Poisson Random Sums Research Articles

Analytic and Asymptotic Properties of the Generalized Student and Generalized Lomax Distributions

Analytic and asymptotic properties of the generalized Student and generalized Lomax distributions are discussed, with the main focus on the representation of these distributions as scale mixtures of the laws that appear as limit distributions in classical limit theorems of probability theory, such as the normal, folded normal, exponential, Weibull, and Fréchet distributions. These representations result in the possibility of proving some limit theorems for statistics constructed from samples with random sizes in which the generalized Student and generalized Lomax distributions are limit laws. An overview of known properties of the generalized Student distribution is given, and some simple bounds for its tail probabilities are presented. An analog of the ‘multiplication theorem’ is proved, and the identifiability of scale mixtures of generalized Student distributions is considered. The normal scale mixture representation for the generalized Student distribution is discussed, and the properties of the mixing distribution in this representation are studied. Some simple general inequalities are proved that relate the tails of the scale mixture with that of the mixing distribution. It is proved that for some values of the parameters, the generalized Student distribution is infinitely divisible and admits a representation as a scale mixture of Laplace distributions. Necessary and sufficient conditions are presented that provide the convergence of the distributions of sums of a random number of independent random variables with finite variances and other statistics constructed from samples with random sizes to the generalized Student distribution. As an example, the convergence of the distributions of sample quantiles in samples with random sizes is considered. The generalized Lomax distribution is defined as the distribution of the absolute value of the random variable with the generalized Student distribution. It is shown that the generalized Lomax distribution can be represented as a scale mixture of folded normal distributions. The convergence of the distributions of maximum and minimum random sums to the generalized Lomax distribution is considered. It is demonstrated that the generalized Lomax distribution can be represented as a scale mixture of Weibull distributions or that of Fréchet distributions. As a consequence, it is demonstrated that the generalized Lomax distribution can be limiting for extreme statistics in samples with random size. The convergence of the distributions of mixed geometric random sums to the generalized Lomax distribution is considered, and the corresponding extension of the famous Rényi theorem is proved. The law of large numbers for mixed Poisson random sums is presented, in which the limit random variable has a generalized Lomax distribution.

Delicate Comparison of the Central and Non-Central Lyapunov Ratios with Applications to the Berry–Esseen Inequality for Compound Poisson Distributions

For each t∈(−1,1), the exact value of the least upper bound H(t)=sup{E|X|3/E|X−t|3} over all the non-degenerate distributions of the random variable X with a fixed normalized first-order moment EX1/EX12=t, and a finite third-order moment is obtained, yielding the exact value of the unconditional supremum M:=supL1(X)/L1(X−EX)=17+77/4, where L1(X)=E|X|3/(EX2)3/2 is the non-central Lyapunov ratio, and hence proving S. Shorgin’s (2001) conjecture on the exact value of M. As a corollary, an analog of the Berry–Esseen inequality for the Poisson random sums of independent identically distributed random variables X1,X2,… is proven in terms of the central Lyapunov ratio L1(X1−EX1) with the constant 0.3031·Ht(1−t2)3/2∈[0.3031,0.4517), t∈[0,1), which depends on the normalized first-moment t:=EX1/EX12 of random summands and being arbitrarily close to 0.3031 for small values of t, an almost 1.5 size improvement from the previously known one.

Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case

In the paper, we apply a new approach to the comparison of the distributions of sums of random variables to the case of Poisson random sums. This approach was proposed in our previous work (Bening, Korolev, 2022) and is based on the concept of statistical deficiency. Here, we introduce a continuous analog of deficiency. In the case under consideration, by continuous deficiency, we will mean the difference between the parameter of the Poisson distribution of the number of summands in a Poisson random sum and that of the compound Poisson distribution providing the desired accuracy of the normal approximation. This approach is used for the solution of the problem of determination of the distribution of a separate term in the Poisson sum that provides the least possible value of the parameter of the Poisson distribution of the number of summands guaranteeing the prescribed value of the (1−α)-quantile of the normalized Poisson sum for a given α∈(0,1). This problem is solved under the condition that possible distributions of random summands possess coinciding first three moments. The approach under consideration is applied to the collective risk model in order to determine the distribution of insurance payments providing the least possible time that provides the prescribed Value-at-Risk. This approach is also used for the problem of comparison of the accuracy of approximation of the asymptotic (1−α)-quantile of the sum of independent, identically distributed random variables with that of the accompanying infinitely divisible distribution.

