A self-correcting point process

Suppose a point process is attempting to operate as closely as possible to a deterministic rate ρ, in the sense of aiming to produce ρt points during the interval (0,t] for all t. This can be modelled by making the instantaneous rate of t of the process a suitable function of n-ρt, n being the number of points in [0, t]. This paper studies such a self-correcting point process in two cases: when the point process is Markovian and the rate function is very general, and when the point process is arbitrary and the rate function is exponential. In each case it is shown that as t→∞ the mean number of points occuring in (0, t] is ρt+O(1) while the variance is bounded further, in the Markov case all the absolute central moments are bounded. An application to the outputs of stationary D/M/s queues is given.     

DIMENSION RESULTS FOR ORNSTEIN-UHLENBECK TYPE MARKOV PROCESS

Let {X_t, t ≥ 0} be an Ornstein-Uhlenbeck type Markov process with Levy process A_t, the authors consider the fractal properties of its ranges, give the upper and lower bounds of the Hausdorff dimensions of the ranges and the estimate of the dimensions of the level sets for the process. The existence of local times and occuption times of X_t are considered in some special situations.     

Convergence of symmetric Markov chains on {\mathbb{Z}^d}

For each n let ${Y^{(n)}_t}$ be a continuous time symmetric Markov chain with state space ${n^{-1} \mathbb{Z}^d}$ . Conditions in terms of the conductances are given for the convergence of the ${Y^{(n)}_t}$ to a symmetric Markov process Y _t on ${\mathbb{R}^d}$ . We have weak convergence of $\{{Y^{(n)}_t: t \leq t_0\}}$ for every t ₀ and every starting point. The limit process Y has a continuous part and may also have jumps.     

On a class of threshold AR(k) processes

We consider the model $$Z_t = \sum\limits_{i = 1}^k {\phi (i,j)Z_{t - i} } + a_t (j)when\left[ {Z_{t - 1} ,Z_{t - 2,...,} Z_{t - k} } \right]^\prime \in R(j),$$ where {R(j);1?j? ?}is a partition of ? ^k , and for each 1?j??,{a _t (j);t? 0} are i.i.d. zero-mean random variables, having a strictly positive density. Sufficient conditions are obtained for this process to be transient. In addition, for a particular class of such models, necessary and sufficient conditions for ergodicity are obtained. Least-squares estimators of the parameters are obtained and are, under mild regularity conditions, shown to be strongly consistent and asymptotically normal.     

On the convergence of sequences of stationary jump Markov processes

This paper presents two main results: first, a Liapunov type criterion for the existence of a stationary probability distribution for a jump Markov process; second, a Liapunov type criterion for existence and tightness of stationary probability distributions for a sequence of jump Markov processes. If the corresponding semigroups T_N(t) converge, under suitable hypotheses on the limit semigroup, this last result yields the weak convergence of the sequence of stationary processes (T_N(t), π_N) to the stationary limit one.     

Control problems with time-lag. II

Consider a continuous time Markov chain with stationary transition probabilities. A function of the state is observed. A regular conditional probability distribution for the trajectory of the chain, given observations up to time t, is obtained. This distribution also corresponds to a Markov chain, but the conditional chain has nonstationary transition probabilities. In particular, computation of the conditional distribution of the state at time s is discussed. For s > t, we have prediction (extrapolation), while s < t corresponds to smoothing (interpolation). Equations for the conditional state distribution are given on matrix form and as recursive differential equations with varying s or t. These differential equations are closely related to Kolmogorov's forward and backward equations. Markov chains with one observed and one unobserved component are treated as a special case. In an example, the conditional distribution of the change-point is derived for a Poisson process with a changing intensity, given observations of the Poisson process.     

Poisson's equation for queues driven by a Markovian marked point process

LetV _t be the virtual waiting time at timet in a queue having marked point process input generated by a finite Markov process {J_t}, such that in addition to Markovmodulated Poisson arrivals there may also be arrivals at jump times of {J_t}. In this setting, Poisson's equation isA g=–f whereA is the infinitesimal generator of {(V_t, J_t)}. It is shown that the solutiong can be expressed asKf for some suitable kernelK, and the explicit form ofK is evaluated. The results are applied to compute limiting variance constants for (normalized) time averages of functionsf(V _t, J_t), in particularf(V _t,J_t)=V_t.     

