Approximation of the Tail Probability of Dependent Random Sums Under Consistent Variation and Applications

In this paper, we consider the random sums of one type of asymptotically quadrant sub-independent and identically distributed random variables {X, X _i , i?=?1, 2,???} with consistently varying tails. We obtain the asymptotic behavior of the tail $\textsf{P}(X_1+\cdots+X_\eta>x)$ under different cases of the interrelationships between the tails of X and η, where η is an integer-valued random variable independent of {X, X _i , i?=?1, 2,???}. We find out that the asymptotic behavior of $\textsf{P}(X_1+\cdots+X_\eta>x)$ is insensitive to the dependence assumed in the present paper. We state some applications of the asymptotic results to ruin probabilities in the compound renewal risk model under dependent risks. We also state some applications to a compound collective risk model under the Markov environment.     

Local probabilities for random walks conditioned to stay positive

Let S ₀ = 0, {S _n , n ≥ 1} be a random walk generated by a sequence of i.i.d. random variables X ₁, X ₂, . . . and let $\tau ^{-}={\rm min} \{ n \geq 1:S_{n}\leq 0 \}$ and $\tau ^{+}={\rm min}\{n\geq1:S_{n} > 0\} $ . Assuming that the distribution of X ₁ belongs to the domain of attraction of an α-stable law we study the asymptotic behavior, as ${n\rightarrow \infty }$ , of the local probabilities ${\bf P}{(\tau ^{\pm }=n)}$ and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities ${\bf P}{(S_{n} \in [x,x+\Delta )|\tau^{-} > n)}$ with fixed Δ and ${x=x(n)\in (0,\infty )}$ .     

Correlation inequalities and a conjecture for permanents

This paper presents conditions on nonnegative real valued functionsf ₁,f ₂,...,f _m andg ₁,g ₂,...g _m implying an inequality of the type

Hitting and martingale characterizations of one-dimensional diffusions

The main theorem of the paper is that, for a large class of one-dimensional diffusions (i. e. strong Markov processes with continuous sample paths): if x(t) is a continuous stochastic process possessing the hitting probabilities and mean exit times of the given diffusion, then x(t) is Markovian, with the transition probabilities of the diffusion. For a diffusion x(t) with natural boundaries at ± ∞, there is constructed a sequence π _n (t, x) of functions with the property that the π _n (t, x (t)) are martingales, reducing in the case of the Brownian motion to the familiar martingale polynomials. It is finally shown that if a stochastic process x (t) is a martingale with continuous paths, with the additional property that

$$\mathop \smallint \limits_0^{x\left( t \right)} m\left( {0,y} \right]dy - t$$     

A Priori Bounds for Periodic Solutions of a Kind of Second Order Neutral Functional Differential Equation with Multiple Deviating Arguments

Thc main aim of this paper is to use the continuation theorem of coincidence degree theory for studying the existence of periodic solutions to a kind of neutral functional differential equation as follows：（x（t）-^n∑i=1cix（t-ri））″=f（x（t））x′＋g（x（t-τ））＋p（t）.In order to do so, we analyze the structure of the linear difference operator A ： C2π→C2π, [Ax]（t） =x（t）-∑^ni=1cix（t-ri）to determine some flmdamental properties first, which we are going to use throughout this paper. Meanwhile, we also prove some new inequalities which are useful for estimating a priori bounds of periodie solutions.     

Über die Funktionalgleichungf(x) = d^{m - 1} \sum\limits_{i = 0}^{d - 1} {f(\frac{{x + i}}{d})} in den Körpern ℤp

Functional equations of the type (1) $$f(x) = d^{m - 1} \sum\limits_{i = 0}^{d - 1} {f(\frac{{x + i}}{d})}$$ for functions f: ?_n→?_n have first been considered in [1] and [2] in the cases m ∈ {0,1} (?_n=Z/n? means the ring of integers modulo n). In this note we give the complete solution of (1) in the case f: ?_p→?_p (p a prime) for each m ∈ ?∪{0}.     

