Strong Law Of Large Numbers Research Articles

Further results on laws of large numbers for the array of random variables under sub-linear expectation

– Motivated by risk measure, super-hedge pricing, and modeling uncertainty in finance, Shige Peng established the theory of sub-linear expectation. In this article, we derive two results of laws of large numbers in the framework of sub-linear expectations. One is the strong law of large numbers for the array of random variables, which satisfies non identical distributed and exponential negatively dependent under sub-linear expectation. The other is the weak law of large numbers for the array of random variables, which satisfies non identical distributed and Φ -negatively dependent under sub-linear expectation. These results include and extend some existing results.

Strong Laws of Large Numbers for Weighted Sums of Hilbert-valued Coordinatewise PNQD Random Variables with Applications

Strong Laws of Large Numbers for Weighted Sums of Hilbert-valued Coordinatewise PNQD Random Variables with Applications

Cyclic Products and Optimal Traps in Cyclic Birth and Death Chains

A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only if $|i-j|=1$. We consider birth-death chains whose birth probabilities $p_{i,i+1}$ form a periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ and $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities $p_0,\ldots,p_{m-1}$ on the velocity $v=\lim_{n\to\infty} X_n/n$. The sign of $v$ is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of $(p_0,\ldots,p_{m-1})$, exactly $(m-1)!/2$ distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all $m\leq 7$. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.

Invariant Measures for Uncountable Random Interval Homeomorphisms

A necessary and sufficient condition for the iterated function system {f(cdot ,omega ),|,omega in Omega } with probability P to have exactly one invariant measure mu _* with mu _*((0,1))=1 is given. The main novelty lies in the fact that we only require the transformations f(cdot ,omega ) to be increasing homeomorphims, without any smoothness condition, neither we impose conditions on the cardinality of Omega . In particular, positive Lyapunov exponents conditions are replaced with the existence of solutions to some functional inequalities. The stability and strong law of large numbers of the considered system are also proven.

The Kolmogorov strong law of large numbers for WOD random variables

The Kolmogorov strong law of large numbers for WOD random variables

Dynamic analysis of the IFCD rumor propagation model under stochastic disturbance on heterogeneous networks

By taking full consideration of the heterogeneity of network users and stochastic disturbance of network environment, a stochastic IFCD (Ignorants-Forwarders-Collectors-Deleters) rumor propagation model is proposed on heterogeneous networks. This study takes MicroBlog as the research background to investigate the dynamical behaviors of rumor spreading on online social networks, which paves the way for controlling rumors effectively and provides positive guidance for the network public opinion. Firstly, some preparations for building a stochastic model are given, including establishing a deterministic IFCD rumor propagation model and obtaining the rumor-free/spreading equilibrium. Secondly, considering the impact of stochastic disturbance on rumor propagation, the corresponding stochastic rumor propagation model is established. Thirdly, a stopping time is defined, which manifests the existence of the unique global positive solution in the stochastic model. Next, sufficient conditions of the rumor extinction and persistence are obtained by using Itô’s formula and strong law of large numbers. Additionally, simulations are operated to validate the results. Finally, the application of the model is shown to bear out the applicability of the proposed model.

Equivalent conditions of complete convergence and Marcinkiewicz–Zygmund-type strong law of large numbers for i.i.d. sequences under sub-linear expectations

Equivalent conditions of complete convergence and Marcinkiewicz–Zygmund-type strong law of large numbers for i.i.d. sequences under sub-linear expectations

Dynamic analysis and optimal control of a stochastic information spreading model considering super-spreader and implicit exposer with random parametric perturbations

Environmental factors in social systems affect information spreading at all times. This paper proposes a stochastic S2EIR model that considers the presence of super-spreaders and implicit exposers in information spreading, as well as the stochastic perturbation of model parameters. The existence of a global positive solution using the Itô′s formula is then demonstrated. Sufficient conditions for information disappearance and smooth distribution of information are calculated by using the Borel–Cantelli lemma and the strong law of large numbers. Furthermore, the optimal control strategy for the stochastic model is proposed using the Hamiltonian function. The results of the theoretical analysis are supported by numerical simulations and compared to the parameter variations of the deterministic model. The results of this study indicate that white noise facilitates the spread of information. The intensity of perturbation is proportional to the fluctuation of information spreading. Controlling random parameters can effectively facilitate the spread of information. For positive information, the randomness and complexity of the social system should be utilized to increase the spread of information. In contrast, for negative information, randomness in the social system should be suppressed to the greatest extent possible to limit information dissemination.

Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space

Abstract In this work, the authors study some convergence results including weak law of large numbers, strong law of large numbers, complete convergence, and complete moment convergence for weighted sums of coordinatewise asymptotically negatively associated random vectors in Hilbert spaces. These results improve or extend some corresponding ones in the literature.

Probabilistic Properties of Zipf Sets and Their Maximal Intersections

In the article, the weak and strong laws of large numbers are obtained for various characteristics of a generalized Zipf set. Based on these results, we investigate the maximal intersection size between a random Zipf set and the elements of a large base of independent random sets of the same type but, eventually, with different parameters. This question is significant for applications.

Capacity inequalities and strong laws for m-widely acceptable random variables under sub-linear expectations

In this work, a generalised concept of dependent random variables, i.e., m-widely acceptable random variables, is put forward and some inequalities such as exponential inequality and Rosenthal-type inequality are established under sub-linear expectations. By virtue of these inequalities, we obtain a general result on complete convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of m-widely acceptable random variables, which extend and improve some existing ones in classical probability space as well as in sub-linear expectation space. As applications of the main results, the complete consistency and strong consistency of weighted estimators in nonparametric regression model under sub-linear expectations are also investigated.

