Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

With the aim of modelling key stylized features of observational series from finance and turbulence a number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed. Ornstein-Uhlenbeck type processes, superpositions of such processes and stochastic volatility models in one and more dimensions are considered in particular, and some discussion is given of the feasibility of making likelihood inference for these models.

Processes of normal inverse Gaussian type

Let S

 n
 =X
1+X
2+⋯+X

 n
 be the sum of i.i.d.r.v.-s, EX
1=0, EX

 1
 2
 =1, and let T

 n
 = Y
1+Y
2+⋯+Y

 n
 be the sum of independent standard normal variables. Strassen proved in [14] that if X
1 has a finite fourth moment, then there are appropriate versions of S

 n
 and T

 n
 (which, of course, are far from being independent) such that ¦S

 n
 -T

 n
 ¦=O(n
1/4(log n)1/1(log log n)1/4) with probability one. A theorem of Bartfai [1] indicates that even if X
1 has a finite moment generating function, the best possible bound for any version of S

 n
 , T

 n
 is O(log n). In this paper we introduce a new construction for the pair S

 n
 , T

 n
 , and prove that if X
1 has a finite moment generating function, and satisfies condition i) or ii) of Theorem 1, then ¦S

 n
 -T

 n
 ¦=O(log n) with probability one for the constructed S

 n
 , T

 n
 . Our method will be applicable for the approximation of sample DF., too.

An Approximation of Partial Sums of Independent RV's, and the Sample DF. II

A comprehensive account of economic size distributions around the world and throughout the years In the course of the past 100 years, economists and applied statisticians have developed a remarkably diverse variety of income distribution models, yet no single resource convincingly accounts for all of these models, analyzing their strengths and weaknesses, similarities and differences. Statistical Size Distributions in Economics and Actuarial Sciences is the first collection to systematically investigate a wide variety of parametric models that deal with income, wealth, and related notions. Christian Kleiber and Samuel Kotz survey, compliment, compare, and unify all of the disparate models of income distribution, highlighting at times a lack of coordination between them that can result in unnecessary duplication. Considering models from eight languages and all continents, the authors discuss the social and economic implications of each as well as distributions of size of loss in actuarial applications. Specific models covered include: Pareto distributions Lognormal distributions Gamma-type size distributions Beta-type size distributions Miscellaneous size distributions Three appendices provide brief biographies of some of the leading players along with the basic properties of each of the distributions. Actuaries, economists, market researchers, social scientists, and physicists interested in econophysics will find Statistical Size Distributions in Economics and Actuarial Sciences to be a truly one-of-a-kind addition to the professional literature.

Statistical Size Distributions in Economics and Actuarial Sciences

Let X,X1,X2,… be a sequence of nondegenerate i.i.d. random variables with zero means. In this paper we show that a self-normalized version of Donsker's theorem holds only under the assumption that X belongs to the domain of attraction of the normal law. A thus resulting extension of the arc sine law is also discussed. We also establish that a weak invariance principle holds true for self-normalized, self-randomized partial sums processes of independent random variables that are assumed to be symmetric around mean zero, if and only if max1≤j≤n|Xj|/Vn→P0, as n→∞, where V2n=∑nj=1X2j.

https://openresearch-repository.anu.edu.au/bitstream/1885/733712354/2/01_Csorgo_Donsker%27s_Theorem_2003.pdf

Donsker's theorem for self-normalized partial sums processes

Publisher Summary Changepoint problems have originally arisen in the context of quality control, where one typically observes the output of a production line and would wish to signal deviation from an acceptable average output level while observing the data. When one observes a random process sequentially and stops observing at a random time of detecting change, then one speaks of a sequential procedure. Otherwise, it is observed that a large finite sequence for the sake of determining possible change during the data collection. Such procedures are described in terms of asymptotic results, and can be called as nonsequential procedures. Sequential and nonsequential procedures are usually based on parametric or nonparametric models for changepoint problems, allowing at most one change (AMOC) or, possibly, more than one change. This chapter focuses on the nonparametric AMOC setting and discusses non-sequential nonparametric AMOC procedures. A large number of nonparametric and parametric modelling of AMOC problems result in the same test statistic, general rank statistics with quantile and Wilcoxon type scores whose asymptotics are described in terms of a two-time parameter stochastic process, U-statistics type processes which are considered for the nonparametric AMOC problem, and detect change in the intensity parameter of a renewal process.

20 Nonparametric methods for changepoint problems

A new method is developed to produce strong laws of invariance principle without making use of the Skorohod representation. As an example, it will be proved that 
$${{\mathop {\lim }\limits_{n \to \infty } \left( {S_n - W(n)} \right)} \mathord{\left/ {\vphantom {{\mathop {\lim }\limits_{n \to \infty } \left( {S_n - W(n)} \right)} {n^{{1 \mathord{\left/ {\vphantom {1 {6 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {6 + \varepsilon }}} }}} \right. \kern-\nulldelimiterspace} {n^{{1 \mathord{\left/ {\vphantom {1 {6 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {6 + \varepsilon }}} }} = 0$$

 with probability 1, for any g3>0, where S

 n
 =X
1 + ... +X

 n
 , X

 i
 is a sequence of i.i.d.r.v.'s with P(X

 i
 <t)=F(t), and F(t) is a distribution function obeying (i), (ii) and W(n) is a suitable Wiener-process. Strassen in [1], proved (under weaker conditions): 
$$S_n - W\left( n \right) = O\left( {\sqrt[4]{{n{\text{ log log }}n}}\sqrt {{\text{log }}n} {\text{ }}} \right)$$

 with probability one. He conjectured that if 
$$S_n - W\left( n \right) = o\left( {\sqrt[4]{{n{\text{ log log }}n}}\sqrt {{\text{log }}n} {\text{ }}} \right)$$

 then F(x) = Φ(x) where Φ(.) is the unit normal distribution function. (See also [2], [6] and [7].) Our result above is a negative answer to this question.

A new method to prove strassen type laws of invariance principle. 1

In our previous sections we have established weak convergence of mean residual life, total time of test, Lorenz and Goldie processes to appropriate Gaussian processes. Apart from a few special cases (cf. Section 8), these limiting processes are functions of the underlying distributions. Consequently when testing for statistical hypotheses for example, one would have to compute the resulting limiting distribution for each F of interest. The same is true when trying to use our results for constructing confidence bands for the theoretical functionals of the said empirical processes. This type of problems can be solved by adapting the bootstrap method, proposed by Efron (1979, 1982), to the present situation. As we will see, the bootstrap method is simply a Monte Carlo simulation determined by the given observations.

/pdf/bootstrapping-empirical-functions-45hcfn8nzu.pdf

Bootstrapping Empirical Functions

Let X1, X2...be a sequence of independent identically distributed random variables with a smooth density function f. We obtain central limit theorems for $\int\limits_\infty ^\infty {|f_n (t) - f_{} (t)|^p d\mu (t)} ,{\text{ }}1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } p < \infty $, where fn is a kernel estimate of f and μ is a measure on the Borel sets of ℝ.

Miklós Csörgő

Papers

Donsker's theorem for self-normalized partial sums processes

20 Nonparametric methods for changepoint problems

A new method to prove strassen type laws of invariance principle. 1

Bootstrapping Empirical Functions

Central limit theorems for Lp-norms of density estimators