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- Probability Approximations and Beyond (Lecture Notes in Statistics - Proceedings)

The Skew-Normal and Related Families. Backward Stochastic Differential Equations. Knowledge Discovery in Spatial Data.

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## Probability Approximations and Beyond - Google Книги

Lecture Notes in Statistics Book How to write a great review. The review must be at least 50 characters long. The title should be at least 4 characters long. Your display name should be at least 2 characters long. At Kobo, we try to ensure that published reviews do not contain rude or profane language, spoilers, or any of our reviewer's personal information.

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Close Report a review At Kobo, we try to ensure that published reviews do not contain rude or profane language, spoilers, or any of our reviewer's personal information. Would you like us to take another look at this review? No, cancel Yes, report it Thanks! Chen Stein's magic method arXiv: Collevecchio General random walk in a random environment defined on Galton-Watson trees Ann. Pollett Connecting deterministic and stochastic metapopulation models J.

Klebaner Escape from the boundary in Markov population processes Adv. Luczak Individual and patch behaviour in structured metapopulation models J. Gan Stein factors for negative binomial approximation in Wasserstein distance Bernoulli 21, Couplings for locally branching epidemic processes J. Nikeghbali Mod-discrete expansions Prob. Reinert Asymptotic behaviour of gossip processes and small world networks Adv. Reinert Approximating the epidemic curve Electr.

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Luczak Central limit approximations for Markov population processes with countably many types Electr. Pollett Total variation approximation for quasi-equilibrium distributions, II Stoch. Luczak A law of large numbers approximation for Markov population processes with countably many types Prob. Torgerson Shot noise processes for clumped infections with time-dependent decay dynamics Biostatistics, Bioinformatics and Biomathematics 2, Interface 8, ; DOI: Reinert The shortest distance in random multi-type intersection graphs Rand.

Pollett Total variation approximation for quasi-equilibrium distributions J. Probability and Mathematical Genetics: Univariate approximations in the infinite occupancy scheme ALEA 6, Janson A functional combinatorial central limit theorem Electronic J. Torgerson Compound processes as models for clumped parasite data Math. Gnedin Small counts in the infinite occupancy scheme Electr.

Luczak Laws of large numbers for epidemic models with countably many types Ann. Torgerson Estimating the Transmission Dynamics of Theileria equi and Babesia caballi in horses Parasitology , Reinert Discrete small world networks Electronic J. Lindvall Translated Poisson approximation for Markov chains J. Gnedin Regenerative compositions in the case of slow variation Stoch. Procs Applics , Xia Normal approximation for random sums Adv. Computing 15, Testing for evolutionary relationship: Chen The permutation distribution of matrix correlation statistics In: Stein's method and applications Eds: Multivariate Poisson-Binomial approximation using Stein's method In: Granovsky Random combinatorial structures: Theory A , Pugliese Asymptotic behaviour of a metapopulation model Ann.

Pugliese Convergence of a structured metapopulation model to Levins's model J. Choi A non-uniform bound for translated Poisson approximation Electronic J.

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Utev Approximating the Reed-Frost epidemic process Stoch. De Boer Procedures for reliable estimation of viral fitness from time series data Proc. B , Cekanavicius Total variation asymptotics for sums of independent integer random variables Ann. Xia Compound Poisson approximation for the distribution of extremes Adv.

Martin Line fitting by rotation: Xia The number of two dimensional maxima Adv. SGSA 33, Chryssaphinou Compound Poisson approximation: Schmidt On Laslett's transform for the Boolean model Adv. Reinert Small worlds Random Struct. Vaggelatou Applications of compound Poisson approximation In: We now use this chance to introduce in Sect.

The basic estimates of exponentially ergodic or decay rate and the principal eigenvalue in different cases are presented. As a consequence, the criteria for the positivity of the rate and the eigenvalue are obtained. The proof of the main result is sketched in Sect. In particular, the basic estimates are refined in Sect. The coincidence of the exponentially decay rate and the corresponding principal eigenvalue is proven in the Appendix for a large class of symmetric Markov processes.

Even though the topologies for these two types of exponential convergence are rather different, but we do have the following result. In our recent study, we go to the opposite direction: We are also going to handle with the non-ergodic case in which 6. However, the totally variational norm in 6. See the Appendix for more details. The state space is E: Then define a function C x as follows: Here and in what follows, the Lebesgue measure dx is often omitted.

