Compound Poisson Approximation for Multiple Runs in a Markov Chain

We consider a sequence X ₁, ..., X _n of r.v.'s generated by a stationary Markov chain with state space A = {0, 1, ..., r}, r 1. We study the overlapping appearances of runs of k _i consecutive i's, for all i = 1, ..., r, in the sequence X ₁,..., X _n. We prove that the number of overlapping appearances of the above multiple runs can be approximated by a Compound Poisson r.v. with compounding distribution a mixture of geometric distributions. As an application of the previous result, we introduce a specific Multiple-failure mode reliability system with Markov dependent components, and provide lower and upper bounds for the reliability of the system.     

On weak asymptotic isomorphy of memoryless correlated sources

Let {(X _i,Z _i)} be an i.i.d. sequence of random pairs in a finite set × x ℒ; we will call it a discrete memoryless stationary correlated (DMSC) source with generic distribution dist(X ₁,Z ₁). Two DMSC sources {(X _i,Z _i)} and {(X _i′,Z _i′)} are called asymptotically isomorphic in the weak sense if for every ε>0 and sufficiently largen, there exists a joint distribution dist(X ⁿ,Z ⁿ,X′ ⁿ,Z′ ⁿ) ofn-length blocks of the two sources such that . For single sources of equal entropy, McMillan’s theorem implies asymptotic isomorphy in the sense suggested by this definition. For correlated sources, however, no nontrivial cases of weak asymptotic isomorphy are known. We show that some spectral properties of the generic distributions are invariant for weak asymptotic isomorphy, and these properties wholly determine the generic distribution in many cases.     

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.     

Approximation for counts of 2-runs in a two state Markov chain

We consider a Markov chain X = {Xi,i = 1,2,...} with the state space {0,1},and define W =∑i=1n XiXi+1,which is the number of 2-runs in X before time n + 1.In this paper,we prove that the negative binomial distribution is an appropriate approximation for LW when VarW is greater than EW.The error estimate obtained herein improves the corresponding result in previous literatures.     

A characterization of limiting distributions of estimators in an autoregressive process

Summary Let the random variablesX ₁,X ₂, ...,X _n be generated by the first-order autoregressive modelX _i =θX _i−1 +e _i wheree _i ,i=1, 2, ...,n, are i.i.d. random variables with mean zero, variance σ², and with unspecified density functiong(·). In the present paper we obtain a characterization of limiting distributions of nonparametric and parametric estimators of θ as well as a local asymptotic minimax bound of the risks of estimators.     

On the Splitting Ring of a Polynomial

Let f(Z) = Zⁿ ? a₁Z^n?1 + … + (?1)^n?1a_n?1Z + (?1)ⁿa_n be a monic polynomial with coefficients in a ring R with identity, not necessarily commutative. We study the ideal I_f of R[X₁,…, X_n] generated by σ_i(X₁,…, X_n) ? a_i, where σ₁,…, σ_n are the elementary symmetric polynomials, as well as the quotient ring R[X₁,…, X_n]/I_f.     

A note on asymptotic expansions for Markov chains using operator theory

We consider asymptotic expansions for sums S_n on the form S_n = ƒ₀(X₀) + ƒ(X₁, X₀) + … + ƒ(X_n, X_n−1), where X_i is a Markov chain. Under different ergodicity conditions on the Markov chain and certain conditional moment conditions on ƒ(X_i, X_i−1), a simple representation of the characteristic function of S_n is obtained. The representation is in term of the maximal eigenvalue of the linear operator sending a function g(x) into the function x → E(g(X_i)exp[itƒ(X_i, x)]|X_i−1 = x).     