Tail Probabilities of St. Petersburg Sums,Trimmed Sums,and Their Limit

We provide exact asymptotics for the tail probabilities \({\mathbb {P}}\{ S_{n,r} > x \}\) as \(x \rightarrow \infty \), for fixed n, where \(S_{n,r}\) is the r-trimmed partial sum of i.i.d. St. Petersburg random variables. In particular, we prove that although the St. Petersburg distribution is only O-subexponential, the subexponential property almost holds. We also determine the exact tail behavior of the r-trimmed limits.     

An Inequality for Large Deviation Probabilities of Sums of Bounded i.i.d. Random Variables

We provide precise bounds for tail probabilities, say {M _n x}, of sums M _n of bounded i.i.d. random variables. The bounds are expressed through tail probabilities of sums of i.i.d. Bernoulli random variables. In other words, we show that the tails are sub-Bernoullian. Sub-Bernoullian tails are dominated by Gaussian tails. Possible extensions of the methods are discussed.     

On the exceptional sets in Sylvester expansions*

For any x ?? (0, 1], let the series \( {\sum}_{n=1}^{\infty }1/{d}_n(x) \) be the Sylvester expansion of x, where {d _j (x),?j?≥?1} is a sequence of positive integers satisfying d₁(x)?≥?2 and d_j?+?1(x)?≥?d _j (x)(d _j (x)???1)?+?1 for j?≥?1. Suppose ? : ? → ?⁺ is a function satisfying ?(n+1) – ? (n) → ∞ as n → ∞. In this paper, we consider the set

$$ E\left(\phi \right)=\left\{x\kern0.5em \in \left(0,1\right]:\kern0.5em \underset{n\to \infty }{\lim}\frac{\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)}{\phi (n)}=1\right\} $$

and quantify the size of the set in the sense of Hausdorff dimension. As applications, for any β > 1 and γ > 0, we get the Hausdorff dimension of the set \( \left\{x\in \kern1em \left(0,1\right]:\kern0.5em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{n}^{\beta }=\upgamma \right\}, \) and for any τ > 1 and η > 0, we get a lower bound of the Hausdorff dimension of the set \( \left\{x\kern0.5em \in \kern0.5em \left(0,1\right]:\kern1em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{\tau}^n=\eta \right\}. \)     

Necessary conditions for renewal processes approximating to Wiener processes

ＮｅｃｅｓｓａｒｙＣｏｎｄｉｔｉｏｎｓｆｏｒＲｅｎｅｗａｌＰｒｏｃｅｓｓｅｓＡｐｐｒｏｘｉｍａｔｉｎｇｔｏＷｉｅｎｅｒＰｒｏｃｅｓｓｅｓ＊ＺｈａｎｇＬｉｘｉｎａｂｓｔｒａｃｔ．Ｌｅｔ｛Ｘ,Ｘｉ;ｉ≥１｝ｂｅａｓｅｑｕｅｎｃｅｏｆｉ．ｉ．ｄ．ｒ．ｖ＇．...     

Ein verschärfter und verallgemeinerter Satz von Alexandrov

In this paper we prove the following theorem: Suppose that n≥3 and 1≤jn $$(\forall a,b) d(a,b) : = \sum\limits_{\nu = 1}^j { (a_\nu - b_\nu )^2 - \sum\limits_{\nu = j + 1}^n { (a_\nu - b_\nu )^2 .} }$$ If a function f:?ⁿ→?ⁿ satisfies the condition: (*) $$(\forall x,y \in \mathbb{R}^n ) d(f(x),f(y)) = 0 \Leftrightarrow d(x,y) = 0,$$ then f is affine. Moreover, f preserves distances up to a constant factor C≠0, i.e. d(f(x),f(y))=C·d(x,y) for every x,y. In contrast to Alexandrov's result [1] we do not assume that f is bijective, and we also do not assume that j=n?1. A very important part of our proof will be the discussion of a functional equation.     

