Ratio Limit Theorems Research Articles

A ratio limit theorem for (sub) Markov chains on {1,2, …} with bounded jumps

We consider positive matricesQ,indexed by {1,2, …}. Assume that there exists a constant 1L <∞ and sequencesu1< u2< · ··andd1d2< · ··such thatQ(i, j) = 0 wheneveri<ur< ur+L < jori > dr+ L > dr> jfor somer. IfQsatisfies some additional uniform irreducibility and aperiodicity assumptions, then fors> 0,Qhas at most one positives-harmonic function and at most ones-invariant measureµ.We use this result to show that ifQis also substochastic, then it has the strong ratio limit property, that isfor a suitableRand someR–1-harmonic functionfandR–1-invariant measureµ.Under additional conditionsµcan be taken as a probability measure on {1,2, …} andexists. An example shows that this limit may fail to exist ifQdoes not satisfy the restrictions imposed above, even thoughQmay have a minimal normalized quasi-stationary distribution (i.e. a probability measureµfor whichR–1µ = µQ).The results have an immediate interpretation for Markov chains on {0,1,2, …} with 0 as an absorbing state. They give ratio limit theorems for such a chain, conditioned on not yet being absorbed at 0 by timen.

A ratio limit theorem for erased branching Brownian motion

In this paper we study a branching Brownian motion inside a bounded, smooth domain with killing at the boundary. The paths of the process are transformed—roughly speaking—so that all the branches which reach the boundary are totally erased with unit speed for a given time ρ,0⩽ρ⩽∞, starting from the tip of the branch. A limit theorem for the ratio of the number of particles in the erased process and the original one is proved. This may be viewed as a generalisation of a result for Galton-Watson processes due to Athreya and Ney (1972).

Addendum: Limit Theorems for Ratios which are Generated by Random Walks in Homogeneous Spaces

Previous article Next article Addendum: Limit Theorems for Ratios which are Generated by Random Walks in Homogeneous SpacesM. G. ShurM. G. Shurhttps://doi.org/10.1137/1135091PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Addendum: Limit Theorems for Ratios which are Generated by Random Walks in Homogeneous Spaces." Theory of Probability & Its Applications, 35(3), p. 618 Previous article Next article FiguresRelatedReferencesCited byDetails Volume 35, Issue 3| 1991Theory of Probability & Its Applications History Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1135091Article page range:pp. 618-618ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Ratio Limit Theorems for Random Walks in Homogeneous Spaces II

Ratio Limit Theorems for Random Walks in Homogeneous Spaces II

A Ratio Limit Theorem for the Tails of Weighted Sums

Let $\{Z_\lambda; \lambda = 0, \pm 1,\ldots\}$ be i.i.d. random variables which have a density $f$ which satisfies $f(z) \sim Kz^\alpha\exp\{-z^p\}$ as $z \rightarrow \infty$ for some constants $p > 1, K > 0$, and $\alpha$. Further let $q$ be defined by $p^{-1} + q^{-1} = 1$, and let $\{c_\lambda\}$ be constants with $c_\lambda = O(|\lambda|^{-\theta})$ for some $\theta > \max\{1,2/q\}$. Then, e.g., if $f$ is symmetric $\frac{P(\sum c_\lambda Z_\lambda > z + x/z^{p/q})}{P(\sum c_\lambda Z_\lambda > z)} \rightarrow \exp \{-p\|c\|^{-p}_q x\}, \text{as} z \rightarrow \infty$, for $\|c\|_q = (\sum|c_\lambda|^q)^{1/q}$, and similar results are obtained also for nonsymmetric cases, under some mild further smoothness restrictions. In addition, an order bound for $P(\sum c_\lambda Z_\lambda > z)$ itself is obtained, and precise estimates of this quantity are found for the special case of finite sums. In the companion paper [7], the results are crucially used to study extreme values of moving average processes.

Some limit theorems for ratios of transition probabilities

In this paper, infinite dimensional forward convergent stochastic chains have been considered in the framework of [1]. The main result of this paper deals with the observation that the total flow of probability from the C‐states to the T‐states is very small compared to that from the T‐states to the C‐states, if the chain is observed for a sufficiently long time. Some examples have been given to justify the assumptions involved.

