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.     

Stationary probability distributions for multiresponse linear learning models

A necessary and sufficient condition is given for the existence of stationary probability distributions of a non-Markovian model with linear transition rule. Similar to the Markovian case, stationary probability distributions are characterized as eigenvectors of nonnegative matrices. The model studied includes as special cases the Markovian model as well as the linear learning model and has applications in psychological and biological research, in control theory, and in adaption theory.     

Weak convergence in applied probability

Weak convergence of probability measures on function spaces has been active area of research in recent years. While the theory has a somewhat abstract base, it is extremely useful in a wide variety of problems and we believe has much to offer to applied probability. Our aim in this survey paper is to discuss those aspects of the theory which are relevant to work in applied probability. After an introduction to the foundations of weak convergence, we shall discuss partial sum, point, Markov and extremal processes. These processes form the building blocks for many of the important models of applied probability.     

Study on patterns and chemical features of NO effect on marine phytoplankton growth

With supernatural bioactivation, nitric oxide (NO), which was first regarded as an endothelium- derived relaxing factor, was recognized by Science as the “Molecule of the Year” in 1992. The Nobel Prize of Physiology & Medicine in 1998 was awarded jointly to Furchgatt, Ignarro, and Murad for their discoveries concerning the NO effects in the cardiovascular sys- tem. From then on, researchers have paid more and more attention to the special and diverse functions of NO in organism[1―18].…     

柴达木盆地西部地区第三系碎屑岩粒度概率累积曲线特征

在岩心观察的基础上,利用主要取心井的粒度分析资料,对柴达木盆地西部地区第三系碎屑岩主要沉积环境的粒度概率曲线进行了研究。研究表明,该地区发育的沉积相有扇三角洲、冲积平原、三角洲、湖泊、湖底扇,每一种沉积相又分为几种沉积亚相类型;该地区碎屑岩的粒度概率累积曲线特征主要包括:“上拱弧形”式、一段式、两段式、三段式、多段式、复杂两段式和复杂三段式、“低斜多段”式、“高斜多跳一悬”式和“高斜多跳一悬夹过渡”式、“高斜多跳”式、“宽缓上拱”式等,每种沉积亚相类型对应其独特的粒度概率累积曲线特征。该研究成果可为新探区的沉积环境判断提供参考。     

Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory

This paper obtains the uniform estimate for maximum of sums of independent and heavy-tailed random variables with nonnegative random weights,which can be arbi- trarily dependent of each other.Then the applications to ruin probabilities in a discrete time risk model with dependent stochastic returns are considered.     

Local asymptotic behavior of the survival probability of the equilibrium renewal model with heavy tails

Recently, Tang established a local asymptotic relation for the ruin probability in the Cramer-Lundberg risk model. In this short note we extend the corresponding result to the equilibrium renewal risk model.     

On tail behavior of nonlinear autoregressive functional conditional heteroscedastic model with heavy-tailed innovations

We study the tail probability of the stationary distribution of nonparametric non- linear autoregressive functional conditional heteroscedastic (NARFCH) model with heavy- tailed innovations.Our result shows that the tail of the stationary marginal distribution of an NARFCH series is heavily dependent on its conditional variance.When the innovations are heavy-tailed,the tail of the stationary marginal distribution of the series will become heavier or thinner than that of its innovations.We give some specific formulas to show how the increment or decrement of tail heaviness depends on the assumption on the con- ditional variance function.Some examples are given.     

Polarization Tomography of Quantum Radiation: Theoretical Aspects and Operator Approach

We present theoretical foundations for the quantum tomography of polarization states of light fields as a method for measuring their polarization density operator , which characterizes only the polarization degrees of freedom of the radiation. We mainly attend to the method in which the tomographic observables (the “measurement instruments”) are polarizable in nature. We show that the quantum nature of this method can be adequately expressed using the quasispectral tomographic decompositions of in special operator bases, which are finite sums of partially orthogonal projection operators determining the probability distributions of tomographic observables as the decomposition coefficients. We obtain the matrix versions of such “tomographic” representations of , in particular, by projecting them on semiclassical operator bases determining the polarization quasiprobability functions. We briefly discuss the information aspects of the schemes considered in the paper. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 3, pp. 344–357, December, 2005.