Some generalized variability orderings among life distributions with reliability applications

We investigate a generalized variability ordering and its weaker versions among non-negative random variables (lifetimes of components). Our results include a necessary and sufficient condition which justifies the generalized variability interpretation of this dominance relation between life distributions, relationships to some weakly aging classes in reliability theory, closure properties and inequalities for the mean life of series and parallel systems under such ordering.

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A Classification and Closure Properties of Languages for Describing Concurrent System Behaviours

Fundamenta Informaticae ◽

10.3233/fi-1981-4305 ◽

1981 ◽

Vol 4 (3) ◽

pp. 531-549 ◽

Cited By ~ 1

Author(s):

Miklós Szijártó

Keyword(s):

Formal Languages ◽

Parallel Program ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Concurrent System ◽

Closure Properties ◽

Sequential Program ◽

Necessary And Sufficient ◽

Special Case ◽

Independence Relations

The correspondence between sequential program schemes and formal languages is well known (Blikle and Mazurkiewicz (1972), Engelfriet (1974)). The situation is more complicated in the case of parallel program schemes, and trace languages (Mazurkiewicz (1977)) have been introduced to describe them. We introduce the concept of the closure of a language on a so called independence relation on the alphabet of the language, and formulate several theorems about them and the trace languages. We investigate the closedness properties of Chomsky classes under closure on independence relations, and as a special case we derive a new necessary and sufficient condition for the regularity of the commutative closure of a language.

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A note on the subcritical generalized age-dependent branching process

Journal of Applied Probability ◽

10.2307/3212536 ◽

1976 ◽

Vol 13 (4) ◽

pp. 798-803 ◽

Cited By ~ 2

Author(s):

R. A. Doney

Keyword(s):

Branching Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Malthusian Parameter ◽

Age Dependent ◽

The Mean ◽

Mean Function ◽

Necessary And Sufficient

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.

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On weak convergence within the hnbue family of life distributions

Journal of Applied Probability ◽

10.1017/s0021900200028850 ◽

1984 ◽

Vol 21 (03) ◽

pp. 654-660 ◽

Cited By ~ 4

Author(s):

Sujit K. Basu ◽

Manish C. Bhattacharjee

Keyword(s):

Weak Convergence ◽

Exponential Distribution ◽

Limiting Distribution ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Moment Sequence ◽

Positive Order ◽

Life Distributions ◽

Necessary And Sufficient

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.

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Subexponential distribution functions

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700021340 ◽

1980 ◽

Vol 29 (3) ◽

pp. 337-347 ◽

Cited By ~ 90

Author(s):

E. J. G. Pitman

Keyword(s):

Distribution Function ◽

Comparison Theorem ◽

Distribution Functions ◽

Sufficient Condition ◽

Important Class ◽

Necessary And Sufficient Condition ◽

Closure Properties ◽

Subexponential Class ◽

Necessary And Sufficient ◽

60 J 80

AbstractA distribution function (F on [0,∞) belongs to the subexponential class if and only if 1−F(2) (x) ~ 2(1−F(x)), as x→ ∞. For an important class of distribution functions, a simple, necessary and sufficient condition for membership of is given. A comparison theorem for membership of and also some closure properties of are obtained.1980 Mathematics subject classification (Amer. Math. Soe.): primary 60 E 05; secondary 60 J 80.

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THE COMPATIBLE BOND-STOCK MARKET WITH JUMPS

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024911006449 ◽

2011 ◽

Vol 14 (05) ◽

pp. 723-755 ◽

Cited By ~ 1

Author(s):

DEWEN XIONG ◽

MICHAEL KOHLMANN

Keyword(s):

Stock Market ◽

Marked Point ◽

Quadratic Functions ◽

Marked Point Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

No Arbitrage ◽

The Mean ◽

Necessary And Sufficient ◽

Mean Variance

We construct a bond-stock market composed of d stocks and many bonds with jumps driven by general marked point process as well as by an ℝn-valued Wiener process. By composing these tools we introduce the concept of a compatible bond-stock market and give a necessary and sufficient condition for this property. We study no-arbitrage properties of the composed market where a compatible bond-stock market is arbitrage-free both for the bonds market and for the stocks market. We then turn to an incomplete compatible bond-stock market and give a necessary and sufficient condition for a compatible bond-stock market to be incomplete. In this market we consider the mean-variance hedging in the special situation where both B(u, T) and eG(u, y, T)-1 are quadratic functions of T - u. So, we need to extend the notion of a variance-optimal martingale (VOM) as in Xiong and Kohlmann (2009) to the more general market. By introducing two virtual stocks [Formula: see text], we prove that the VOM for the bond-stock market is the same as the VOM for the new stock market [Formula: see text]. The mean-variance hedging problem in this incomplete bond-stock market for a contingent claim [Formula: see text] is solved by deriving an explicit solution of the optimal measure-valued strategy and the optimal cost induced by the optimal strategy of MHV for the stocks [Formula: see text] is computed.