On the strong laws of large numbers for pairwise NQD random variables

Let { X , X n , n ≥ 1 } be a sequence of pairwise NQD identically distributed random variables and { b n , n ≥ 1 } be a sequence of positive constants. In this article, we study the strong laws of large numbers for the sequence { X , X n , n ≥ 1 } , under the general moment condition ∑ n = 1 ∞ P ( | X | > b n / log n ) < ∞ , which improve some known results.

Degree distributions in recursive trees with fitnesses

AbstractWe study a general model of recursive trees where vertices are equipped with independent weights and at each time-step a vertex is sampled with probability proportional to its fitness function, which is a function of its weight and degree, and connects to $\ell$ new-coming vertices. Under a certain technical assumption, applying the theory of Crump–Mode–Jagers branching processes, we derive formulas for the limiting distributions of the proportion of vertices with a given degree and weight, and proportion of edges with endpoint having a certain weight. As an application of this theorem, we rigorously prove observations of Bianconi related to the evolving Cayley tree (Phys. Rev. E66, paper no. 036116, 2002). We also study the process in depth when the technical condition can fail in the particular case when the fitness function is affine, a model we call ‘generalised preferential attachment with fitness’. We show that this model can exhibit condensation, where a positive proportion of edges accumulates around vertices with maximal weight, or, more drastically, can have a degenerate limiting degree distribution, where the entire proportion of edges accumulates around these vertices. Finally, we prove stochastic convergence for the degree distribution under a different assumption of a strong law of large numbers for the partition function associated with the process.

Multilevel bootstrap particle filter

We consider situations where the applicability of sequential Monte Carlo particle filters is compromised due to the expensive evaluation of the particle weights. To alleviate this problem, we propose a new particle filter algorithm based on the multilevel approach. We show that the resulting multilevel bootstrap particle filter (MLBPF) retains the strong law of large numbers as well as the central limit theorem of classical particle filters under mild conditions. Our numerical experiments demonstrate up to 85% reduction in computation time compared to the classical bootstrap particle filter, in certain settings. While it should be acknowledged that this reduction is highly application dependent, and a similar gain should not be expected for all applications across the board, we believe that this substantial improvement in certain settings makes MLBPF an important addition to the family of sequential Monte Carlo methods.

On the long time behaviour of single stochastic Hodgkin-Huxley neurons with constant signal, and a construction of circuits of interacting neurons showing self-organized rhythmic oscillations

The stochastic Hodgkin-Huxley neurons considered in this paper replace time-constant deterministic input $a dt$ of the classical deterministic model by increments $\vartheta dt + dX_t$ of a stochastic process: $X$ is Ornstein-Uhlenbeck with volatility $\sigma&gt;0$ and backdriving force $\tau&gt;0$, and we call $\vartheta&gt;0$ the signal. We have ergodicity and strong laws of large numbers for various functionals of the process, and characterize 'quiet behaviour' and 'regular spiking' as events whose probability depends on the parameters $(\tau,\sigma)$ and on the signal $\vartheta$. The notions of quiet behaviour and regular spiking allow for a construction of circuits of interacting stochastic Hodgkin-Huxley neurons, combining excitation with inhibition according to a bloc structure along the circuit, on which self-organized rhythmic oscillations can be observed.

Random Euclidean coverage from within

Let X_1,X_2, ldots be independent random uniform points in a bounded domain A subset mathbb {R}^d with smooth boundary. Define the coverage thresholdR_n to be the smallest r such that A is covered by the balls of radius r centred on X_1,ldots ,X_n. We obtain the limiting distribution of R_n and also a strong law of large numbers for R_n in the large-n limit. For example, if A has volume 1 and perimeter |partial A|, if d=3 then mathbb {P}[npi R_n^3 - log n - 2 log (log n) le x] converges to exp (-2^{-4}pi ^{5/3} |partial A| e^{-2 x/3}) and (n pi R_n^3)/(log n) rightarrow 1 almost surely, and if d=2 then mathbb {P}[n pi R_n^2 - log n - log (log n) le x] converges to exp (- e^{-x}- |partial A|pi ^{-1/2} e^{-x/2}). We give similar results for general d, and also for the case where A is a polytope. We also generalize to allow for multiple coverage. The analysis relies on classical results by Hall and by Janson, along with a careful treatment of boundary effects. For the strong laws of large numbers, we can relax the requirement that the underlying density on A be uniform.

Exponential inequalities and a strong law of large numbers for END random variables under sub-linear expectations

&lt;abstract&gt;&lt;p&gt;The focus of our work is to investigate exponential inequalities for extended negatively dependent (END) random variables in sub-linear expectations. Through these exponential inequalities, we were able to establish the strong law of large numbers with convergence rate $ O\left(n^{-1/2}\ln^{1/2}n\right) $. Our findings in sub-linear expectation spaces have extended the corresponding results previously established in probability space.&lt;/p&gt;&lt;/abstract&gt;

Strong law of large numbers for the stochastic six vertex model

Strong law of large numbers for the stochastic six vertex model

Complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations

&lt;abstract&gt;&lt;p&gt;In this article, we study the complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations. We also give some sufficient assumptions for the convergence. Moreover, we get the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of extended negatively dependent random variables. The results obtained in this paper generalize the relevant ones in probability space.&lt;/p&gt;&lt;/abstract&gt;

A strong law of large numbers under sublinear expectations

A strong law of large numbers under sublinear expectations