An inequality for a metric in a random collision process

1975 ◽  
Vol 12 (2) ◽  
pp. 239-247 ◽  
Author(s):  
Shōichi Nishimura
Keyword(s):  
Characteristic Function ◽  
Transition Probabilities ◽  
Initial Distribution ◽  
Radius Of Convergence ◽  
Collision Process ◽  
Kac’S Model

A random collision process with transition probabilities belonging to the same type of distribution is considered. It was proved that if the characteristic function of the initial distribution has a positive radius of convergence, then the sequence {Fn(x)} converges weakly to a distribution G(x) [9]. We define a metric e1[Fn;G], which is analogous to the functional e[f] introduced by Tanaka to Kac's model of Maxwellian gas [10]. We prove that e1[Fn; G] is monotone non-increasing as n → ∞, and also the convergence of the sequence {Fn(x)} under the weaker assumption that for some a > 1 the initial distribution has an ath moment.

An inequality for a metric in a random collision process

1975 ◽  
Vol 12 (02) ◽  
pp. 239-247
Author(s):  
Shōichi Nishimura
Keyword(s):  
Characteristic Function ◽  
Transition Probabilities ◽  
Initial Distribution ◽  
Radius Of Convergence ◽  
Collision Process ◽  
Kac’S Model

A random collision process with transition probabilities belonging to the same type of distribution is considered. It was proved that if the characteristic function of the initial distribution has a positive radius of convergence, then the sequence {Fn(x)} converges weakly to a distributionG(x) [9]. We define a metrice1[Fn;G], which is analogous to the functionale[f] introduced by Tanaka to Kac's model of Maxwellian gas [10]. We prove thate1[Fn;G] is monotone non-increasing asn→ ∞, and also the convergence of the sequence {Fn(x)} under the weaker assumption that for somea >1 the initial distribution has anath moment.

Random collision processes with transition probabilities belonging to the same type of distribution

1974 ◽  
Vol 11 (4) ◽  
pp. 703-714 ◽  
Author(s):  
Shōichi Nishimura
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Probability Distributions ◽  
Collision Process ◽  
Continuous States ◽  
Collision Processes ◽  
Kac’S Model

By analogy with statistical mechanics we consider a random collision process with discrete time wand continuous states x ∈ [0, ∞). We assume three conditions (i), (ii) and (iii), which can be applied to Kac's model of a Maxwellian gas, and show that the sequence of probability distributions converges to a probability distribution using their moments.

Random collision processes with transition probabilities belonging to the same type of distribution

1974 ◽  
Vol 11 (04) ◽  
pp. 703-714 ◽  
Author(s):  
Shōichi Nishimura
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Probability Distributions ◽  
Collision Process ◽  
Continuous States ◽  
Collision Processes ◽  
Kac’S Model

By analogy with statistical mechanics we consider a random collision process with discrete time wand continuous states x ∈ [0, ∞). We assume three conditions (i), (ii) and (iii), which can be applied to Kac's model of a Maxwellian gas, and show that the sequence of probability distributions converges to a probability distribution using their moments.

scholarly journals Probabilistic Investigations on the Explosion of Solutions of the KAC Equation with Infinite Energy Initial Distribution

2008 ◽  
Vol 45 (1) ◽  
pp. 95-106 ◽  
Author(s):  
Eric Carlen ◽  
Ester Gabetta ◽  
Eugenio Regazzini
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Initial Distribution ◽  
Initial Energy ◽  
Infinite Variance ◽  
Borel Sets ◽  
Second Moment ◽  
Quantitative Estimates ◽  
Kac Equation ◽  
Kac’S Model

Gabetta and Regazzini (2006b) have shown that finiteness of the initial energy (second moment) is necessary and sufficient for the solution of the Kac's model Boltzmann equation to converge weakly (Cb-convergence) to a probability measure on R. Here, we complement this result by providing a detailed analysis of what does actually happen when the initial energy is infinite. In particular, we prove that such a solution converges vaguely (C0-convergence) to the zero measure (which is identically 0 on the Borel sets of R). More precisely, we prove that the total mass of the limiting distribution splits into two equal masses (of value ½ each), and we provide quantitative estimates on the rate at which such a phenomenon takes place. The methods employed in the proofs also apply in the context of sums of weighted independent and identically distributed random variables x̃1, x̃2, …, where these random variables have an infinite second moment and zero mean. Then, with Tn := ∑j=1ηnλj,nx̃j, with max1 ≤ j ≤ ηnλj,n → 0 (as n → +∞), and ∑j=1ηnλj,n2 = 1, n = 1, 2, …, the classical central limit theorem suggests that T should in some sense converge to a ‘normal random variable of infinite variance’. Again, in this setting we prove quantitative estimates on the rate at which the mass splits into adherent masses to -∞ and +∞, or to ∞, that are analogous to those we have obtained for the Kac equation. Although the setting in this case is quite classical, we have not uncovered any previous results of a similar type.

scholarly journals Probabilistic Investigations on the Explosion of Solutions of the KAC Equation with Infinite Energy Initial Distribution

2008 ◽  
Vol 45 (01) ◽  
pp. 95-106 ◽  
Author(s):  
Eric Carlen ◽  
Ester Gabetta ◽  
Eugenio Regazzini
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Initial Distribution ◽  
Initial Energy ◽  
Infinite Variance ◽  
Borel Sets ◽  
Second Moment ◽  
Quantitative Estimates ◽  
Kac Equation ◽  
Kac’S Model