Transition Semi–groups on a Local Field Induced by Galois Group and their Representation

For a given map defined on the field of p-adic numbers satisfying

Limiting distributions of functionals of Markov chains

Let \s{X_n, n ? 0\s} and \s{Y_n, n ? 0\s} be two stochastic processes such that Y_n depends on X_n in a stationary manner, i.e. P(Y_n ? A\vbX_n) does not depend on n. Sufficient conditions are derived for Y_n to have a limiting distribution. If X_n is a Markov chain with stationary transition probabilities and Y_n = f(X_n,..., X_n+k) then Y_n depends on X_n is a stationary way. Two situations are considered: (i) \s{X_n, n ? 0\s} has a limiting distribution (ii) \s{X_n, n ? 0\s} does not have a limiting distribution and exits every finite set with probability 1. Several examples are considered including that of a non-homogeneous Poisson process with periodic rate function where we obtain the limiting distribution of the interevent times.     

一个关于马尔可夫链的极限定理

给出一个关于可列非齐次马尔可夫链M元状态序组出现频率的新形式的强极限定理,所得结论对任意可列非齐次马尔可夫链普遍成立.     

Probabilistic metric spaces induced by Markov chains

Let C be a collection of particles, each of which is independently undergoing the same Markov chain, and let d be a metric on the state space. Then, using transition probabilities, for distinct p, q in C, any time t and real x, we can calculate F _pq ^(t) (x) = Pr [d (p,q)t]. For each time t 0, the collection C is shown to be a probabilistic metric space under the triangle function . In this paper we study the structure and limiting behavior of PM spaces so constructed. We show that whenever the transition probabilities have non-degenerate limits then the limit of the family of PM spaces exists and is a PM space under the same triangle function. For an irreducible, aperiodic, positive recurrent Markov chain, the limiting PM space is equilateral. For an irreducible, positive recurrent Markov chain with period p, the limiting PM space has at most only [p/2]+2 distinct distance distribution functions. Finally, we exhibit a class of Markov chains in which all of the states are transient, so that P _ij(t)0 for all states i, j, but for which the {F _pq ^tt } all have non-trivial limits and hence a non-trivial limiting PM space does exist.     

Weak Invariance Principle for the Local Times of Partial Sums of Markov Chains

Let {X _n } be an integer-valued Markov chain with finite state space. Let $S_{n}=\sum_{k=0}^{n}X_{k}$ and let L _n (x) be the number of times S _k hits x∈? up to step n. Define the normalized local time process l _n (t,x) by The subject of this paper is to prove a functional weak invariance principle for the normalized sequence l _n (t,x), i.e., we prove under the assumption of strong aperiodicity of the Markov chain that the normalized local times converge in distribution to the local time of the Brownian motion.     

Limit points badly approximable by horoballs

For a proper, geodesic, Gromov hyperbolic metric space X, a discrete subgroup of isometries Γ whose limit set is uniformly perfect, and a disjoint collection of horoballs {H _j }, we show that the set of limit points badly approximable by {H _j } is absolutely winning in the limit set Λ(Γ). As an application, we deduce that for a geometrically finite Kleinian group acting on ${\mathbb{H}^{n+1}}$ , the limit points badly approximable by parabolics, denoted BA(Γ), is absolutely winning, generalizing previous results of Dani and McMullen. As a consequence of winning, we show that BA(Γ) has dimension equal to the critical exponent of the group. Since BA(Γ) can alternatively be described as the limit points representing bounded geodesics in the quotient ${\mathbb{H}^{n+1}/\Gamma}$ , we recapture a result originally due to Bishop and Jones.     

Asymptotic Normality for Non-linear Functionals of Non-causal Linear Processes with Summable Weights

Let be a non-causal linear process with weights a_js satisfying certain summability conditions, and the iid sequence of innovation {_i} having zero mean and finite second moment. For a large class of non-linear functional K which includes indicator functions and polynomials, the present paper develops the central limit theorem for the partial sums      

Convergence in probability and almost sure with applications

A necessary and sufficient condition is given for the convergence in probability of a stochastic process {X_t}. Moreover, as a byproduct, an almost sure convergent stochastic process {Y_t} with the same limit as {X_t} is identified. In a number of cases {Y_t} reduces to {X_t} thereby proving a.s. convergence. In other cases it leads to a different sequence but, under further assumptions, it may be shown that {X_t} and {Y_t} are a.s. equivalent, implying that {X_t} is a.s. convergent. The method applies to a number of old and new cases of branching processes providing an unified approach. New results are derived for supercritical branching random walks and multitype branching processes in varying environment.     