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The weak law of large numbers for nonnegative summands

Advances in Applied Probability ◽

10.1017/apr.2018.83 ◽

2018 ◽

Vol 50 (A) ◽

pp. 241-252

Author(s):

Eugene Seneta

Keyword(s):

Branching Process ◽

Law Of Large Numbers ◽

Random Variables ◽

Sufficient Condition ◽

Specific Function ◽

Slowly Varying Function ◽

Necessary And Sufficient Condition ◽

Large Numbers ◽

Norming Sequence ◽

Necessary And Sufficient

Abstract Khintchine's (necessary and sufficient) slowly varying function condition for the weak law of large numbers (WLLN) for the sum of n nonnegative, independent and identically distributed random variables is used as an overarching (sufficient) condition for the case that the number of summands is more generally [cn],cn→∞. Either the norming sequence {an},an→∞, or the number of summands sequence {cn}, can be chosen arbitrarily. This theorem generalizes results from a motivating branching process setting in which Khintchine's sufficient condition is automatically satisfied. A second theorem shows that Khintchine's condition is necessary for the generalized WLLN when it holds with cn→∞ and an→∞. Theorem 3, which is known, gives a necessary and sufficient condition for Khintchine's WLLN to hold with cn=n and an a specific function of n; it is extended to general cn subject to a growth restriction in Theorem 4. Section 6 returns to the branching process setting.

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Some Reliability and Other Properties of Beta-Transformed Random Variables

Stochastics and Quality Control ◽

10.1515/eqc-2018-0017 ◽

2018 ◽

Vol 33 (2) ◽

pp. 83-92

Author(s):

M. Sreehari ◽

E. Sandhya ◽

V. K. Mohamed Akbar

Keyword(s):

Power Series ◽

Geometric Distribution ◽

Random Variables ◽

Random Variable ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Power Series Distribution ◽

Geometric Random Variable ◽

Necessary And Sufficient

Abstract The reliability properties of beta-transformed random variables are discussed. A necessary and sufficient condition for a beta-transformed geometric random variable to follow a power series distribution is derived. It is shown that a beta-transformed member of the Katz family does not belong to the Katz family unless it is a geometric distribution, thereby getting a characterization.

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A non-uniform rate of convergence in a local limit theorem

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055171 ◽

1978 ◽

Vol 84 (2) ◽

pp. 351-359 ◽

Cited By ~ 4

Author(s):

Sujit K. Basu

Keyword(s):

Limit Theorem ◽

Rate Of Convergence ◽

Local Limit Theorem ◽

Random Variables ◽

Local Limit ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Standard Normal ◽

The Common ◽

Necessary And Sufficient

AbstractLet {Xn} be a sequence of iid random variables. If the common charac-teristic function is absolutely integrable in mth power for some integer m ≥ 1, then Zn = n−½(X1 + … + Xn) has a pdf fn for all n ≥ m. Here we give a necessary and sufficient condition for sup as n. → ∞, where φ (x) is the standard normal pdf and M(x) is a non-decreasing function of x ≥ 0 such that M(0) > 0 and M(x)/xδ is non-increasing for 0 < δ ≤ 1.

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STABILITY OF HALF-LINEAR NEUTRAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAYS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972709000422 ◽

2009 ◽

Vol 80 (3) ◽

pp. 369-383 ◽

Cited By ~ 4

Author(s):

MENG WU ◽

NAN-JING HUANG ◽

CHANG-WEN ZHAO

Keyword(s):

Fixed Point Theory ◽

Point Theory ◽

Stability Theorem ◽

Sufficient Condition ◽

Mean Square ◽

Necessary And Sufficient Condition ◽

Variable Delays ◽

Mean Square Stability ◽

The Mean ◽

Necessary And Sufficient

AbstractIn this paper, we study the mean square asymptotic stability of a generalized half-linear neutral stochastic differential equation with variable delays applying fixed point theory. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton, Zhang and Luo. Two examples are given to illustrate our results.

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