Gabetta and Regazzini (2006b) have shown that finiteness of the initial energy (second moment) is necessary and sufficient for the solution of the Kac's model Boltzmann equation to converge weakly (Cb-convergence) to a probability measure onR. Here, we complement this result by providing a detailed analysis of what does actually happen when the initial energy is infinite. In particular, we prove that such a solution converges vaguely (C0-convergence) to the zero measure (which is identically 0 on the Borel sets ofR). More precisely, we prove that the total mass of the limiting distribution splits into two equal masses (of value ½ each), and we provide quantitative estimates on the rate at which such a phenomenon takes place. The methods employed in the proofs also apply in the context of sums of weighted independent and identically distributed random variablesx̃1,x̃2, …, where these random variables have an infinite second moment and zero mean. Then, withTn:= ∑j=1ηnλj,nx̃j, with max1 ≤j≤ ηnλj,n→ 0 (asn→ +∞), and ∑j=1ηnλj,n2= 1,n= 1, 2, …, the classical central limit theorem suggests thatTshould in some sense converge to a ‘normal random variable of infinite variance’. Again, in this setting we prove quantitative estimates on the rate at which the mass splits into adherent masses to -∞ and +∞, or to ∞, that are analogous to those we have obtained for the Kac equation. Although the setting in this case is quite classical, we have not uncovered any previous results of a similar type.

Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

1991 ◽  
Vol 23 (4) ◽  
pp. 683-700 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
State Space ◽  
Spectral Representation ◽  
Transition Probabilities ◽  
Initial Distribution ◽  
Death Process ◽  
Finite Support ◽  
Stationary Distributions ◽  
Absorbing State ◽  
Initial Distributions ◽  
Birth Death

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for birth–death processes.

A Conditional-Probability Approach to the Development of Spatial Pattern

1979 ◽  
Vol 11 (3) ◽  
pp. 313-326 ◽  
Author(s):  
J Odland
Keyword(s):  
Spatial Patterns ◽  
Transition Probabilities ◽  
Random Variables ◽  
Random Variable ◽  
Initial Distribution ◽  
American City ◽  
Spatial Lag ◽  
City Space ◽  
Lag Structure ◽  
Selection Of

A stochastic model for the development of spatial patterns is introduced and used to investigate the process of housing deterioration in an American city. Space is treated as a sequence of discrete locations and a spatial-lag structure is incorporated in the model by defining multivalued random variables whose values indicate conditions at a central location and at a series of spatial lags. The possible combinations of these values define the states of a Markov process, and a description of this process can be obtained by estimating probabilities for the transitions from state to state. Qualitative inferences about the effects of a process on existing spatial patterns are obtained by comparing an initial distribution, for the multivalued random variables, with the limiting distribution implied by the process description. Application of the model involves selection of an appropriate random variable as well as estimation of a set of transition probabilities. Results for Indianapolis in 1977 indicate that the probability of housing deterioration is strongly associated with the presence of deteriorated structures in nearby locations.

The Gillis–Domb–Fisher correlated random walk

1992 ◽  
Vol 29 (04) ◽  
pp. 792-813 ◽  
Author(s):  
Anyue Chen ◽  
Eric Renshaw
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Characteristic Function ◽  
Simple Proof ◽  
Complete Information ◽  
Initial Distribution ◽  
Fundamental Importance ◽  
Correlated Random Walk ◽  
Scientific Disciplines ◽  
Probability Structure

Correlated random walk models figure prominently in many scientific disciplines. Of fundamental importance in such applications is the development of the characteristic function of then-step probability distribution since it contains complete information on the probability structure of the process. Using a simple algebraic lemma we derive then-step characteristic function of the Gillis correlated random walk together with other related results. In particular, we present a new and simple proof of Gillis's conjecture, consider the generalization to the Gillis–Domb–Fisher walk, and examine the effect of including an arbitrary initial distribution.

scholarly journals Reduction Techniques for Model Checking and Learning in MDPs

2017 ◽  
Author(s):  
Suda Bharadwaj ◽  
Stephane Le Roux ◽  
Guillermo Perez ◽  
Ufuk Topcu
Keyword(s):  
Model Checking ◽  
Transition Probabilities ◽  
Initial Distribution ◽  
Decision Processes ◽  
Probabilistic Model Checking ◽  
Model Learning ◽  
Reduction Techniques ◽  
Learning Techniques ◽  
Markov Decision ◽  
Target Set

Omega-regular objectives in Markov decision processes (MDPs) reduce to reachability: find a policy which maximizes the probability of reaching a target set of states. Given an MDP, an initial distribution, and a target set of states, such a policy can be computed by most probabilistic model checking tools. If the MDP is only partially specified, i.e., some prob- abilities are unknown, then model-learning techniques can be used to statistically approximate the probabilities and enable the computation of the de- sired policy. For fully specified MDPs, reducing the size of the MDP translates into faster model checking; for partially specified MDPs, into faster learning. We provide reduction techniques that al- low us to remove irrelevant transition probabilities: transition probabilities (known, or to be learned) that do not influence the maximal reachability probability. Among other applications, these reductions can be seen as a pre-processing of MDPs before model checking or as a way to reduce the number of experiments required to obtain a good approximation of an unknown MDP.

Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

1991 ◽  
Vol 23 (04) ◽  
pp. 683-700 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
State Space ◽  
Spectral Representation ◽  
Transition Probabilities ◽  
Initial Distribution ◽  
Death Process ◽  
Finite Support ◽  
Stationary Distributions ◽  
Absorbing State ◽  
Initial Distributions ◽  
Birth Death

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for birth–death processes.

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