Mathematics

Research

13 pages, 305 KiB

Open AccessArticle

Limit Distributions of Products of Independent and Identically Distributed Random 2 × 2 Stochastic Matrices: A Treatment with the Reciprocal of the Golden Ratio

by Santanu Chakraborty

Mathematics 2023, 11(24), 4993; https://doi.org/10.3390/math11244993 - 18 Dec 2023

Viewed by 611

Abstract

Consider a sequence

{(X_{n})}_{n \geq 1}

of i.i.d.

2 \times 2

stochastic matrices with each

X_{n}

distributed as

μ

. This

μ

is described as follows. Let

{(C_{n}, D_{n})}^{T}

denote the first [...] Read more.

Consider a sequence

{(X_{n})}_{n \geq 1}

of i.i.d.

2 \times 2

stochastic matrices with each

X_{n}

distributed as

μ

. This

μ

is described as follows. Let

{(C_{n}, D_{n})}^{T}

denote the first column of

X_{n}

and for a given real r with

0 < r < 1

, let

r^{- 1} C_{n}

and

r^{- 1} D_{n}

each be Bernoulli distributions with parameters

p_{1}

and

p_{2}

, respectively, and

0 < p_{1}, p_{2} < 1

. Clearly, the weak limit of the sequence

μ^{n}

, namely

λ

, is known to exist, whose support is contained in the set of all

2 \times 2

rank one stochastic matrices. In a previous paper, we considered

0 < r \leq \frac{1}{2}

and obtained

λ

explicitly. We showed that

λ

is supported countably on many points, each with positive

λ

-mass. Of course, the case

0 < r \leq \frac{1}{2}

is tractable, but the case

r > \frac{1}{2}

is very challenging. Considering the extreme nontriviality of this case, we stick to a very special such r, namely,

r = \frac{\sqrt{5} - 1}{2}

(the reciprocal of the golden ratio), briefly mention the challenges in this nontrivial case, and completely identify

λ

for a very special situation. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications, 2nd Edition)

10 pages, 285 KiB

Open AccessArticle

Estimates of the Convergence Rate in the Generalized Rényi Theorem with a Structural Digamma Distribution Using Zeta Metrics

by Alexey Kudryavtsev and Oleg Shestakov

Mathematics 2023, 11(21), 4477; https://doi.org/10.3390/math11214477 - 29 Oct 2023

Viewed by 701

Abstract

This paper considers a generalization of the Rényi theorem to the case of a structural distribution with a scale parameter. In terms of the zeta metric, some estimates of the convergence rate in the generalized Rényi theorem are obtained when the structural mixed [...] Read more.

This paper considers a generalization of the Rényi theorem to the case of a structural distribution with a scale parameter. In terms of the zeta metric, some estimates of the convergence rate in the generalized Rényi theorem are obtained when the structural mixed Poisson distribution of the summation index is a scale mixture of the generalized gamma distribution. Estimates of the convergence rate for the structural digamma distribution are given as a special case. The paper extends the results previously obtained for the generalized gamma distribution. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications, 2nd Edition)

19 pages, 687 KiB

Open AccessArticle

A Rényi-Type Limit Theorem on Random Sums and the Accuracy of Likelihood-Based Classification of Random Sequences with Application to Genomics

by Leonid Hanin and Lyudmila Pavlova

Mathematics 2023, 11(20), 4254; https://doi.org/10.3390/math11204254 - 11 Oct 2023

Viewed by 819

Abstract

We study classification of random sequences of characters selected from a given alphabet into two classes characterized by distinct character selection probabilities and length distributions. The classification is based on the sign of the log-likelihood score (LLS) consisting of a random sum and [...] Read more.

We study classification of random sequences of characters selected from a given alphabet into two classes characterized by distinct character selection probabilities and length distributions. The classification is based on the sign of the log-likelihood score (LLS) consisting of a random sum and a random term depending on the length distributions for the two classes. For long sequences selected from a large alphabet, computing misclassification error rates is not feasible either theoretically or computationally. To mitigate this problem, we computed limiting distributions for two versions of the normalized LLS applicable to long sequences whose class-specific length follows a translated negative binomial distribution (TNBD). The two limiting distributions turned out to be plain or transformed Erlang distributions. This allowed us to establish the asymptotic accuracy of the likelihood-based classification of random sequences with TNBD length distributions. Our limit theorem generalizes a classic theorem on geometric random sums due to Rényi and is closely related to the published results of V. Korolev and coworkers on negative binomial random sums. As an illustration, we applied our limit theorem to the classification of DNA sequences contained in the genome of the bacterium Bacillus subtilis into two classes: protein-coding genes and standard noncoding open reading frames. We found that TNBDs provide an excellent fit to the length distributions for both classes and that the limiting distributions capture essential features of the normalized empirical LLS fairly well. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications, 2nd Edition)

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16 pages, 294 KiB

Open AccessArticle

Equivalent Conditions of Complete p-th Moment Convergence for Weighted Sum of ND Random Variables under Sublinear Expectation Space

by Peiyu Sun, Dehui Wang and Xili Tan

Mathematics 2023, 11(16), 3494; https://doi.org/10.3390/math11163494 - 13 Aug 2023

Viewed by 582

Abstract

We investigate the complete convergence for weighted sums of sequences of negative dependence (ND) random variables and p-th moment convergence for weighted sums of sequences of ND random variables under sublinear expectation space. Using moment inequality and truncation methods, we prove the equivalent [...] Read more.

