scholarly journals Study on asymptotic behavior of stochastic Lotka–Volterra system in a polluted environment

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Li Wang
Keyword(s):  
Asymptotic Behavior ◽  
Lyapunov Function ◽  
Law Of Large Numbers ◽  
Sufficient Condition ◽  
Polluted Environment ◽  
Positive Periodic Solutions ◽  
Volterra System ◽  
Strong Law ◽  
Large Numbers ◽  
Extinction Property

AbstractA three-species non-autonomous stochastic Lotka–Volterra food web system in a polluted environment is proposed, and the existence of positive periodic solutions of this system is established by constructing a proper Lyapunov function. Then the extinction property and its threshold between persistence and extinction are discussed by using Itô’s formula and the strong law of large numbers of martingale, and the sufficient condition of a.s. exponential stability of equilibrium point is obtained. Finally, the conclusions are tested by several numerical simulations.

Random splittings of an interval

1993 ◽  
Vol 30 (01) ◽  
pp. 131-152
Author(s):  
T. S. Mountford ◽  
S. C. Port
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Law Of Large Numbers ◽  
Unit Interval ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Strong Law ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Necessary And Sufficient

Points are independently and uniformly distributed onto the unit interval. The first n—1 points subdivide the interval into n subintervals. For 1 we find a necessary and sufficient condition on {ln } for the events [Xn belongs to the ln th largest subinterval] to occur infinitely often or finitely often with probability 1. We also determine when the weak and strong laws of large numbers hold for the length of the ln th largest subinterval. The strong law of large numbers and the central limit theorem are shown to be valid for the number of times by time n the events [Xr belongs to l r th largest subinterval] occur when these events occur infinitely often.

Random splittings of an interval

1993 ◽  
Vol 30 (1) ◽  
pp. 131-152
Author(s):  
T. S. Mountford ◽  
S. C. Port
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Law Of Large Numbers ◽  
Unit Interval ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Strong Law ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Necessary And Sufficient

Points are independently and uniformly distributed onto the unit interval. The first n—1 points subdivide the interval into n subintervals. For 1 we find a necessary and sufficient condition on {ln} for the events [Xn belongs to the ln th largest subinterval] to occur infinitely often or finitely often with probability 1. We also determine when the weak and strong laws of large numbers hold for the length of the ln th largest subinterval. The strong law of large numbers and the central limit theorem are shown to be valid for the number of times by time n the events [Xr belongs to lr th largest subinterval] occur when these events occur infinitely often.

scholarly journals The asymptotic behavior for Markov chains in a finite i.i.d random environment indexed by cayley trees

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 273-283 ◽  
Author(s):  
Huilin Huang
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Markov Chains ◽  
Random Environment ◽  
Law Of Large Numbers ◽  
Homogeneous Tree ◽  
Strong Law ◽  
Cayley Trees ◽  
Large Numbers ◽  
Finite Markov Chains

We firstly define a Markov chain indexed by a homogeneous tree in a finite i.i.d random environment. Then, we prove the strong law of large numbers and Shannon-McMillan theorem for finite Markov chains indexed by a homogeneous tree in the finite i.i.d random environment.

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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