### Discretization of Processes: 67 (Stochastic Modelling and Applied Probability)

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## Monte Carlo Methods and Applications

Very minimal wear and tear. See all condition definitions - opens in a new window or tab Read more about the condition. About this product. Discretization of Processes, Paperback by Jacod, Jean; Protter, Philip, ISBN , ISBN , Like New Used, Free shipping in the US Using classic statistical tools, this book synthesizes ten years of research to establish a sohisticated theory of how to go about estimating not just scalar parameters of a proposed model, but also the underlying structure of the model itself. Shipping and handling. The seller has not specified a shipping method to Germany.

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## Stochastic Processes: From Applications to Theory

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## AMS eBooks: Memoirs of the American Mathematical Society

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- Stochastic Processes: From Applications to Theory - CRC Press Book?
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Other offers may also be available. Interest will be charged to your account from the purchase date if the balance is not paid in full within 6 months. Minimum monthly payments are required. Subject to credit approval. For example, if a point process, other than a Poisson, has its points randomly and independently displaced, then the process would not necessarily be a Poisson point process.

However, under certain mathematical conditions for both the original point process and the random displacement, it has been shown via limit theorems that if the points of a point process are repeatedly displaced in a random and independent manner, then the finite-distribution of the point process will converge weakly to that of a Poisson point process.

Similar convergence results have been developed for thinning and superposition operations [] that show that such repeated operations on point processes can, under certain conditions, result in the process converging to a Poisson point processes, provided a suitable rescaling of the intensity measure otherwise values of the intensity measure of the resulting point processes would approach zero or infinity. Such convergence work is directly related to the results known as the Palm—Khinchin [g] equations, which has its origins in the work of Conny Palm and Aleksandr Khinchin , [] and help explains why the Poisson process can often be used as a mathematical model of various random phenomena.

The Poisson point process can be generalized by, for example, changing its intensity measure or defining on more general mathematical spaces. These generalizations can be studied mathematically as well as used to mathematically model or represent physical phenomena. For mathematical models the Poisson point process is often defined in Euclidean space, [1] [46] but has been generalized to more abstract spaces and plays a fundamental role in the study of random measures, [] [] which requires an understanding of mathematical fields such as probability theory, measure theory and topology.

In general, the concept of distance is of practical interest for applications, while topological structure is needed for Palm distributions, meaning that point processes are usually defined on mathematical spaces with metrics.

### Alle Bücher der Reihe Stochastic Modelling and Applied Probability

The process is named after David Cox who introduced it in , though other Poisson processes with random intensities had been independently introduced earlier by Lucien Le Cam and Maurice Quenouille. For example, if the logarithm of the intensity measure is a Gaussian random field , then the resulting process is known as a log Gaussian Cox process. Cox point processes exhibit a clustering of points, which can be shown mathematically to be larger than those of Poisson point processes.

The generality and tractability of Cox processes has resulted in them being used as models in fields such as spatial statistics [] and wireless networks. For a given point process, each random point of a point process can have a random mathematical object, known as a mark , randomly assigned to it.

These marks can be as diverse as integers, real numbers, lines, geometrical objects or other point processes. If a general point process is defined on some mathematical space and the random marks are defined on another mathematical space, then the marked point process is defined on the Cartesian product of these two spaces. For a marked Poisson point process with independent and identically distributed marks, the marking theorem [] [] states that this marked point process is also a non-marked Poisson point process defined on the aforementioned Cartesian product of the two mathematical spaces, which is not true for general point processes.

The compound Poisson point process or compound Poisson process is formed by adding random values or weights to each point of Poisson point process defined on some underlying space, so the process is constructed from a marked Poisson point process, where the marks form a collection of independent and identically distributed non-negative random variables.

In other words, for each point of the original Poisson process, there is an independent and identically distributed non-negative random variable, and then the compound Poisson process is formed from the sum of all the random variables corresponding to points of the Poisson process located in some region of the underlying mathematical space. The failure process with the exponential smoothing of intensity functions FP-ESI is an extension of the nonhomogeneous Poisson process.

The intensity function of an FP-ESI is an exponential smoothing function of the intensity functions at the last time points of event occurrences and outperforms other nine stochastic processes on 8 real-world failure datasets when the models are used to fit the datasets [] , where the model performance is measured in terms of AIC Akaike information criterion and BIC Bayesian information criterion. From Wikipedia, the free encyclopedia. It has been suggested that Complete spatial randomness be merged into this article.

Discuss Proposed since August Main article: Point process notation. Main article: Point process operation. Main article: Mapping theorem point process.

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Main article: Cox process. See also: Markov renewal process. Kingman 17 December Poisson Processes. Clarendon Press.

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Babu and E. Spatial point processes in astronomy. Journal of statistical planning and inference , 50 3 —, Othmer, S.

Dunbar, and W. Models of dispersal in biological systems. Journal of mathematical biology , 26 3 —, Spatial point processes, with applications to ecology. Connor and B. Three nonhomogeneous poisson models for the probability of basaltic volcanism: application to the yucca mountain region, nevada.

Bulletin of the Seismological Society of America. Studies in astronomical time series analysis. The Astrophysical Journal , 1 , Aghion and P. A Model of Growth through Creative Destruction.