Range-Renewal Processes:SLLNs and Power Laws

Given n samples(viewed as an n-tuple)of aγ-regular discrete distributionπ,in this article the authors concern with the weighted and unweighted graphs induced by the n samples.They first prove a series of SLLN results(of Dvoretzky-Erdos'type).Then they show that the vertex weights of the graphs under investigation obey asymptotically power law distributions with exponent 1+γThey also give a conjecture that the degrees of unweighted graphs would exhibit asymptotically power law distributions with constant exponent 2.This exponent is obviously independent of the parameterγ∈(0,1),which is a surprise to us at first sight.

Probability Distributions Arising in Connection with the Inspection Paradox for the Poisson Process

The Inspection Paradox refers to the fact that in a Renewal Process, the length of the interarrival period which contains a fixed time is stochastically larger than the length of a typical interarrival period. To provide a more complete understanding of this phenomenon, conditioning arguments are used to obtain the distributions and moments of the lengths of the interarrival periods other than the one containing this fixed time for the case of the time-homogeneous Poisson Process. Distributions of the waiting times for events that occur both before and after this fixed time are derived. This provides a fairly complete probabilistic analysis of the Inspection Paradox.

Necessary Conditions for Renewal Processes Approximating to Wiener Processes

Let {X, X i ; i≥1} be a sequence of i.i.d.r.v’.s, Z (t)= t i=1 X i,t≥0. Consider the renewal process N(t) =inf{x; Z(x) 】t}. In this paper we study the necessary momentconditions for { N(t) ; t≥0} approximating to Wiener processes.Our results have improved those of Csrg, Horváth and Stei, Horváth and Steinebach

Precise large deviations for sums of random vectors in a multidimensional size-dependent renewal risk model

Consider a multidimensional renewal risk model, in which the claim sizes {Xk, k ≥1} form a sequence of independent and identically distributed random vectors with nonnegative components that are allowed to be dependent on each other. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Suppose that the claim sizes and inter-arrival times correspondingly form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure. A precise large deviation for the multidimensional renewal risk model is obtained.

Large Deviations for a Generalized Compound Renewal Risk Model

This paper extends the ordinary renewal risk model to the case where the premium income process,based on a renewal counting process,is no longer a linear function;and the total claim amount process is described by a compound renewal process.For this realistic risk model,the large deviations for the claim surplus process is investigated.

Probability Distributions Arising in Connection with the Inspection Paradox for Bernoulli Trials

In renewal theory, the Inspection Paradox refers to the fact that an interarrival period in a renewal process which contains a fixed inspection time tends to be longer than one for the corresponding uninspected process. We focus on the paradox for Bernoulli trials. Probability distributions and moments for the lengths of the interarrival periods are derived for the inspected process, and we compare them to those for the uninspected case.

Precise large deviation result for heavy-tailed random sums and applications to risk theory

The differences between two sequences of nonnegative independent and identically distributed random variables with sub-exponential tails and the random index are studied. The random index is a strictly stationary renewal counting process generated by some negatively associated random variables. Using a revised large deviation result of partial sums, the elementary renewal theorem and the central limit theorem of negatively associated random variables, a precise large deviation result is derived for the random sums. The result is applied to the customer-arrival-based insurance risk model. Some uniform asymptotics for the ruin probabilities of an insurance company are obtained as the number of customers or the time tends to infinity.

On Closure Properties of NBU(2) Class of Life Distributions

Some new results about the NBU(2) class of life distributions were obtained. Fir stly, it was proved that the decrease with time of the increasing concave orderi ng of the excess lifetime in a renewal process leads to the NBU(2) property of t he interarrival times. Secondly, the NBU(2) class of life distributions is prove d to be closed under the formation of series systems. Finally, it was also shown that the NBU(2) class is closed under convolution operation.

Evolution of Damage on Historical Heritage Buildings in Presence of Catastrophic Events and Aggressive Natural Phenomena

The feasibility analysis of projects for the preservation of the historical heritage buildings is an important problem concerning the evaluation of ＂the total cost of intervention＂, which includes all the future damage costs. The total cost of intervention represents a suitable measure of the expected deterioration risk and its evolution obviously depends on the damage process which buildings are subjected to. That damage phenomena affecting masonry buildings pleased into an aggressive environment are suitably modelled by renewal processes： this happens both in the case of catastrophic events, or in the case of the so-called ＂natural aging＂, in which damage comes off gradually in time. In the hypothesis ofa Markovian renewal process （Mrp） describing the damage process, the total cost of all the future damage is evaluated taking into account both the damage aspects： damages due to catastrophic aspects and damages due to aggressive environment, supposing different maintenance and/or rehabilitation scenarios. A semi-Markov process （s-Mp） is defined to model the damage rehabilitation history of buildings in presence of seismic events, natural ageing and rehabilitation strategies. The expected rewards connected to the process are defined; they represent a significant measure of the risk.