Unbeschränkte Randwerte bei parabolischen Differentialgleichungen

In this paper we will prove the existence of classical solutions u for a quasilinear parabolic differential equation of type $$(1)u_t = \sum\limits_{i = 1}^n {\tfrac{d}{{dx_i }}a_i (x,t,u,u_x )} + a(x,t,u,u_x )$$ in a cylindrical domain G, whereby unbounded boundary values g may be given on the parabolic boundary R_p of G. In general, u is unbounded and u_x?L₂(G). Under certain conditions (see Satz 3) we nevertheless get a reasonable boundary-behavior of u, that is: \(u(Q) \to g(\bar Q)\) when g is continuous in \(\bar Q \in {\text{R(}}Q \to \bar Q{\text{)}}\) , and u(Γ)→g in the L₂-sense for Γ→R_p where Γ means a parallel surface to R_p.     

Estimation of a multivariate Box-Cox transformation to elliptical symmetry via the empirical characteristic function

Let X=(X ₁, X ₂,..., X _d ) ^t be a random vector of positive entries, such that for some =(₁,₂,..., _d ) ^t , the vector X ⁽⁾ defined by % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamiwamaaDaaaleaamiaadMgaaSqaaWGaaiikaiabeU7aSnaaBaaa% baGaamyAaiaacMcaaeqaaaaakiabg2da9iaacIcadaWcgaqaaiaadI% fadaqhaaWcbaadcaWGPbaaleaamiabeU7aSnaaBaaabaGaamyAaaqa% baaaaOGaeyOeI0IaaGymaiaacMcaaeaacqaH7oaBdaWgaaWcbaadca% WGPbGaaiilaaWcbeaakiaadMgacqGH9aqpcaaIXaGaeSOjGSKaaiil% aiaadsgaaaaaaa!53BB!\[X_i^{(\lambda _{i)} } = ({{X_i^{\lambda _i } - 1)} \mathord{\left/ {\vphantom {{X_i^{\lambda _i } - 1)} {\lambda _{i,} i = 1 \ldots ,d}}} \right. \kern-\nulldelimiterspace} {\lambda _{i,} i = 1 \ldots ,d}}\]is elliptically symmetric. We describe a procedure based on the multivariate empirical characteristic function for estimating the _i's. Asymptotic results regarding consistency of the estimators are given and we evaluate their performance in simulated data. In a one-dimensional setting, comparisons are made with other available transformations to symmetry.Adolfo Quiroz and Miguel Nakamura's research was partially supported by CONACYT (Mexico) grants numbers 1858E9219 and 4224E9405, while Dr. Quiroz was visiting Centro de Investigación en Matemáticas at Guanajuato, Mexico.     

Integral Van Vleck’s and Kannappan’s functional equations on semigroups

In this paper we study the solutions of the integral Van Vleck’s functional equation for the sine

$$\begin{aligned} \int _{S}f(x\tau (y)t)d\mu (t)-\int _{S}f(xyt)d\mu (t) =2f(x)f(y),\; x,y\in S \end{aligned}$$

and the integral Kannappan’s functional equation

$$\begin{aligned} \int _{S}f(xyt)d\mu (t)+\int _{S}f(x\tau (y)t)d\mu (t) =2f(x)f(y),\; x,y\in S, \end{aligned}$$

where S is a semigroup, \(\tau \) is an involution of S and \(\mu \) is a measure that is a linear combination of Dirac measures \((\delta _{z_{i}})_{i\in I}\), such that for all \(i\in I\), \(z_{i}\) is contained in the center of S. We express the solutions of the first equation by means of multiplicative functions on S, and we prove that the solutions of the second equation are closely related to the solutions of d’Alembert’s classic functional equation with involution.

     

Strong consistency of Kernel density estimates for Markov chains failure rates

For a homogeneous and uniformly ergodic Markov chain, with transition kernel , we analyse some reliability measures and failure rates associated with the transition probabilities. Sufficient conditions for strong consistency are obtained for estimates based on kernel density estimators.      

