Bounds for some general sums of random variables

Rogers and Shi (1995) have used the technique of conditional expectations to derive approximations for the distribution of a sum of lognormals. In this paper we extend their results to more general sums of random variables. In particular we study sums of functions of dependent random variables that are multivariate normally distributed and also derive results for sums of functions of dependent random variables from the additive exponential dispersion family. The usefulness of our results for practical applications is also discussed.     

Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables

In this paper we extend some results about the probability that the sum of n dependent subexponential random variables exceeds a given threshold u. In particular, the case of non-identically distributed and not necessarily positive random variables is investigated. Furthermore we establish criteria how far the tail of the marginal distribution of an individual summand may deviate from the others so that it still influences the asymptotic behavior of the sum. Finally we explicitly construct a dependence structure for which, even for regularly varying marginal distributions, no asymptotic limit of the tail of the sum exists. Some explicit calculations for diagonal copulas and t-copulas are given. Dominik Kortschak was supported by the Austrian Science Fund Project P18392.     

Decompounding random sums: a nonparametric approach

A compound distribution is the distribution of a random sum, which consists of a random number N of independent identically distributed summands, independent of N. Buchmann and Grübel (Ann Stat 31:1054–1074, 2003) considered decompounding a compound Poisson distribution, i.e. given observations on a random sum when N has a Poisson distribution, they constructed a nonparametric plug-in estimator of the underlying summand distribution. This approach is extended here to that of general (but known) distributions for N. Asymptotic normality of the proposed estimator is established, and bootstrap methods are used to provide confidence bounds. Finally, practical implementation is discussed, and tested on simulated data. In particular we show how recursion formulae can be inverted for the Panjer class in general, as well as for an example drawn from the Willmot class.     

The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2^{−n(n−1)/4}.     

Asymptotic expansions for potential functions of i.i.d. random fields

Summary For sums of finite range potential functions of an iid random field we derive the validity of formal expansions of length two. Under standard conditions, formal expansions are valid if and only if the characteristic functions of the sum converge to zero for all nonzero frequency parameters. If this convergence fails, the distribution of the sum can be approximated by a mixture of lattice distributions. The result applies to m-dependent random fields generated by independent random variables.     

On sums of two counter-monotonic risks

In risk management, capital requirements are most often based on risk measurements of the aggregation of individual risks treated as random variables. The dependence structure between such random variables has a strong impact on the behavior of the aggregate loss. One finds an extensive literature on the study of the sum of comonotonic risks but less, in comparison, has been done regarding the sum of counter-monotonic risks. A crucial result for comonotonic risks is that the Value-at-risk and the Tail Value-at-risk of their sum correspond respectively to the sum of the Value-at-risk and Tail Value-at-risk of the individual risks. In this paper, our main objective is to derive such simple results for the sum of counter-monotonic risks. To do so, we examine separately different contexts in the class of bivariate strictly continuous distributions for which we obtain closed-form expressions for the Value-at-risk and Tail Value-at-risk of the sum of two counter-monotonic risks. The expressions for the subadditive Tail Value-at risk allow us to quantify the maximal diversification benefit. Also, our findings allow us to analyze the tail of the distribution of the sum of two identically subexponentially distributed counter-monotonic random variables.     

On the probabilistic analysis of an approximation algorithm for solving the <Emphasis Type="Italic">p</Emphasis>-median problem

In order to solve the location problem in the p-median form we present an approximation algorithm with time complexity O(n ₂) and the results of its probabilistic analysis. The input data are defined on a complete graph with distances between the vertices expressed by the independent random variables with the same uniform distribution. The value of the objective function produced by the algorithm amounts to a certain sum of the random variables that we analyze basing on an estimate for the probabilities of large deviations of these sums. We use a limit theorem in the form of the Petrov inequalities, taking into account the dependence of the random variables in the sum. The probabilistic analysis yields some estimates for the relative error and the failure probability of our algorithm, as well as conditions for its asymptotic exactness.     

