Abstract. This chapter provides a survey of the main methods in non-uniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.

Let f be a density on R&quot;, and let f, be the kernel estimate off, fn(x) _ ( nh d)- ' ~ = 1 K((x- X1)lh) where h = h n is a sequence of positive numbers, and K is an absolutely integrable function with f K(x) dx =1. Let J, = f l f,(x)- f (x) ( dx. We show that when limnh = 0 and limn nh d = oo, then for every e> 0 there exist constants r, no> 0 such that P(Jn> e) < _ exp(-rn), n? no. Also, when J,-p 0 in probability as n--p oo and K is a density, then lim nh = 0 and limnnh d = oo. 1. Introduction. The

Probabilistic optimization algorithms that mimic the process of biological evolution are usually subsumed under the term `evolutionary algorithms.' This work extends the convergence theory of evolutionary algorithms by presenting a sufficient convergence condition for those evolutionary algorithms that do not necessarily generate a sequence of feasible points such that the associated objective function values decrease monotonically to the global minimum. Moreover, it is investigated how fast the sequence of objective function values generated by an evolutionary algorithm approaches the minimum of strongly convex functions in a probabilistic sense. The theoretical analysis presented here distinguishes from related studies in three points: First, it does not require advanced calculus. Second, only the first partial derivatives of the objective function are assumed to exist. Third, one obtains sharp bounds on the convergence rates for a class of functions being a superset of the class of ...

The standard choice for mutating an individual of an evolutionary algorithm with continuous variables is the normal distribution; however other distributions, especially some versions of the multivariate Cauchy distribution, have recently gained increased popularity in practical applications. Here the extent to which Cauchy mutation distributions may affect the local convergence behavior of evolutionary algorithms is analyzed. The results show that the order of local convergence is identical for Gaussian and spherical Cauchy distributions, whereas nonspherical Cauchy mutations lead to slower local convergence. As a by--product of the analysis some recommendations for the parametrization of the self-adaptive step size control mechanism can be derived.

When evaluating the performance of computer systems, often uncertainties or variabilities in service demands may be observed. Applying well known mean value analysis (MVA) for single- or multiclass queueing network models of such systems is inappropriate and ineffective, because these models fail to represent variations within a class. This paper proposes to use histograms for characterizing model parameters that are associated with uncertainty or variability and presents an adaptation of the single class MVA algorithm, which traditionally accepts single (mean) values for service demands, so that one or more input parameters can be specified as a histogram. The adapted algorithm generates a histogram output for the performance measures, thus providing a more detailed information (e.g. percentile values) than the mean values obtained from conventional MVA. The proposed technique is demonstrated on selected examples in different problem domains. It is shown, that the computational comple...

A sample X,, •.., X, ~ of i.i.d. R d-valued random vectors with common density f is used to construct the density estimate fn(x) _ ( 1/n) ~n, HndK((x- Xi)/Hm), where K is a given density on Rd, and the H's are positive functions of n, i and X,, • • • , X „ (but not of x). The H, 's can be thought of as locally adapted smoothing parameters. We give sufficient conditons for the weak convergence to 0 of f I fn- f I for all f. This is illustrated for the estimate of Breiman, Meisel and Purcell (1977).

A conventional analytic model used for evaluating the performance of computer and communication systems accepts single values as model inputs and computes a single value for each performance measure of interest. However uncertainties regarding parameter values exist in different situations such as during early stages of system design. Although the clients in a system are statistically identical factors such as the time of the day and the current size of the data files can introduce variabilities in service demands for the server devices. Existence of uncertainties or variabilities in service demands makes the use of a single mean value for each model parameter inappropriate causing the conventional modelling approach to become ineffective. This paper proposes to use histograms for characterizing one or more model parameters that are associated with such uncertainty or variability and demonstrates its application with separable queueing network models. A histogram consists of a number o...

Let f, be the kernel density estimate with arbitrary smoothing factor h and arbitrary (absolutely integrable) kernel K, based upon an i.i.d. sample of size n drawn from a density f. It is shown that ) inf E([If,-fl > , h,K,f " 52V and that liminf// inf E([If.-fl)>.

For any sequence of empirical probability measures {p) on the Borel sets of the real line and any 6> 0, there exists a singular continuous probability measure µ such that inf sup µ n (A)- µ (A) I> 2- 6 almost surely. n A We consider a probability measure µ on the Borel sets of the real line, from which we draw an i.i.d. sample X 1,..., X,. An empirical probability measure µ n is a probability measure on the same Borel sets and for a fixed set A, µ(A) is a measurable function of the data X 1,..., X n. In particular, we are interested in the total variation Tn 4 suPll-t n (A) A-(A)I, where the supremum is over all the Borel sets. By considering suprema over left infinite intervals only, it is easy to see that Tn> sup x IFn(x)- F(x)I, the Glivenko-Cantelli norm, where Fn and F are the distribution functions corresponding to µ and µ, respectively. The standard empirical measure, defined by 0 1 µn ( A)- ~ I[Xl E AJ, is atomic in nature. Hence, whenever µ is continuous, we have Tn =- 1 almost surely for all n. This is in stark contrast with the Glivenko-Cantelli norm, which is known to converge to zero almost surely as n-- ~ oo (by the Glivenko--Cantelli theorem). If µ is atomic, it is quite obvious that Tn-- ~ 0 almost surely as n-- ~ 0. In order for Tn to be small when µ is nonatomic, we should not use the standard empirical measure. For example, for absolutely continuous µ (with density f), SchefF 's lemma [SchefF (1947) states that Tn n n L=i 2 fIfn- f l,

Almost thirty years ago, D.G. Kendall [8] considered diffusions of shape induced by independent Brownian motions in Euclidean space. In this paper, we consider a different class of diffusions of shape, induced by the projections of a randomly rotating labelled ensemble. In particular, we study diffusions of shapes induced by projections of planar triangular configurations of labelled points onto a fixed straight line. That is, we consider the process in Σ 3 1 (the shape space of labelled triads in R 1) that results from extracting the “shape information ” from the projection of a given labelled planar triangle, as this evolves under the action of a Brownian motion in SO(2). We term the thus defined diffusions Radon diffusions and derive explicit stochastic differential equations and stationary distributions. The latter belong to the family of angular central Gaussian distributions. In addition, we discuss how these Radon diffusions and their limiting distributions are related to the shape of the initial triangle, and explore whether the relationship is bijective. The triangular case is then used as a pivot for the study of processes in Σ k 1 arising from projections of an arbitrary number k of labelled points on the plane. Finally, we discuss the problem of Radon diffusions in general shape-space Σ k n. Keywords: Single particle Biophysics; circular Brownian motion; D.G. Kendall’s shape theory; angular central Gaussian distribution; integral geometry; stochastic geometry; random processes of geometrical