On the limiting behavior of randomly weighted partial sums

We study the almost sure limiting behavior and convergence in probability of weighted partial sums of the form where {W_nj, 1jn, n1} and {X_nj, 1jn, n1} are triangular arrays of random variables. The results obtain irrespective of the joint distributions of the random variables within each array. Applications concerning the Efron bootstrap and queueing theory are discussed.     

A quasi random search technique for functional fitting of detector responses

The problem of nonlinear optimization is encountered while fitting the observed response of certain radiation detectors. In this paper the development of an optimizing technique based on quasi random search is described. The performance of this technique is compared with other existing methods of gradient search and linear Taylor differential correction technique in the case of fitting the photopeak responses of Nal(Tl) detectors. Our results are favourable especially when the observed data are associated with poor statistics.     

中子诱发233,235,238U裂变机制的多通道理论研究

用改进的多通道和无规颈断裂模型,计算了能量E_n=0—6MeV的中子诱发²³³U、²³⁵U裂变和E_n=1.3—5.3MeV中子诱发²³⁸U裂变的碎片的质量分布、动能分布和瞬发中子数分布,理论计算在定量上与实验符合较好.     

On the exact asymptotic behaviour of the distribution of ladder epochs

Let T₊ denote the first increasing ladder epoch in a random walk with a typical step-length X. It is known that for a large class of random walks with E(X)=0,E(X²)=∞, and the right-hand tail of the distribution function of X asymptotically larger than the left-hand tail, $\text{P}\text{T}{}_{\mathrm{+}}\text{?n∽n}{}^{\mathrm{<mtext>1</mtext><mtext>\beta ?1</mtext>}}\text{L}{}_{\mathrm{+}}\text{(n)}$ as n→∞, with 1<β<2 and L₊ slowly varying, if and only if$\text{P}\text{{X?x}∽ 1/{x}{}^{\mathrm{\beta }}\text{L(x)}}$ as x→+∞, with L slowly varying. In this paper it is shown how the asymptotic behaviour of L determines the asymptotic behaviour of L₊ and vice versa. As a by-product, it follows that a certain class of random walks which are in the domain of attraction of one-sided stable laws is such that the down-going ladder height distribution has finite mean.     

Inference for earthquake models: A self-correcting model

Questions of asymptotic inference are discussed for a point process model in which the conditional intensity function increases monotonically between events and drops by determined (nonrandom) amounts after each event. Parameter estimates are shown to be consistent and, except under the null hypothesis of a Poisson process, normally distributed. Under the null hypothesis, however, the Hessian matrix is not asymptotically constant, and the limiting distribution of the likelihood ratio statistics is not χ², but has a form related to that of the Cramer-von Mises ω² statistic for the test of goodness of fit.     

Exact kinetics for “almost random” irreversible filling of lattices

A variety of processes can be modeled by the irreversible cooperative filling of lattice sites. We consider here cases where the filling rate is independent of the state of the surrounding sitesexcept when all nearest neighbors are already filled (where a generally different rate operates) and indicate some applications for such choices. Exact solutions are obtained to the hierarchical form of the master equations for subconfiguration probabilities for a lattice of arbitrary dimension or coordination number. Analogous irreversible coadsorption processes and processes with longer-range cooperative effects, also amenable to exact solution, are discussed.Ames Laboratory is operated for the U.S. Department of Energy under Contract No. W-7405-ENG-82. This work was supported by the Office of Basic Energy Sciences.     

Extendibility of spherical matrix distributions

We investigate the structure of distributions for matrices which can be embedded in arbitrarily large matrices whose distributions have properties of invariance under orthogonal rotations.     

Properties of fourth-order strong mixing rates and its application to random set theory

We shall define the concept of fourth-order strong mixing rates and study their properties. Results are useful for establishing a condition of the form (*) Σ_a,b,c |cum(X_o, X_a, X_b, X_c)| < ∞ or ∫|cum(X_o, X_a, X_b, X_c)| da db dc < ∞ for dependent random variables {X_a}. As an application we shall consider an evaluation of a fourth-order strong mixing rate for a random closed set Z (in the sense of Matheron) and derive the condition (*) for {X_a}, X_a being an outcome of a local measurement upon Z. The result is also applicable to point processes which admit clustering representations.     

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.