The first one has different names: The second one is called scale measure. Neumann at 0 and Dirichlet at N. We call them the first non-trivial or the principal eigenvalue. Certainly, this classification is still meaningful if M or N is infinite. In other 78 M. More seriously, the spectrum of the operator may be continuous for unbounded intervals. This is the reason why we need the L 2 -spectral theory. Then the principal eigenvalues are defined as follows.

Chen rule, we obtain 6. We mention that 6. Clearly, we are in the typical situation of the Sturm—Liouville eigenvalue problem — From which, we learn the general properties of the eigenfunction: Except some very specific cases, the problem is usually not solvable analytically. This leads to the theory of special functions used widely in sciences.

To see this, rewrite 6. To which, several famous mathematicians Weyl, Wiener, Schur et al. After a half century, the basic estimates in the DN-case were finally obtained by several mathematicians, for instance Muckenhoupt [20]. The reason should be now clear why 6. In the DD-case, the problem was begun by Gurka, A better estimate can be done in terms of variational formulas given in [4; Theorem 3. It is surprising that in the more complicated DD- and NN-cases, by adding one more parameter only, we can still obtain a compact expression 6.

Note that these two formulas have the following advantage: Between the nearest neighbors i and j in Zd , there is an interaction. The figure says that in the gray region, the system has a positive principal eigenvalue and so is ergodic; but in the region which is a little away above the curve, 82 M.

Chen the eigenvalue vanishes. The picture exhibits a phase transition. The key to prove Theorem 6. Having onedimensional result at hand, as far as we know, there are at least three different ways to go to the higher or even infinite dimensions: This explains our original motivation and shows the value of a sharp estimate for the leading eigenvalue in dimension one.

The application of the present result to this model should be clear now. Here we sketch its proof. The proof for the first assertion consists mainly of three steps by using three methods: Note that f is regarded as a mimic of the eigenfunction g. The difficulty is that we do not know where the zero-point of g is located. In the new formula 6. This is the advantage of formula 6.

Now, a new problem appears: The dual now we adopted is very simple: Chen we have thus obtained the following identity. The rule mentioned in the remark after Theorem 6. In contract to what we have talked so far, this time we extend the inequalities to the higher dimensional situation. This leads to a use of the capacity since in the higher dimensions, the boundary may be very complicated. Note that we have the universal factor 4 here and the isoperimetric constant BB has a very compact form. We now need to compute the capacity only. The problem is that 6 Basic Estimates of Stability Rate for One-Dimensional Diffusions 85 the capacity is usually not computable explicitly.

Luckily, we are able to compute the capacity for the one-dimensional elliptic operators. The result has a simple expression: It looks strange to have double inverse here. This finishes the proof of the main assertion of Theorem 6. Finally, we 86 M. It is very interesting that we now have an opposite interaction.

We use the main tools coupling and duality developed in the study on IPS to investigate a very classical problem and produce an interesting result. For half-line at least, we have actually an approximating procedure for each of the principal eigenvalues. Refer to [6, 9] and references therein. Moreover, one may approach the whole line by half-lines. As will be seen soon, the resulting bounds are much more complicated, less simple and less symmetry, than those given in Theorem 6.

Let us begin with a simper but effective result. Refer to the remark after the proof of [9; Theorem 8. Actually, the discrete case is even more complex since the eigenfunction can be a simple echelon, not necessarily unimodal. We now introduce a typical application of Theorem 6. Now, by Theorem 6. Here we omit all of the details. The new difficulty of 6. Finally, we mention that the method used here is meaningful for birth—death processes, refer to [9; Lemma 7.

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We have for z: The ratio becomes 2. However, as an illustration of Theorem 6. It is a good chance to discuss the approximating procedure remarked after Corollary 6. Here we consider the lower estimate only. The DN-case is actually a special one of the last example. Note that the present case can be reduced to Example 6.

The author is fortunate to have been invited by Professor Louis Chen three times with financial support to visit Singapore. He is deep appreciative of his continuous encouragement and friendship in the past 30 years. The author acknowledges the organizers of the conferences: Hsu and Dayue Chen, in particular , for their kind invitation and financial support. Appendix The next result is a generalization of [9; Proposition 1. Proof The proof is similar to the ergodic case cf. The technique used here goes back to [17].