Asymptotic independence and individual ratio limit theorems

Let X n be a discrete parameter Markov process on general state space with transition functions P n (x, A). Individual ratio limit theorems are results about convergence as n→∞ of the ratios $$\frac{{P^{n + k} (x,A)}}{{P^n (y,B)}}$$ (1) where k is a fixed integer, A and B are subsets of state space, and x and y are starting points. This paper investigates the relationship of individual ratio limit theorems to various forms of asymptotic independence. When the state space is denumerable it is shown that, subject to standard auxiliary conditions, the much studied Strong Ratio Limit Property is equivalent to a form of asymptotic independence related to mixing conditions of Rosenblatt and Blum, Hanson, and Koopmans.

Ratio limit theorems for branching Ornstein-Uhlenbeck processes

We prove ratio limit theorems for critical ano supercritical branching Ornstein-Uhlenbeck processes. A finite first moment of the offspring distribution { p n } assures convergence in probability for supercritical processes and conditional convergence in probability for critical processes. If even Σp n nlog +log + n< ∞, then almost sure convergence obtains in the supercritical case.

Ratio limit theorems and applications to ergodic theory

Ratio limit theorems and applications to ergodic theory

Access-control disciplines for multi-access communication channels: Reservation and TDMA schemes

Reservation and TDMA schemes are studied for governing the access-control discipline for a network of terminals communicating through a multi-access broadcast channel. A single repeater is employed to allow a fully connected network structure. A channel can be characterized as inducing a Iow propagation-delay value, as for terrestrial radio or fine networks, or as being associated with a higher propagation-delay value, as for a satellite communication channel. A synchronized (slotted) communication medium is considered. Messages are composed of a random number of packets, governed by an arbitrary message-length distribution. The process describing the number of reserved message arrivals within each time frame is assumed to be a sequence of i.i.d, random variables, governed by an arbitrary distribution. (A Poisson arrival stream thus becomes a special case.) The reservation access-control disciplines studied in this paper employ message-switching distributed-control procedures. The performance of each access-control scheme is evaluated according to its delay-throughput function. In particular, schemes are developed to adapt their structure, or protocol, dynamically to the underlying fluctuating network traffic-flow values. A fixed-reservation access-control (FRAC) discipline is studied, employing a fixed periodic pattern of reservation and service periods. The reservation periods are used for the transmission of reservation packets as well as for the integrated service of other groups of network stations. The latter stations can access the channel during these periods, using any proper access control procedure. As a special case, message-delay distributions and moments under a TDMA scheme are obtained. Using dynamic estimates of the underlying message traffic parameters, a dynamic fixed-reservation access-control (DFRAC) scheme is obtained. An analytical technique, which employs a Markov ratio limit theorem, is presented for the derivation of the delay-throughput performance curves of dynamic demand-assignment reservation schemes. To illustrate its application, asynchronous reservation demand-assignment (ARDA) schemes are developed to adapt automatically to the underlying network traffic characteristics. Such schemes establish reservation slots dynamically according to observed network service demands and queue sizes.

Group random-access disciplines for multi-access broadcast channels

A group random-access (GILA) control discipline for a multi-access communication channel is presented and studied. A GILA scheme uses only certain channel time periods to allow some network terminals to transmit their information-bearing packets on a random-access basis. The channel can thus be utilized at other times to grant access to other terminals or other message types, by applying as appropriate group random-access, reservation, or fixed access-control procedures. GILA schemes could also be utilized to provide channel access to various network protocol packets. The average packet delay under a GRA discipline is evaluated by a Markov ratio limit theorem. To stabilize the channel, the GRA procedure is controlled dynamically by a control policy that rejects any newly arriving packets within certain time periods. Studying the associated Markov decision problem, the optimal control policy is characterized as yielding a minimal average packet delay under a prescribed maximal packet probability of rejection. This policy is shown to be represented by a single-threshold scheme. For such a scheme, a threshold value that attains the minimum probability of rejection is shown to exist and to yield a desirable control procedure. Performance curves are presented to demonstrate the delay-throughput characteristics induced by GRA procedures.

A Ratio Limit Theorem for Subterminal Times

Consider a recurrent random walk $\{X_n\}$ with state space $S \subseteqq R^d (d \leqq 2)$. A stopping time $T$ is called subterminal if it satisfies a technical condition which essentially states that it is the first time a path possesses some property which does not depend on how long the process has been running. Suppose $T$ is a subterminal time which can occur only when $\{X_n\}$ is in a bounded set; then under an additional assumption a ratio limit theorem (as $n \rightarrow \infty$) is obtained for $P(T > n \mid X_0 = x) (x\in S)$. The theorem applies in particular to the hitting time of a bounded set with nonempty interior in the general case, and to the hitting time of a bounded set with nonzero Haar measure in the nonsingular case.