相互独立的m值随机变量和的极限分布定理

利用ｍ值随机变量的特征函数,在一定条件下,得到了相互独立的ｍ值随机变量和的极限分布均匀的充要条件;再结合无穷乘积的有关性质,给出了相互独立的ｍ值随机变量和极限分布均匀分布的充分条件,特别当ｍ为素数Ｐ时,所得的充分条件易于验证,且不难满足。     

Lower-Bound Estimates for Poisson-Type Approximations

Let S = w ₁ S ₁ + w ₂ S ₂ + ⋯ + w _N S _N . Here S _j is a sum of identically distributed random variables with weight w _j > 0. We consider the cases where S _j is a sum of independent random variables, the sum of independent lattice variables, or has the Markov binomial distribution. Apart from the general case, we investigate the case of symmetric random variables. Distribution of S is approximated by a compound Poisson distribution, by a second-order asymptotic expansion, and by a signed exponential measure. Lower bounds for the accuracy of approximations in uniform metric are established. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 501–524, October–December, 2005.     

From moments of sum to moments of product

We provide an identity that relates the moment of a product of random variables to the moments of different linear combinations of the random variables. Applying this identity, we obtain new formulae for the expectation of the product of normally distributed random variables and the product of quadratic forms in normally distributed random variables. In addition, we generalize the formulae to the case of multivariate elliptically distributed random variables. Unlike existing formulae in the literature, our new formulae are extremely efficient for computational purposes.     

Approximating Probability Distributions Using Small Sample Spaces

We formulate the notion of a "good approximation" to a probability distribution over a finite abelian group ?. The quality of the approximating distribution is characterized by a parameter ɛ which is a bound on the difference between corresponding Fourier coefficients of the two distributions. It is also required that the sample space of the approximating distribution be of size polynomial in and 1/ɛ. Such approximations are useful in reducing or eliminating the use of randomness in certain randomized algorithms. We demonstrate the existence of such good approximations to arbitrary distributions. In the case of n random variables distributed uniformly and independently over the range , we provide an efficient construction of a good approximation. The approximation constructed has the property that any linear combination of the random variables (modulo d) has essentially the same behavior under the approximating distribution as it does under the uniform distribution over . Our analysis is based on Weil's character sum estimates. We apply this result to the construction of a non-binary linear code where the alphabet symbols appear almost uniformly in each non-zero code-word. Received: September 22, 1990/Revised: First revision November 11, 1990; last revision November 10, 1997     

What portion of the sample makes a partial sum asymptotically stable or normal?

Summary Let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0<2 be given. We show that if at each stage n a number k _n depending on n of the lower and upper order statistics are removed from the n-th partial sum of the given random variables then under appropriate conditions on k _n the remaining sum can be normalized to converge in distribution to a standard normal random variable. A further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour. Our main tool is a new Brownian bridge approximation to the uniform empirical process in weighted supremum norms.Work done while visiting the Bolyai Institute, Szeged University, partially supported by a University of Delaware Research Foundation Grant     

Limit theorems of general functions of independent and identically distributed random variables

In this paper we derive limit theorems of some general functions of independent and identically distributed random variables. A stability property is used to derive the limit theory for general functions. A procedure followed in de Haan (1976) and Leadbetter et al. (1983) is used to prove the main result. The limit theorems for the maximum, minimum and sum of fixed sample sizes and random sample sizes are derived as special cases of the main result.     

On the convolution of Mittag–Leffler distributions and its applications to fractional point processes

We obtain the distribution of the sum of independent Mittag–Leffler (ML) random variables which are not necessarily identically distributed. Firstly we discuss the corresponding known result for independent and identically distributed ML random variables which follows as a special case of our result. Some applications of the obtained result to fractional point processes are also discussed.     