Note that here the semigroup is allowed to be subMarkovian. Combining this with a , we complete the proof. Acta Math Sin New Ser 7 1: Chen MF Explicit bounds of the first eigenvalue. Sci China A 43 Chen MF Variational formulas and approximation theorems for the first eigenvalue. Sci China A 44 4: Chen MF a Eigenvalues, inequalities, and ergodic theory. Potential Anal 23 4: Chen MF Spectral gap and logarithmic Sobolev constant for continuous spin systems. Acta Math Sin NS 24 5: Chen MF Speed of stability for birth—death processes. Front Math China 5 3: Trans Amer Math Soc 3: Stoch Proc Appl Feller W On second order differential operators.

Ann Math 2nd Ser 61 1: Fukushima M, Uemura T Capacitary bounds of measures and ultracontracitivity of time changed processes. J Math Pure et Appliquees 82 5: Imbedding theorems of Sobolev type in potential theory. Hardy GH Note on a theorem of Hilbert. Hartman P Ordinary differential equations, 2nd edn. Ann Appl Prob 15 2: Markov Processes Relat Fields 5: Opic B, Kufner A Hardy-type inequalities. Longman, New York Siegmund D The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. Ann Prob 4 6: Wang FY Functional inequalities, semigroup properties and spectrum estimates.

This short article illustrates with an explicit example that in some cases the extreme value index seems to change gradually rather than instantaneously. To this end a moving Hill estimator is introduced. Further a change point analysis and a trend analysis are performed. With this last analysis it is investigated whether a linear trend appears in the extreme value index. Therefore, Swiss Re, one of the leading global reinsurance companies, lists every year the biggest disasters of different types hurricanes, earthquakes, floods, The losses have been calibrated to January 1, As one can see, the types of disaster are quite different: The indices to the quantities X are made for convenience for those disasters arranged in time that did not get a name attached.

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The listed events form themselves a set of extreme values that can be analyzed in its own right. The variable Y represents the losses of the catastrophes in million US-dollars. Note that to avoid empty places and erratic behavior, we rescale the time axis so that one unit represents 5 years. Moreover, and this offers another aspect of Fig. We will investigate this phenomenon in more detail. More specifically, we will focus on the extreme value index of the losses. When using the Hill estimator, we choose k to be the number of data that minimizes the Empirical Mean Square Error. From classical extreme value theory we know that such a large value is associated with Pareto type distributions with a finite mean but with an infinite variance.

The Pareto QQ plot of the data in Fig. When the time is taken into account the data tells a slightly different story. We will study this in three different ways: Note that change point models using a simple change-point and change-point models in a regression context have been studied before. Loader [6] considered a regression model in which the mean function might have a discontinuity at an unknown point and proposes a change-point estimate with confidence regions for the location and the size of the change.

Kim and Siegmund [5] considered likelihood ratio tests to b 8 9 10 11 0. Cox [2] commented on the choice between a simple change-point model without covariates and a regression model with no change-point.

Model selection for changepoint-like problems was also discussed by Siegmund [7]. Unlike these authors we focus on the special features of the extreme value index. Several choices can be made regarding the size of the moving window. One can choose a fixed number of data in each window or a fixed length of the time window. This is illustrated in Fig.

Around each xi , the 10 nearest x-values are selected in a time window. Clearly, the length of the time window is not constant when moving. One can see that a length of two time units, that is 10 years, results in reasonable plots. Around each xi , the x-values lying within a range of two time units are selected in the time window. One can see that the choice of the size of the window and the number of data taken into account in each window has some effect on the results.

But overall an increasing trend seems to prevail. This figure nicely illustrates that the frequency of catastrophes also seems to increase with time. To test whether 15 0 5 10 Fig. Teugels 20 0 2 4 6 8 the extreme value index changes at some point, a likelihood-based test statistic T is used. In that procedure, each point is investigated as a potential change point and the data set is split up into two groups.

The test statistic compares the log-likelihoods of the two groups with the one obtained for the entire data set. A large difference leads to the conclusion that the extreme value index is not constant overall. In mathematical terms this means that the relative excesses over some threshold u x follow approximately a Pareto distribution, i. It is natural to determine the estimated values of these two parameters by maximum likelihood. The outcome of this estimation is illustrated in Fig. The estimators are plotted as a function of k, the number of data taken into account.

Note that the choice of k does not seem to be crucial as the estimations are remarkably stable over the broad set of k-values. The obtained linear trend model summarizes well what can be seen in Fig. The above conclusion is strengthened by looking at an adapted exponential quantile plot. It is somewhat surprising that the largest data point that corresponds to hurricane Katrina Time: Assuming some climatological models for the natural disasters, several climatological studies have detected an increase in severity.

