• Journal of applied mathematics & informatics
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• v.33 no.5_6
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• pp.607-625
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• 2015

Recently, Wang et al. [Computers and Mathematics with Ap-plications 57 (2009) 1232-1248.] and Wang and Watada [Information Sci-ences 179 (2009) 4057-4069.] studied the renewal process and renewal reward process with fuzzy random inter-arrival times and rewards under the T-independence associated with any continuous Archimedean t-norm. But, their main results do not cover the classical theory of the random elementary renewal theorem and random renewal reward theorem when fuzzy random variables degenerate to random variables, and some given assumptions relate to the membership function of the fuzzy variable and the Archimedean t-norm of the results are restrictive. This paper improves the results of Wang and Watada and Wang et al. from a mathematical per-spective. We release some assumptions of the results of Wang and Watada and Wang et al. and completely generalize the classical stochastic renewal theorem and renewal rewards theorem.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1459-1463
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• 2009

Recently, Zhao et.al [European Journal of Operational Research 169 (2006) 189-201] discussed a random fuzzy renewal process based on random fuzzy theory. They considered the rate of the random fuzzy renewal process and presented a random fuzzy elementary renewal theorem. They also established Blackwell's theorem in random fuzzy sense. But all these results are invalid. We give a counter example in this note.

• Journal of the Korean Statistical Society
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• v.25 no.1
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• pp.153-159
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• 1996

Sums of independent random variables $S_n = X_1 + X_ + cdots + X_n$ are considered, where the X$_{n}$ are chosen according to a stationary process of distributions. Given the time t .geq. O, let N (t) be the number of indices n for which O < $S_n$ $\geq$ t. In this set up we prove that N (t)/t converges almost surely and in $L^1$ as t longrightarrow $\infty$, which generalizes classical renewal theorem.m.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.9 no.3
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• pp.219-223
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• 2009

Recently, Zhao et al. [Fuzzy Optimization and Decision Making (2007) 6, 279-295] characterized the interarrival times as fuzzy random variables and presented a fuzzy random elementary renewal theorem on the limit value of the expected renewal rate of the process in the fuzzy random renewal process. They also depicted both the interarrival times and rewards are depicted as fuzzy random variables and provided fuzzy random renewal reward theorem on the limit value of the long run expected reward per unit time in the fuzzy random renewal reward process. In this note, we simplify the proofs of two main results of the paper.

• Journal of applied mathematics & informatics
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• v.34 no.3_4
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• pp.355-357
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• 2016

Recently, some results of Blackwell's Theorem in which interarrival times are characterized as fuzzy variables under t-norm-based fuzzy operations are discussed by Hong. However, these results are invalid. In this note, we give counter examples of these results.

• Journal of the Korean Data and Information Science Society
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• v.19 no.4
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• pp.1345-1352
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• 2008

An inventory with constant demand is studied. We adopt a renewal argument to obtain the transient and stationary distribution of the level of the inventory. We show that the stationary distribution can be also derived by making use of either the level crossing technique or the renewal reward theorem. After assigning several managing costs to the inventory, we calculate the long-run average cost per unit time. A numerical example is illustrated to show how we optimize the inventory.

• Journal of the Korean Data and Information Science Society
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• v.16 no.1
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• pp.165-172
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• 2005

In recently, Popova and Wu(1999) proved a theorem which presents the long-run average fuzzy reward per unit time. In this note, we improve this result. Indeed we will show uniform convergence of a renewal reward processes with respect to the level ${\alpha}$ modeled as a fuzzy random variables.

• Journal of the Korean Mathematical Society
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• v.36 no.6
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• pp.1115-1132
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• 1999

We consider a supercritical multitype age dependent branching process. We define a stochastic process Zf(t) which is a functional of the empirical age distribution. When the limit of the expectation of this functional vanishes we4 find some sufficient conditions for the asymptotic normality of the mean of f with respect to the empirical age distribution at time t.

• Journal of the Korean Mathematical Society
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• v.56 no.4
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• pp.935-946
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• 2019

Hawkes process is a self-exciting simple point process with clustering effect whose jump rate depends on its entire past history. We consider Hawkes processes with uniform immigrants which is a special case of the Hawkes processes with renewal immigrants. We study the limit theorems for Hawkes processes with uniform immigrants. In particular, we obtain a law of large number, a central limit theorem, and a large deviation principle.

• Nuclear Engineering and Technology
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• v.51 no.6
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• pp.1616-1625
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• 2019

Measuring occurrence times of random events, aimed to determine the statistical properties of the governing stochastic process, is a basic topic in science and engineering, and has been the subject of numerous mathematical modeling approaches. Often, true statistical properties deviate from measured properties due to the so called dead time phenomenon, where for a certain time period following detection, the detection system is not operational. Understanding the dead time effect is especially important in radiation measurements, often characterized by high count rates and a non-reducible detector dead time (originating in the physics of particle detection). The effect of dead time can be interpreted as a suitable rarefied sequence of the original time sequence. This paper provides a limit theorem for a high rate (diffusion-scale) counter with extendable (Type II) dead time, where the underlying counting process is a renewal process with finite second moment for the inter-event distribution. The results are very general, in the sense that they refer to a general inter arrival time and a random dead time with general distribution. Following the theoretical results, we will demonstrate the applicability of the results in three applications: serially connected components, multiplicity counting and measurements of aerosol spatial distribution.