Ergodic Theorem for Bounded Random Walks

In summary, the conversation discusses the search for resources on ergodic theorem for bounded random walks. It also mentions the concept of agent-based modeling and its applications in simulating interactions between individual agents in a system, such as in cellular automata theory. Other related topics mentioned include the Ergodic Theorem and Metropolis-Hastings algorithms.
  • #1
hatelise
1
0
Could anyone point me towards, articles, or any other reading material regarding ergodic theorem for Bounded Random Walks.

1) Consider an simple bounded (x,y) plane on which several 2-dimensional random walker are randomly assigned an (x,y) coordinate.

2) Furthermore, if 2 random walkers collide 1 of them dies off.

I have tried time series, Markov Chains etc...

But I can't seem to be able to factor in the interaction between 2 elements.

http://demonstrations.wolfram.com/PredatorPreyEcosystemARealTimeAgentBasedSimulation/

I'm working on a simpler version of this simulation.
 
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  • #2
You may want to look into the field of agent-based modeling, which is a type of modeling used to simulate the interactions between individual agents within a system. Specifically, you can look into cellular automata theory, which is a type of agent-based modeling used to simulate the behavior of particles in a two-dimensional grid. The most famous example is John Conway's Game of Life. Other related topics include the Ergodic Theorem, which describes the behavior of random walks, and Metropolis-Hastings algorithms, which are used to solve certain types of optimization problems.
 

FAQ: Ergodic Theorem for Bounded Random Walks

What is the Ergodic Theorem for Bounded Random Walks?

The Ergodic Theorem for Bounded Random Walks is a mathematical principle that states that for a bounded random walk, the time average of a function of the walker's position will converge to the space average of that function as the number of steps approaches infinity.

How does the Ergodic Theorem for Bounded Random Walks differ from the regular Ergodic Theorem?

The Ergodic Theorem for Bounded Random Walks is a modification of the regular Ergodic Theorem to account for bounded random walks, where the walker's position is limited within a certain range. This means that the time average and space average may not converge for bounded random walks, unlike in the regular Ergodic Theorem.

What are some real-world applications of the Ergodic Theorem for Bounded Random Walks?

The Ergodic Theorem for Bounded Random Walks has applications in various fields such as physics, economics, and biology. It can be used to model the behavior of particles in a confined space, the movement of stock prices, and the spread of diseases, among others.

Can the Ergodic Theorem for Bounded Random Walks be applied to unbounded random walks?

No, the Ergodic Theorem for Bounded Random Walks is specifically designed for bounded random walks and may not hold true for unbounded random walks. The regular Ergodic Theorem can be applied to unbounded random walks.

How is the Ergodic Theorem for Bounded Random Walks proven?

The Ergodic Theorem for Bounded Random Walks is proven using mathematical techniques such as measure theory and probability theory. It involves showing that as the number of steps approaches infinity, the time average and space average of the function of the walker's position converge to the same value.

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