t) is determined in the case $\text{3(t + 1)}\text{2}\text{< C(n, k, t) ?}\text{3(t + 2)}\text{2}$.     

A covering theorem for quasi-groups

It is shown that if Q is a quasi-group of order n and k is moderately large, there exists a subset A of Q of size k such that if t is the least number of left translates of A needed to cover Q, then $\text{t >}\text{c(n}\text{log}\text{n)}\text{k}$.