Superposition of Spatially Interacting Aggregated Continuous Time Markov Chains

A system s{ X(t)} = {X ₁(t),X ₂(t),..., X _N(t)} of N interacting time reversible continuous time Markov chains is considered. The state space of each of the processes {X _i(t)} (i = 1, 2,...,N) is partitioned into two aggregates. Interaction between the processes {X _i(t)},{X ₂(t)},...,{X _N(t)} is introduced by allowing the transition rates of an individual process at time t to depend on the configuration of aggregates occupied by the other N - 1 processes at that time. The motivation for this work comes from ion channel modeling, where {(X}(t)} describes the gating mechanisms of N channels and the partitioning of the state space of {X _i(t)} correspond to whether the channel is conducting or not. Let S(t) denote the number of conducting channels at time t. For a time-reversible class of such processes, expressions are derived for the mean and probability density function of the sojourns of {S(t)} at its different levels when {X(t)} is in equilibrium. Particular attention is paid to the situation when the N channels are located on a circle with nearest neighbor interaction. Necessary and sufficient conditions for a general co-operative multiple channel system to be time reversible are derived.     

On calculation of moments of ladder heights

Let X,X ₁,X ₂, … be independent identically distributed random variables, F(x) = P{X < x}, S ₀ = 0, and S _n =Σ _i=1 ⁿ X _i . We consider the random variables, ladder heights Z ₊ and Z ₋ that are respectively the first positive sum and the first negative sum in the random walk {S _n }, n = 0, 1, 2, …. We calculate the first three (four in the case EX = 0) moments of random variables Z ₊ and Z ₋ in the qualitatively different cases EX > 0, EX < 0, and EX = 0. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 159–179, April–June, 2006.     

Random markov processes and uniform martingales

It is shown here that a certain generalization of ann-step Markov chain is equivalent to the uniform convergence of the martingale {P(X ₀|X ₋₁ X ₋₂···X _−n)} _n=1 ^∞ . Ergodic and probabilistic properties of this process are explored. This work was initiated at SUNY/Albany.     

Limit Theorems for Randomly Selected Ratios of Order Statistics from a Pareto Distribution

Abstract

Consider independent and identically distributed random variables {X _nk , 1 ≤ k ≤ m, n ≥ 1} from the Pareto distribution. We randomly select a pair of order statistics from each row, X _n(i) and X _n(j), where 1 ≤ i < j ≤ m. Then we test to see whether or not Strong and Weak Laws of Large Numbers with nonzero limits for weighted sums of the random variables X _n(j)/X _n(i) exist where we place a prior distribution on the selection of each of these possible pairs of order statistics.     

Joint asymptotic distribution of exceedances point process and partial sum of stationary Gaussian sequence

Let {X _i } _i=1^∞ be a standardized stationary Gaussian sequence with covariance function r(n) = EX ₁ X _n+1, S _n = Σ _i=1 ⁿ X _i , and $\bar X_n = \tfrac{{S_n }} {n} $\bar X_n = \tfrac{{S_n }} {n} . And let N _n be the point process formed by the exceedances of random level $(\tfrac{x} {{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}} {{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n $(\tfrac{x} {{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}} {{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n by X ₁,X ₂,…, X _n . Under some mild conditions, N _n and S _n are asymptotically independent, and N _n converges weakly to a Poisson process on (0,1].     