Other reports however threw doubts on such conclusions as the conclusions are based on model assumptions. In this article, we study the severity of the catastrophes by looking at a secondary measurement, namely losses corresponding to major natural disasters. The conclusions, based on a change-point and trend analysis of the extreme value index of the G. Teugels losses from onward, indicate that the catastrophes are becoming more and more severe over time. Cox DR Tests of separate families of hypotheses. Proceedings of 4th Berkeley symposium, vol 1. Dierckx G, Teugels J Change point analysis of extreme values.

Dierckx G Trend analysis of extreme values. Loader CR Change point estimation using nonparametric regression.

## Probability Approximations and Beyond (Lecture Notes in Statistics - Proceedings)

Siegmund D Model selection in irregular problems: Swiss re catastrophe database. Also, Tomonaga said in his series of lectures at Nagoya University, in Nov. We present here a realization of such continuously many, linearly independent vectors with infinite length in terms of white noise. They are defined T. We shall overcome this by using the method of renormalization which is the main topic of this paper.

The system can, therefore, be taken to be the variable system of random functions. To establish the differential and integral calculus on the space of those functionals. We shall, however, not discuss this topic, since there is no direct connection with the renormalization. We note, in fact, two ways, i. Here and also in what follows the constant n! The idea is to have an infinite dimensional analog of the Schwartz space S and the space S of Schwartz distributions: However, it is known that polynomials are not always ordinary generalized white noise functionals, although they are most basic functions.

This can be done by, so-to-speak, renormalization which is going to be explained in what follows. The grade is obviously the degree of a polynomial. It is important to note that differential operators acting on white noise functionals cannot be analogous to the differential operators acting on sure functions. In the case of random functions the definition may seem to be rather simple, however not quite. We have to be careful for the definition. Many reasons suggest us to propose the annihilation operator in place of differential operator, if, in particular, the algebra A is concerned, where the grade is defined.

If the duality is taken into account, it is natural to introduce creation operator. Thus, we have to deal with a non-commutative algebra generated by annihilation and creation operators in line with the study of operator algebra. Each B t tion of random functions. The differential operator should, therefore, dt be quite different from non-random calculus.

Formal observations are given, if permitted: Is it sure variable or random? If the B t is understood to be a multiplication variable, it has to be the sum of creation and annihilation: With these facts in mind we shall propose a method of renormalization starting from the algebra A in a framework as general as possible.

The key role is played by the so-called S -transform in white noise theory. We use the notation in white noise analysis see [4]. The S-transform is defined by the following formula: The inner product of the exponential functional and a polynomial defines a projection, since the exponentials generate the space of test functionals. With these considerations, we define the renormalization via the following proposition.

One thing to take note of is how to interpret the symbol d1t. We define the renormalization using the notation: Linear exponent is easily dealt with, so that we shall be involved only in exponential functionals with quadratic exponent. Then, we have from [4]: For the proof of Theorem 8. The answer is worth to be mentioned. Roughly speaking, f x is not quite equal to a quadratic form, but it is a renormalized quadratic form.

The diagonal term is subtracted off from the ordinary expression of quadratic form.

This fact is related to the modification of the determinants that appear in the expansion of the Fredholm determinant. We now have the theorem: In 1 such a case we have: Hida T Analysis of Brownian functionals. Carleton Math Notes 2. Hida T Brownian motion. Ricciardi LM et al. Mikusinski J On the square of the dirac delta-distribution, Bull. Si SI Introduction to hida distributions. Smithies F Integral equations. Cambridge University Press, Cambridge 9.

Tomonaga S The theory of Spin. Japanese original Japanese translation by Uchiyama T , , Chikuma Chapter 9 M-Dependence Approximation for Dependent Random Variables Zheng-Yan Lin and Weidong Liu Abstract The purpose of this paper is to describe the m-dependence approximation and some recent results obtained by using the m-dependence approximation technique.

In particular, we will focus on strong invariance principles of the partial sums and empirical processes, kernel density estimation, spectral density estimation and the theory on periodogram. Varieties of important theory were proposed and developed in the last century. Among them, the law of large numbers, central limit theorem, the moderate and large deviation, weak and strong invariance principle and lots of their variation dominate the development in limiting theory.

Many classical theorems were first proved under the independent and identically distributed i. The dependence can often arise in practical and statistical problems such as time series analysis, finance and economy. There is a large literature on the properties of mixing random variables and we refer to [17] for an excellent review. Although the mixing condition is general, there are still many random sequences Z. Liu in time series which do not satisfy mixing conditions.