Bounds for the Rate of Convergence in the Generalized Rényi Theorem

In the paper, an overview is presented of the results on the convergence rate bounds in limit theorems concerning geometric random sums and their generalizations to mixed Poisson random sums, including the case where the mixing law is itself a mixed exponential distribution. The main focus is on the upper bounds for the Zolotarev ζ-metric as the distance between the pre-limit and limit laws. New results are presented that extend existing estimates of the rate of convergence of geometric random sums (in the well-known Rényi theorem) to a considerably more general class of random indices whose distributions are mixed Poisson, including generalized negative binomial (e.g., Weibull-mixed Poisson), Pareto-type (Lomax)-mixed Poisson, exponential power-mixed Poisson, Mittag-Leffler-mixed Poisson, and one-sided Linnik-mixed Poisson distributions. A transfer theorem is proven that makes it possible to obtain upper bounds for the rate of convergence in the law of large numbers for mixed Poisson random sums with mixed exponential mixing distribution from those for geometric random sums (that is, from the convergence rate estimates in the Rényi theorem). Simple explicit bounds are obtained for ζ-metrics of the first and second orders. An estimate is obtained for the stability of representation of the Mittag-Leffler distribution as a geometric convolution (that is, as the distribution of a geometric random sum).

Formulation and Investigation of an Integral Equation for Characteristic Functions of Positive Random Variables

Functional equations of characteristic functions constitute power research tools for establishing new results in several significant areas of probability theory. The present paper makes use of the characteristic functions of two Poisson random sums and the concept of equality in distribution for introducing an important selfdecomposable distribution.

Equality in Distribution of Random Sums for Introducing Selfdecomposability

It constitutes a general recognition that discrete Poisson random sums are very strong tools of probability theory with significant applications in a very wide variety of important practical disciplines. The paper makes use of an equality in distribution for the investigation of the structure of a particularly significant class of discrete Poisson random sums.

Bounds for convergence rate in laws of large numbers for mixed Poisson random sums

In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev ζ-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous Rényi theorem) to a considerably wider class of random indices with mixed Poisson distributions including, e.g., those with the (generalized) negative binomial distribution.

Bounds for the Concentration Functions of Random Sums under Relaxed Moment Conditions

Estimates are constructed for the deviation of the concentration functions of sums of independent random variables with finite variances from the folded normal distribution function without any assumptions concerning the existence of the moments of summands of higher orders. The results obtained are extended to Poisson-binomial, binomial, and Poisson random sums. Under the same assumptions, bounds are obtained for the approximation of the concentration functions of mixed Poisson random sums by the corresponding limit distributions. In particular, bounds are put forward for the accuracy of approximation of the concentration functions of geometric, negative binomial, and Sichel random sums by exponential, folded variance gamma, and folded Student distributions. Numerical estimates of all the constants involved are written down explicitly.

Convergence Rate Estimates in the Global CLT for Compound Mixed Poisson Distributions

Using the estimates of the accuracy of the normal approximation to distributions of Poisson-binomial random sums from [I. G. Shevtsova, Theory Probab. Appl., 62 (2018), pp. 278--294], we obtain mom...

Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions∗

Bounds of the accuracy of the normal approximation to the distribution of a sum of independent random variables are improved under relaxed moment conditions, in particular, under the absence of moments of orders higher than the second. These results are extended to Poisson-binomial, binomial and Poisson random sums. Under the same conditions, bounds are obtained for the accuracy of the approximation of the distributions of mixed Poisson random sums by the corresponding limit law. In particular, these bounds are constructed for the accuracy of approximation of the distributions of geometric, negative binomial and Poisson-inverse gamma (Sichel) random sums by the Laplace, variance gamma and Student distributions, respectively. All absolute constants are written out explicitly.

On the normal approximation under some condition for a differential equation of the characteristic function

We present upper bounds of Ls norms (1 ≪ s ≪ ∞) of the normal approximation of random variables the characteristic functions of which satisfy some linear homogeneous differential equation. Note that this differential equation is satisfied, for example, for Poisson, Gamma, binomial, negative binomial random variables, Poisson random sums, binomial random sums, negative binomial random sums, and so on. The upper bounds of Ls norms obtained are of the optimal order with respect to certain parameters of the examined distributions for all 1 ≪ s ≪ ∞.