We investigate the complete convergence for weighted sums of sequences of negative dependence (ND) random variables and p-th moment convergence for weighted sums of sequences of ND random variables under sublinear expectation space. Using moment inequality and truncation methods, we prove the equivalent conditions of complete convergence for weighted sums of sequences of ND random variables and p-th moment convergence for weighted sums of sequences of ND random variables under sublinear expectation space. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications, 2nd Edition)

27 pages, 404 KiB

Open AccessArticle

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

by Victor Korolev

Mathematics 2023, 11(13), 2890; https://doi.org/10.3390/math11132890 - 27 Jun 2023

Cited by 3 | Viewed by 606

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications, 2nd Edition)

30 pages, 556 KiB

Open AccessArticle

Quick and Complete Convergence in the Law of Large Numbers with Applications to Statistics

by Alexander G. Tartakovsky

Mathematics 2023, 11(12), 2687; https://doi.org/10.3390/math11122687 - 13 Jun 2023

Viewed by 787

Abstract

In the first part of this article, we discuss and generalize the complete convergence introduced by Hsu and Robbins in 1947 to the r-complete convergence introduced by Tartakovsky in 1998. We also establish its relation to the r-quick convergence first introduced [...] Read more.

In the first part of this article, we discuss and generalize the complete convergence introduced by Hsu and Robbins in 1947 to the r-complete convergence introduced by Tartakovsky in 1998. We also establish its relation to the r-quick convergence first introduced by Strassen in 1967 and extensively studied by Lai. Our work is motivated by various statistical problems, mostly in sequential analysis. As we show in the second part, generalizing and studying these convergence modes is important not only in probability theory but also to solve challenging statistical problems in hypothesis testing and changepoint detection for general stochastic non-i.i.d. models. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications, 2nd Edition)

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13 pages, 334 KiB

Open AccessArticle

Limit Distributions for the Estimates of the Digamma Distribution Parameters Constructed from a Random Size Sample

by Alexey Kudryavtsev and Oleg Shestakov

Mathematics 2023, 11(8), 1778; https://doi.org/10.3390/math11081778 - 7 Apr 2023

Cited by 1 | Viewed by 957

Abstract

In this paper, we study a new type of distribution that generalizes distributions from the gamma and beta classes that are widely used in applications. The estimators for the parameters of the digamma distribution obtained by the method of logarithmic cumulants are considered. [...] Read more.

In this paper, we study a new type of distribution that generalizes distributions from the gamma and beta classes that are widely used in applications. The estimators for the parameters of the digamma distribution obtained by the method of logarithmic cumulants are considered. Based on the previously proved asymptotic normality of the estimators for the characteristic index and the shape and scale parameters of the digamma distribution constructed from a fixed-size sample, we obtain a statement about the convergence of these estimators to the scale mixtures of the normal law in the case of a random sample size. Using this result, asymptotic confidence intervals for the estimated parameters are constructed. A number of examples of the limit laws for sample sizes with special forms of negative binomial distributions are given. The results of this paper can be widely used in the study of probabilistic models based on continuous distributions with an unbounded non-negative support. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications, 2nd Edition)

16 pages, 318 KiB

Open AccessArticle

Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point

by Elena Filichkina and Elena Yarovaya

Mathematics 2023, 11(7), 1676; https://doi.org/10.3390/math11071676 - 31 Mar 2023

Viewed by 863

Abstract

We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only the absorption [...] Read more.

We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only the absorption of particles can occur. The asymptotic behavior of the integer moments of both the total number of particles and the number of particles at a lattice point is studied depending on the relationship between the model parameters. In the case of the existence of an isolated positive eigenvalue of the evolution operator of the average number of particles, a limit theorem is obtained on the exponential growth of both the total number of particles and the number of particles at a lattice point. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications, 2nd Edition)

32 pages, 519 KiB

Open AccessArticle

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

by Vladimir Makarenko and Irina Shevtsova

Mathematics 2023, 11(3), 625; https://doi.org/10.3390/math11030625 - 26 Jan 2023

Viewed by 914

Abstract

For each

t \in (- 1, 1)

, the exact value of the least upper bound

H (t) = sup {E | X |^{3} / E | X - t |^{3}}

over all the [...] Read more.

For each

t \in (- 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

E X_{1} / \sqrt{E X_{1}^{2}} = t

, and a finite third-order moment is obtained, yielding the exact value of the unconditional supremum

M : = sup L_{1} (X) / L_{1} (X - E X) = (\sqrt{17 + 7 \sqrt{7}}) / 4

, where

L_{1} (X) = E {| X |}^{3} / {(E X^{2})}^{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

X_{1}, X_{2}, \dots

is proven in terms of the central Lyapunov ratio

L_{1} (X_{1} - E X_{1})

with the constant

0.3031 \cdot H (t) {(1 - t^{2})}^{3 / 2} \in [0.3031, 0.4517)

,

t \in [0, 1)

, which depends on the normalized first-moment

t : = E X_{1} / \sqrt{E X_{1}^{2}}

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. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications, 2nd Edition)

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12 pages, 305 KiB

Open AccessFeature PaperArticle

Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case

by Vladimir Bening and Victor Korolev

Mathematics 2022, 10(24), 4712; https://doi.org/10.3390/math10244712 - 12 Dec 2022

Viewed by 1165

Abstract

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 [...] Read more.

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 \in (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. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications, 2nd Edition)

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