A perturbative approach is used to derive approximations of arbitrary order to estimate high percentiles of sums of positive independent random variables that exhibit heavy tails. Closed-form expressions for the successive approximations are obtained both when the number of terms in the sum is deterministic and when it is random. The zeroth order approximation is the percentile of the maximum term in the sum. Higher orders in the perturbative series involve the right-truncated moments of the individual random variables that appear in the sum. These censored moments are always finite. As a result, and in contrast to previous approximations proposed in the literature, the perturbative series has the same form regardless of whether these random variables have a finite mean or not. The accuracy of the approximations is illustrated for a variety of distributions and a wide range of parameters. The quality of the estimate improves as more terms are included in the perturbative series, specially for higher percentiles and heavier tails.
Besides the heuristic derivation given in @cite_38 and the perturbative expansion proposed in this work, higher order corrections to the single-loss approximation can be derived in at least three different ways: Using the second order asymptotic approximations introduced in @cite_19 @cite_2 @cite_27 @cite_25 , from the asymptotic expansion analyzed in @cite_31 @cite_40 @cite_35 or from asymptotic approximations based on evaluations of @math at different arguments @cite_32 .
@cite_38 @cite_35 @cite_32 @cite_19 @cite_27 @cite_40 @cite_2 @cite_31 @cite_25
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Let X i (i=1,2, …) be a sequence of subexponential positive independent and identically distributed random variables. In this paper, we offer two alternative approaches to obtain higher-order expansions of the tail of and subsequently for ruin probabilities in renewal risk models with claim sizes X i . In particular, these emphasize the importance of the term for the accuracy of the resulting asymptotic expansion of . Furthermore, we present a more rigorous approach to the often suggested technique of using approximations with shifted arguments. The cases of a Pareto type, Weibull and Lognormal distribution for X i are discussed in more detail and numerical investigations of the increase in accuracy by including higher-order terms in the approximation of ruin probabilities for finite realistic ranges of s are given. This monograph presents a unified treatment of single- and multi-user problems in Shannon's information theory where we depart from the requirement that the error probability decays asymptotically in the blocklength. Instead, the error probabilities for various problems are bounded above by a non-vanishing constant and the spotlight is shone on achievable coding rates as functions of the growing blocklengths. This represents the study of asymptotic estimates with non-vanishing error probabilities.In Part I, after reviewing the fundamentals of information theory, we discuss Strassen's seminal result for binary hypothesis testing where the type-I error probability is non-vanishing and the rate of decay of the type-II error probability with growing number of independent observations is characterized. In Part II, we use this basic hypothesis testing result to develop second- and sometimes, even third-order asymptotic expansions for point-to-point communication. Finally in Part III, we consider network information theory problems for which the second order asymptotics are known. These problems include some classes of channels with random state, the multiple-encoder distributed lossless source coding (Slepian-Wolf) problem and special cases of the Gaussian interference and multiple-access channels. Finally, we discuss avenues for further research. We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions. In this paper we construct asymptotic expansions for the tail area of a compound sum, when the distribution of the summands belongs to a class of rapidly varying subexponential distributions. To be more precise, let X?, i > 1, be a sequence of independent random variables, all having the same distribution, F. For any positive integer n the distribution of the partial sums Sn = X + + Xn is the n-fold convolution F*n. We set So = 0 and therefore F*? is defined as the distribution of the point mass at the origin. Let N be a nonnegative integer-valued random variable, independent of the X s. We consider the distribution G of the compound sum Sn, that is E F*N, and we seek an asymptotic expansion for its tail area G = 1 ? G. First-order asymptotic results for G have been obtained by (1979), Cline (1987), and Embrechts (1985). A second-order formula may be found in Griibel (1987) and Omey and Willekens (1987). Compound sums or subordinated distributions arise as distributions of interest in several stochastic models. In insurance risk theory, it models the total claim amount. For a discussion of issues related to random sums and insurance risk, we refer the reader to Embrechts et We apply the exponential weight algorithm, introduced and Littlestone and Warmuth [26]and by Vovk [35]to the problem of predicting a binary sequence almost as well as the best biased coin. We first show that for the case of the logarithmic loss, the derived algorithm is equivalent to the Bayes algorithm with Jeffreys prior, that was studied by Xie and Barron [38]under probabilistic assumptions. We derive a uniform bound on the regret which holds for any sequence. We also show that if the empirical distribution of the sequence is bounded away from 0 and from 1, then, as the length of the sequence increases to infinity, the difference between this bound and a corresponding bound on the average case regret of the same algorithm (which is asymptotically optimal in that case) is only 1 2. We show that this gap of 1 2 is necessary by calculating the regret of the min–max optimal algorithm for this problem and showing that the asymptotic upper bound is tight. We also study the application of this algorithm to the square loss and show that the algorithm that is derived in this case is different from the Bayes algorithm and is better than it for prediction in the worstcase.