On the Accuracy of the Normal Approximation to Compound Poisson Distributions

The value of the asymptotically exact constant is found in the analogue of the Berry--Esseen inequality for Poisson random sums of independent identically distributed random variables $X_1,X_2,\ldots$ possessing the third order moments. Moreover, for the uniform distance $\Delta_\lambda$ between the distribution function of the standard normal law and that of the centered and normalized random sum $S_\lambda=X_1+\cdots+X_{N_\lambda}$, ${N_\lambda}$ following the Poisson distribution with parameter $\lambda>0$ and being independent of $X_1,X_2,\ldots,$ the estimate $ \Delta_\lambda\le \frac{2\ell_\lambda}{3\sqrt{2\pi}} + 0.5\cdot\ell_\lambda^2< 0.2660\cdot\ell_\lambda+ 0.5\cdot\ell_\lambda^2$, where $\ell_\lambda = \frac{\E|X_1|^3}{\sqrt\lambda(\E X_1^2)^{3/2}}, $ is proved. It is demonstrated that this estimate is unimprovable regarding the factor $2/(3\sqrt{2\pi})=0.2659\ldots$ at $\ell_\lambda$. For the case of symmetric distribution of $X_1$ an improved bound $ \Delta_\lambda\le \frac{1+2\varkappa}{2\sqrt{2\pi}}\,\ell_\lambda + 0.4\cdot \ell_\lambda^2< 0.2391\cdot\ell_\lambda + 0.4\cdot \ell_\lambda^2 $ is obtained, where $\varkappa=\sup_{x>0}(\cos x-1+x^2/2)/x^3=0.0991\ldots\,.$ It is also shown that the value of the factor at $\ell_\lambda$ in this estimate cannot be made less than $(2\sqrt{2\pi})^{-1}=0.1994\ldots\,.$ Similar estimates are obtained under weakened moment conditions ${\bf E}|X_1|^{2+\delta}<\infty$ for some $0<\delta<1.$

Sharpened upper bounds for the absolute constant in the Berry-Esseen inequality for mixed Poisson random sums

Sharpened upper bounds for the absolute constant in the Berry-Esseen inequality for mixed Poisson random sums

An improvement of the Berry–Esseen inequality with applications to Poisson and mixed Poisson random sums

By a modification of the method that was applied in study of Korolev & Shevtsova (2009), here the inequalities and are proved for the uniform distance ρ(F n ,Φ) between the standard normal distribution function Φ and the distribution function F n of the normalized sum of an arbitrary number n≥1 of independent identically distributed random variables with zero mean, unit variance, and finite third absolute moment β3. The first of these two inequalities is a structural improvement of the classical Berry–Esseen inequality and as well sharpens the best known upper estimate of the absolute constant in the classical Berry–Esseen inequality since 0.33477(β3+0.429)≤0.33477(1+0.429)β3<0.4784β3 by virtue of the condition β3≥1. The latter inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry–Esseen inequality for Poisson random sums to 0.3041 which is strictly less than the least possible value 0.4097… of the absolute constant in the classical Berry–Esseen inequality. As corollaries, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.

On the accuracy of the normal approximation to the distributions of Poisson random sums

AbstractThe analogue of the Berry–Esseen inequality for Poisson random sums is improved in 4–11 times depending on the maximal order 2 &lt; s ≤ 3 of the finite absolute moment of the distribution of random summands. For the case, when s = 3, under additional assumption concerning the smoothness of distribution of summands we also obtained in some sense unimprovable estimate consisting of two summands: the first summand being the noncentral Lyapunov fraction with the coefficient equal to the unimprovable asymptotically exact constant and the second one decreases faster. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Decompounding poisson random sums: Recursively truncated estimates in the discrete case

Given a sample from a discrete compound Poisson distribution, we consider variants of plug-in and likelihood estimators for the corresponding base distribution. These proceed recursively with an intermediate truncation step. We discuss the asymptotic behaviour of the estimators and give some numerical examples. Both procedures compare favourably with the straightforward and the naively projected plug-in estimator that we introduced in Buchmann and Grubel (2003,The Annals of Statistics,31, 1054–1074).

Convergence rate estimates in local limit theorems for poisson random sums

The asymptotic properties of Poisson random sums are investigated. A series of convergence-rate estimates in local limit theorems for those sums are obtained.

Exponential bounds for generalized poisson distributions

Exponential estimates are constructed for the probabilities of large deviations of Poisson random sums.

Some Limit Theorems for Large Deviations

Some Limit Theorems for Large Deviations