Calculation of reliability function and remaining useful life for a Markov failure time process

^** Email: banjev{at}mie.utoronto.ca Reliability analysts are interested in calculating a reliabilityfunction (RF), e.g. in order to establish an optimal replacementpolicy. To implement this policy, it is often important to includemeasured condition information, such as those from oil or vibrationanalysis. Information from condition monitoring can be includedin reliability analysis by considering the hazard rate functionas a function of a stochastic covariate process. In this paper,the failure process along with the covariate process is representedby a discrete Markov process. Methods are designed for calculatingthe conditional and unconditional RFs and for computing theremaining useful life (RUL) as a function of the current conditions.It is shown that a function that appears in the computationcan be obtained as a solution to a Kolmogorov-type system ofdifferential equations. The product-integration method is suggestedas the main general method for numerical calculation. The samemethod is also used to calculate the RUL. Illustration of themain concepts is given using field data from a transmission'soil analysis histories.     

Structural properties of reflected Lévy processes

This paper considers a number of structural properties of reflected Lévy processes, where both one-sided reflection (at 0) and two-sided reflection (at both 0 and K>0) are examined. With V _t being the position of the reflected process at time t, we focus on the analysis of $\zeta(t):=\mathbb{E}V_{t}$ and $\xi(t):=\mathbb{V}\mathrm{ar}V_{t}$ . We prove that for the one- and two-sided reflection, ζ(t) is increasing and concave, whereas for the one-sided reflection, ξ(t) is increasing. In most proofs we first establish the claim for the discrete-time counterpart (that is, a reflected random walk), and then use a limiting argument. A key step in our proofs for the two-sided reflection is a new representation of the position of the reflected process in terms of the driving Lévy process.     

The mean of the running maximum of an integrated Gauss–Markov process and the connection with its first-passage time

We find explicit formulae for the mean of the running maximum of conditional and unconditional Brownian motion; they are used to obtain the mean, a(t), of the running maximum of an integrated Gauss–Markov process. Then, we deal with the connection between the moments of its first-passage-time and a(t). As explicit examples, we consider integrated Brownian motion and integrated Ornstein–Uhlenbeck process.     

Gaussian fluctuation for linear eigenvalue statistics of large dilute Wigner matrices

This paper focuses on the dilute real symmetric Wigner matrix Mn=1√n(aij)n×n,whose offdiagonal entries aij(1 i=j n)have mean zero and unit variance,Ea4ij=θnα(θ0)and the fifth moments of aij satisfy a Lindeberg type condition.When the dilute parameter 0α13and the test function satisfies some regular conditions,it proves that the centered linear eigenvalue statistics of Mn obey the central limit theorem.     

GI-graphs: a new class of graphs with many symmetries

The class of generalized Petersen graphs was introduced by Coxeter in the 1950s. Frucht, Graver and Watkins determined the automorphism groups of generalized Petersen graphs in 1971, and much later, Nedela and ?koviera and (independently) Lovre?i?-Sara?in characterised those which are Cayley graphs. In this paper we extend the class of generalized Petersen graphs to a class of GI-graphs. For any positive integer n and any sequence j ₀,j ₁,…,j _t?1 of integers mod n, the GI-graph GI(n;j ₀,j ₁,…,j _t?1) is a (t+1)-valent graph on the vertex set \(\mathbb{Z}_{t} \times\mathbb{Z}_{n}\) , with edges of two kinds:

• an edge from (s,v) to (s′,v), for all distinct \(s,s' \in \mathbb{Z}_{t}\) and all \(v \in\mathbb{Z}_{n}\) ,
• edges from (s,v) to (s,v+j _s ) and (s,v?j _s ), for all \(s \in\mathbb{Z}_{t}\) and \(v \in\mathbb{Z}_{n}\) .

By classifying different kinds of automorphisms, we describe the automorphism group of each GI-graph, and determine which GI-graphs are vertex-transitive and which are Cayley graphs. A GI-graph can be edge-transitive only when t≤3, or equivalently, for valence at most 4. We present a unit-distance drawing of a remarkable GI(7;1,2,3).