Random dense bipartite graphs and directed graphs with specified degrees

Let s and t be vectors of positive integers with the same sum. We study the uniform distribution on the space of simple bipartite graphs with degree sequence s in one part and t in the other; equivalently, binary matrices with row sums s and column sums t . In particular, we find precise formulae for the probabilities that a given bipartite graph is edge‐disjoint from, a subgraph of, or an induced subgraph of a random graph in the class. We also give similar formulae for the uniform distribution on the set of simple directed graphs with out‐degrees s and in‐degrees t . In each case, the graphs or digraphs are required to be sufficiently dense, with the degrees varying within certain limits, and the subgraphs are required to be sufficiently sparse. Previous results were restricted to spaces of sparse graphs. Our theorems are based on an enumeration of bipartite graphs avoiding a given set of edges, proved by multidimensional complex integration. As a sample application, we determine the expected permanent of a random binary matrix with row sums s and column sums t . © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Formula for the Laplace transform of the projection of a distribution on the positive semiaxis and some of its applications

We obtain a formula for the Laplace transform of the restriction of an arbitrary probability distribution on the positive semiaxis in the form of a Cauchy-type integral in infinite limits of the characteristic function of this distribution. Using this result and the estimates of the concentration function of the sum of independent random variables, we derive a representation for the Laplace transform of the restriction of the harmonic measure on the positive semiaxis. In conclusion, we present an estimate of the lower ladder height distribution for the case in which the distribution of the value of the jump in a random walk is normal.     

Rate of convergence of the price of European option on a market for which the jump of stock price is uniformly distributed over an interval

We consider a model of market for which the jump of the stock price is uniformly distributed over a certain symmetric interval. By using the theorem on asymptotic expansions of the distribution function of the sum of independent identically distributed random variables, we determine the rate of convergence of fair prices for the European options. It is shown that, in the prelimit model, there exists a martingale measure on the market such that the rate of convergence of the prices of European options to the Black-Scholes price has an order of 1/n ^1/2. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1075–1086, August, 2008.     

Statistical modeling for discrete patterns in a sequence of exchangeable trials

This paper proposes a new method for constructing a sequence of infinitely exchangeable uniform random variables on the unit interval. For constructing the sequence, we utilize a Pólya urn partially. The resulting exchangeable sequence depends on the initial numbers of balls of the Pólya urn. We also derive the de Finetti measure for the exchangeable sequence. For an arbitrarily given one-dimensional distribution function, we generate sequences of exchangeable random variables with the one-dimensional marginal distribution by transforming the exchangeable uniform sequences with the inverse function of the distribution function. Among them we mainly investigate sequences of exchangeable discrete random variables. They differ from the well-known exchangeable sequence generated only by the Pólya urn scheme. Some examples are also given as applications of the results to exact distributions of some statistics based on sequences of exchangeable trials. Further, from the above exchangeable uniform sequence we construct partial or Markov exchangeable sequences. We also provide numerical examples of statistical inference based on the exchangeable and Markov exchangeable sequences.     

On the distribution of the first exit time and overshoot in a two-sided boundary crossing problem

We consider a random walk generated by a sequence of independent identically distributed random variables. We assume that the distribution function of a jump of the random walk equals an exponential polynomial on the negative half-axis. For double transforms of the joint distribution of the first exit time from an interval and overshoot, we obtain explicit expressions depending on finitely many parameters that, in turn, we can derive from the system of linear equations. The principal difference of the present article from similar results in this direction is the rejection of using factorization components and projection operators connected with them.     

On the DFR property of the compound geometric distribution with applications in risk theory

In 1988, Shanthikumar proved that the sum of a geometrically distributed number of i.i.d. DFR random variables is also DFR. In this paper, motivated by the inverse problem, we study monotonicity properties related to defective renewal equations, and obtain that if a compound geometric distribution is DFR, then the random variables of the sums are NWU (a class that contains DFR). Furthermore, we investigate some applications of risk theory and give a characterization of the exponential distribution.