Large Deviations for Random Sums on Some Kind of Heavy-tailed Classes in Risk Models

This paper is a further investigation into the large deviations for random sums of heavy-tailed,we extended and improved some results in ref. [1] and [2]. These results can applied to some questions in Insurance and Finance.

A Class of Delayed Renewal Risk Processes with a Threshold Dividend Strategy

This paper considers a class of delayed renewal risk processes with a threshold dividend strategy. The main result is an expression of the Gerber-Shiu expected discounted penalty function in the delayed renewal risk model in terms of the corresponding Cerber-Shiu function in the ordinary renewal model. Subsequently, this relationship is considered in more detail in both the stationary renewal risk model and the ruin probability.

A Novel Control Strategy of PWM Rectifier with LCL Filter Under Unbalanced Grid Condition

The paper presents a successful design and good result of grid voltage feed-forward control strategy with no damping regulator of LCL filter applied in Three-phase Pulse Width Modulation( PWM) rectifier under unbalance grid condition for renewable energy processing. It is demonstrated that the closed-loop current control is decoupled with the grid voltage feed-forward control,and good waveform of grid current with low order harmonics is obtained under unbalance grid condition. This novel strategy is simple and reliable,applied with PI regulator but no resonant controller in α,β two-phase stationary frame. Moreover,the results of experiment and simulation are also illustrated to validate the novel strategy well applied in the closed-loop current control system under unbalanced grid condition.

The Ruin Probability of the Renewal Model with Constant Interest Force and Upper-tailed Independent Heavy-tailed Claims

In this paper, we obtain the uniform estimate for discounted aggregate claims in the continuous-time renewal model of upper-tailed independent and heavy-tailed random variables. With constant interest force and constant premium rate, we establish a uniform simple asymptotic formula for ruin probability of the renewal model in the case where the initial surplus is large.

Analysis of Data from a Series of Events by a Geometric Process Model

Geometric process was first introduced by Lam.A stochastic process {X_i,i=1,2,...} iscalled a geometric process (GP) if,for some a>0,{a^(i-1)X_i,i=1,2,...} forms a renewal process.In thispaper,the GP is used to analyze the data from a series of events.A nonparametric method is introduced forthe estimation of the three parameters in the GP.The limiting distributions of the three estimators are studied.Through the analysis of some real data sets,the GP model is compared with other three homogeneous andnonhomogeneous Poisson models.It seems that on average the GP model is the best model among these fourmodels in analyzing the data from a series of events.

Signal-to-noise ratio gain of an adaptive neuron model with Gamma renewal synaptic input

We take an adaptive leaky integrate-and-fire neuron model to explore the effect of non-Poisson neurotransmitter on stochastic resonance and its signal-to-noise ratio(SNR)gain.Event triggered algorithm is adopted to speed up the simulating process.It is revealed that both the output SNR and the SNR gain can be monotonically improved when increasing the shape parameter for Gamma distribution.Particularly,for large signal coupling strength,the 1:1 stochastic phase locking induced by Gamma noise is responsible for the frequency matching stochastic resonance,and the output signal-to-noise ratio can surpass the input signal-to-noise ratio,which is significantly different with Poisson case,while for extremely weak signal coupling strength,the SNR gain peak,which is far larger than unity,is due to noise induced resonance.The observations are meaningful in understanding the neural processing mechanisms from a more realistic viewpoint of synaptic modeling.

Strong Approximation Method and the(Functional)Law of Iterated Logarithm for GI/G/1 Queue

In this paper, a unified method based on the strong approximation(SA) of renewal process(RP) is developed for the law of the iterated logarithm(LIL) and the functional LIL(FLIL), which quantify the magnitude of the asymptotic rate of the increasing variability around the mean value of the RP in numerical and functional forms respectively. For the GI/G/1 queue, the method provides a complete analysis for both the LIL and the FLIL limits for four performance functions: The queue length, workload, busy time and idle time processes, covering three regimes divided by the traffic intensity.

THE MATCHED QUEUEING SYSTEM GI。PH/PH/1

We study the matched queueing system GIoPH/PH/1, where the type-I input is a renewal process, the type-II input is a PH renewal process, and the service times are i. i. d. random variables with PH-distributions. First, a condition is given for the stationarity of the system. Then the distributions of the number of type-I customers at the arrival epoches of type-I customers and the number of type-I customers at an arbitrary epoch are derived. We also discuss the occupation time and the waiting time. Their L. S. transforms are derived. Finally, we discuss some problems in numerical computation.

A contribution to large deviations for heavy-tailed random sums

In this paper we consider the large deviations for random sums $S(t) = \sum _{i = t}^{N(t)} X_i ,t \geqslant 0$ , whereX n,n?1 are independent, identically distributed and non-negative random variables with a common heavy-tailed distribution function F, andN(t), t?0 is a process of non-negative integer-valued random variables, independent ofX n,n?1. Under the assumption that the tail of F is of Pareto’s type (regularly or extended regularly varying), we investigate what reasonable condition can be given onN(t), t?0 under which precise large deviation for S( t) holds. In particular, the condition we obtain is satisfied for renewal counting processes.