A strong law of Erdös-Rényi type for cumulative processes in renewal theory

1978 ◽  
Vol 15 (1) ◽  
pp. 96-111 ◽  
Author(s):  
Josef Steinebach
Keyword(s):  
Law Of Large Numbers ◽  
Renewal Theory ◽  
Strong Law ◽  
Failure Times ◽  
Renewal Counting Process ◽  
Large Numbers ◽  
Wide Range ◽  
Cumulative Processes ◽  
Cumulative Process ◽  
Maximum Increment

Let {Nt}t >0 be a renewal counting process (cf. Parzen (1962), p. 160) with underlying failure times let be a sequence of non-negative random variables and {Zt}t >0 an associated cumulative process, i.e. if Nt = 1, 2, …, and Zt = 0, if Nt = 0. By convention set Z0 = 0. Consider the maximum increment of the process {Zt}t >0 in [0, T] over a time K, 0 < K < T, divided by K. Under appropriate conditions it is shown that for a wide range of numbers a there exist constants C(a), uniquely determined by a and the distributions of the Xi's and Yj's, such that D(T, C log T) converges to a with probability 1. This result provides a renewal theoretic variant of Erdös and Rényi's (1970) ‘new law of large numbers’.

A strong law of Erdös-Rényi type for cumulative processes in renewal theory

1978 ◽  
Vol 15 (01) ◽  
pp. 96-111 ◽  
Author(s):  
Josef Steinebach
Keyword(s):  
Law Of Large Numbers ◽  
Renewal Theory ◽  
Strong Law ◽  
Failure Times ◽  
Renewal Counting Process ◽  
Large Numbers ◽  
Wide Range ◽  
Cumulative Processes ◽  
Cumulative Process ◽  
Maximum Increment

Let {Nt } t &gt;0 be a renewal counting process (cf. Parzen (1962), p. 160) with underlying failure times let be a sequence of non-negative random variables and {Zt } t &gt;0 an associated cumulative process, i.e. if Nt = 1, 2, …, and Zt = 0, if Nt = 0. By convention set Z 0 = 0. Consider the maximum increment of the process {Zt } t &gt;0 in [0, T] over a time K, 0 &lt; K &lt; T, divided by K. Under appropriate conditions it is shown that for a wide range of numbers a there exist constants C(a), uniquely determined by a and the distributions of the X i's and Yj 's, such that D(T, C log T) converges to a with probability 1. This result provides a renewal theoretic variant of Erdös and Rényi's (1970) ‘new law of large numbers’.

scholarly journals The Strong Law of Large Numbers for Extended Negatively Dependent Random Variables

2010 ◽  
Vol 47 (04) ◽  
pp. 908-922 ◽  
Author(s):  
Yiqing Chen ◽  
Anyue Chen ◽  
Kai W. Ng
Keyword(s):  
Law Of Large Numbers ◽  
Large Deviation ◽  
Random Variables ◽  
Renewal Theory ◽  
Independent Random Variables ◽  
Strong Law ◽  
Finite Dimensional ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
New Inequalities

A sequence of random variables is said to be extended negatively dependent (END) if the tails of its finite-dimensional distributions in the lower-left and upper-right corners are dominated by a multiple of the tails of the corresponding finite-dimensional distributions of a sequence of independent random variables with the same marginal distributions. The goal of this paper is to establish the strong law of large numbers for a sequence of END and identically distributed random variables. In doing so we derive some new inequalities of large deviation type for the sums of END and identically distributed random variables being suitably truncated. We also show applications of our main result to risk theory and renewal theory.

scholarly journals The Strong Law of Large Numbers for Extended Negatively Dependent Random Variables

2010 ◽  
Vol 47 (4) ◽  
pp. 908-922 ◽  
Author(s):  
Yiqing Chen ◽  
Anyue Chen ◽  
Kai W. Ng
Keyword(s):  
Law Of Large Numbers ◽  
Large Deviation ◽  
Random Variables ◽  
Renewal Theory ◽  
Independent Random Variables ◽  
Strong Law ◽  
Finite Dimensional ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
New Inequalities

A sequence of random variables is said to be extended negatively dependent (END) if the tails of its finite-dimensional distributions in the lower-left and upper-right corners are dominated by a multiple of the tails of the corresponding finite-dimensional distributions of a sequence of independent random variables with the same marginal distributions. The goal of this paper is to establish the strong law of large numbers for a sequence of END and identically distributed random variables. In doing so we derive some new inequalities of large deviation type for the sums of END and identically distributed random variables being suitably truncated. We also show applications of our main result to risk theory and renewal theory.

scholarly journals On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xiaochen Ma ◽  
Qunying Wu
Keyword(s):  
Probability Space ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Sublinear Expectation ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
End Random Variables

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

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