On the distribution of powers of a complex number

Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α ⁿ  + ν, n = 0, 1, 2, . . . , modulo ${\mathbb{Z}[i],}Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α ⁿ  + ν, n = 0, 1, 2, . . . , modulo \mathbbZ[i],{\mathbb{Z}[i],} where i=?{-1}{i=\sqrt{-1}} and \mathbbZ[i]=\mathbbZ+i\mathbbZ{\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}} is the ring of Gaussian integers. For any z ? \mathbbC,{z\in \mathbb{C},} one may naturally call the quantity z modulo \mathbbZ[i]{\mathbb{Z}[i]} the fractional part of z and write {z} for this, in general, complex number lying in the unit square S:={z ? \mathbbC:0 ￡ \mathfrakR(z),\mathfrakJ(z) < 1 }{S:=\{z\in\mathbb{C}:0\leq \mathfrak{R}(z),\mathfrak{J}(z) <1 \}}. We first show that if α is a complex non-real number which is algebraic over \mathbbQ{\mathbb{Q}} and satisfies |α| > 1 then there are two limit points of the sequence {ξ α ⁿ  +ν}, n = 0, 1, 2, . . . , which are ‘far’ from each other (in terms of α only), except when α is an algebraic integer whose conjugates over \mathbbQ(i){\mathbb{Q}(i)} all lie in the unit disc |z| ≤  1 and x ? \mathbbQ(a,i).{\xi\in\mathbb{Q}(\alpha,i).} Then we prove a result in the opposite direction which implies that, for any fixed a ? \mathbbC{\alpha\in\mathbb{C}} of modulus greater than 1 and any sequence z_n ? \mathbbC,n=0,1,2,...,{z_n\in\mathbb{C},n=0,1,2,\dots,} there exists x ? \mathbbC{\xi \in \mathbb{C}} such that the numbers ξ α ⁿ −z _n , n = 0, 1, 2, . . . , all lie ‘far’ from the lattice \mathbbZ[i]{\mathbb{Z}[i]}. In particular, they all can be covered by a union of small discs with centers at (1+i)/2+\mathbbZ[i]{(1+i)/2+\mathbb{Z}[i]} if |α| is large.     

Multiplicative infinite loop space theory

Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of G,γn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.     

Adaptive estimation of the transition density of a particular hidden Markov chain

We study the following model of hidden Markov chain: with (X_i) a real-valued positive recurrent and stationary Markov chain, and (?_i)_1?i?n+1 a noise independent of the sequence (X_i) having a known distribution. We present an adaptive estimator of the transition density based on the quotient of a deconvolution estimator of the density of X_i and an estimator of the density of (X_i,X_i+1). These estimators are obtained by contrast minimization and model selection. We evaluate the L² risk and its rate of convergence for ordinary smooth and supersmooth noise with regard to ordinary smooth and supersmooth chains. Some examples are also detailed.     

Generating Uniform Random Vectors in Z p k : The General Case

We consider the rate of convergence of the Markov chain X _n+1=A X _n +B _n (mod p), where A is an integer matrix with nonzero eigenvalues, and {B _n } _n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Q ^k invariant under A. If for all eigenvalues λ _i of A, then n=O((ln p)²) steps are sufficient and n=O(ln p) steps are necessary to have X _n sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λ _i that are roots of positive integer numbers, |λ ₁|=1 and |λ _i |>1 for all , then O(p ²) steps are necessary and sufficient.      

Stochastically monotone Markov Chains

Summary A real-valued discrete time Markov Chain {X _n} is defined to be stochastically monotone when its one-step transition probability function pr {X _n+1y¦ X _n=x} is non-increasing in x for every fixed y. This class of Markov Chains arises in a natural way when it is sought to bound (stochastically speaking) the process {X _n} by means of a smaller or larger process with the same transition probabilities; the class includes many simple models of applied probability theory. Further, a given stochastically monotone Markov Chain can readily be bounded by another chain {Y _n}, with possibly different transition probabilities and not necessarily stochastically monotone, and this is of particular value when the latter process leads to simpler algebraic manipulations. A stationary stochastically monotone Markov Chain {X _n} has cov(f(X ₀), f(X _n)) cov(f(X ₀), f(X _n+1))0 (n =1, 2,...) for any monotonic function f(·). The paper also investigates the definition of stochastic monotonicity on a more general state space, and the properties of integer-valued stochastically monotone Markov Chains.     

Limit theorems for a recursive maximum process with location-dependent periodic intensity-parameter

We investigate the recursive sequence Z _n : =  max {Z _n − 1,λ(Z _n − 1)X _n } where X _n is a sequence of iid random variables with exponential distributions and λ is a periodic positive bounded measurable function. We prove that the Césaro mean of the sequence λ(Z _n ) converges toward the essential minimum of λ. Subsequently we apply this result and obtain a limit theorem for the distributions of the sequence Z _n . The resulting limit is a Gumbel distribution.