Polynomial Properties on Large Symmetric Association Schemes

In this paper we characterize “large” regular graphs using certain entries in the projection matrices onto the eigenspaces of the graph. As a corollary of this result, we show that “large” association schemes become P-polynomial association schemes. Our results are summarized as follows. Let G = (V, E) be a connected k-regular graph with d +1 distinct eigenvalues \({k = \theta_{0} > \theta_{1} > \cdots > \theta_{d}}\). Since the diameter of G is at most d, we have the Moore bound

$$|V| \leq M(k,d) = 1 + k \sum^{d-1}_{i=0} (k-1)^{i}.$$

Note that if |V| > M(k, d ? 1) holds, the diameter of G is equal to d. Let E _i be the orthogonal projection matrix onto the eigenspace corresponding to θ _i . Let ?(u, v) be the path distance of u, v ∈V.

Theorem. Assume \({|V| > M(k, d - 1)}\) holds. Then for x, y ∈V with \({\partial (x, y) = d}\), the (x, y) -entry of E _i is equal to

$$-\frac{1}{|V|} \prod _{j=1,2,...,d, j \neq i} \frac{\theta_{0}-\theta_{j}}{\theta_{i}-\theta_{j}}.$$

If a symmetric association scheme \({\mathfrak{X} = (X, \{R_{i}\}^{d}_{i=0})}\) has a relation R _i such that the graph (X, R _i ) satisfies the above condition, then \({\mathfrak{X}}\) is P-polynomial. Moreover we show the “dual” version of this theorem for spherical sets and Q-polynomial association schemes.

     

Polar Decompositions of C 0(N) Contractions

Let A be a bounded linear operator on a complex separable Hilbert space H. We show that A is a C₀(N) contraction if and only if , where U is a singular unitary operator with multiplicity and x₁, . . . , x_d are orthonormal vectors satisfying . For a C₀(N) contraction, this gives a complete characterization of its polar decompositions with unitary factors.     

A mixed boundary-value problem for a hyperbolic-parabolic equation

Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W ₂ ¹ (QT) and W ₂ ² (QT).     

Negative entropy,zero temperature and Markov chains on the interval

We consider ergodic optimization for the shift map on the modified Bernoulli space σ: [0, 1]^? → [0, 1]^?, where [0, 1] is the unit closed interval, and the potential A: [0, 1]^? → ? considered depends on the two first coordinates of [0, 1]^?. We are interested in finding stationary Markov probabilities µ_∞ on [0, 1]^? that maximize the value ∫ Adµ, among all stationary (i.e. σ-invariant) probabilities µ on [0, 1]^?. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential A. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities µ _β which weakly converges to µ_∞. The probabilities µ _β are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure in Thermodynamic Formalism. As the potential A depends only on the first two coordinates, instead of the probability µ on [0, 1]^?, we can consider its projection ν on [0, 1]². We look at the problem in both ways. If µ_∞ is the maximizing probability on [0, 1]^?, we also have that its projection ν _∞ is maximizing for A. The hypothesis about stationarity on the maximization problem can also be seen as a transhipment problem. Under the hypothesis of A being C ² and the twist condition, that is,

$\frac{{\partial ^2 A}}{{\partial x\partial y}}(x,y) \ne 0, for all (x,y) \in [0,1]^2 ,$

we show the graph property of the maximizing probability ν on [0, 1]². Moreover, the graph is monotonous. An important result we get is: the maximizing probability is unique generically in Mañé’s sense. Finally, we exhibit a separating sub-action for A.

     

A new characterization of the Poisson distribution

In this note we show that an infinitely divisible (i.d.) distribution function F is Poisson if and only if it satisfies the conditions F(+0) > 0, for any 0 < ∈ < 1 $$\int_{ - \infty }^{I - E} {\frac{{\left| x \right|}}{{1 + \left| x \right|}}} dF = 0$$ and for any 0 < β < 1 $$\int_0^\infty {e^{\alpha xln(x + 1)} } dF< \infty $$