Ratio limit theorems

Ratio limit theorems

A ratio limit theorem for a strongly subadditive set function in a locally compact amenable group

A ratio limit theorem for a strongly subadditive set function in a locally compact amenable group

A ratio limit theorem for the finite nonhomogeneous Markov chains

A new simple proof concerning the structure of the tail σ-field of a nonhomogeneous Markov chain as well as a result regarding the asymptotic behaviour of the ratio between the transition probabilities and the absolute probabilities of the chain are given. It is shown that the latter result can be used to characterize the atomic sets of the tail σ-field.

A Ratio Limit Theorem for Approximate Martingales

It has been proved [3, p. 630] that the martingale convergence theorem obtained by Andersen and Jessen [1, p. 5] follows from the classical theory developed by Doob. By using some results of Yosida and Hewitt [9] on finitely additive set functions, Johansen and Karush [7] proved that the identification of the limit function as a derivative in the approach of Andersen and Jessen can be obtained in the general case. In this paper we sharpen the methods of Andersen and Jessen to obtain a ratio limit theorem for “approximate martingales”.

A ratio limit theorem for contraction projections and applications

The similarities between martingale convergence theory and pointwise ergodic theory are now well known [5, 7, 9, 10]. In [5] the similarity between the proofs of the Hopf– Dunford–Schwartz individual ergodic theorem and the martingale convergence theorem is systematically exploited to produce very general ” maximal ergodic ” inequalities for certain sequences of contractions on L1-spaces. A different approach by Rota [10] and Rao [9] leads to a unified convergence theory for martingales and Abel limits. Bishop [1] has produced ” upcrossing” inequalities which yield both theChacon-Ornstein theorem [4] and the martingale convergence theorem.

The 1971 Rietz Lecture Sums of Independent Random Variables--Without Moment Conditions

Analogues of classical limit laws for sums of independent random variables (central limit theorem, laws of large numbers and law of the iterated logarithm) are discussed. We stress results which go through without moment or smoothness assumptions on the underlying distributions. These include (i) estimates for the spread of the distribution of $S_n = \sum_1^nX_i$ in terms of concentration functions (Levy-Rogozin inequality), (ii) comparison of the distribution of $S_n$ on different intervals (ratio limit theorems and Spitzer's theorem for the existence of the potential kernel for recurrent random walk), (iii) study of the set of accumulation points of $S_n/\Upsilon(n)$ for suitable $\Upsilon (n) \uparrow \infty$. Only the following parallel to the law of the iterated logarithm is new: If $X_1, X_2, \cdots$ are independent random variables all with distribution $F, S_n = \sum_1^nX_1, m_n = \operatorname{med} (S_n)$, then there exists a sequence $\{\Upsilon (n)\}$ such that $\Upsilon (n) \rightarrow \infty$ and $- \infty < \lim \inf (S_n - m_n)/\Upsilon (n) < \lim \sup (S_n - m_n)/\Upsilon (n) < \infty$ w.p. 1, if and only if $F$ belongs to the domain of partial attraction of the normal law.

Strong Ratio Limit Theorems for Markov Processes

Let $P$ be a conservative and ergodic Markov operator on $L_\infty(X, \Sigma, m)$ (where $m(X) = 1$). It is proved that if for $A \in \Sigma$ with $m(A) > 0$ and $\mu \ll m$ a finite measure with $\mu(A) > 0 \lim_{n\rightarrow\infty} < \mu, P^{n+1}1_B >/<\mu, P^n 1_A>$ exists for every $B \subset A$, then $P$ has a $\sigma$-finite invariant measure $\lambda$ and there is a sequence $A_k \uparrow X$ with $A_0 = A$ such that for $0 \leqq f, g \in L_\infty(A_k) < \mu, P^n f>/<\mu, P^n g>\rightarrow < \lambda, f>/< \lambda, g>$. The result is used to study the convergence of $< \mu, P^n f>/< \eta, P^n g>$ for $\mu, \eta \ll m$, with applications to Harris processes and strong mixing point transformations. An analogous result for a positive contraction of $C(X)$ is given.

Some ratio limit theorems for Markov operators

Some ratio limit theorems for Markov operators