Stability of Stochastic Systems
The stochastic systems demonstrate random probability distributions. At the same time they are amenable to statistical analysis. We categorise numerous processes into stochastic processes. There are different categories of stochastic processes such as Markov processes, Poisson processes, Random walks, Brownonian motion, Levy processes etc. The basic premise that they are amenable to statistical analysis makes them quite resourceful to many domains and models. At the sametime their randomness results in countless possibilities and promises.
As per Wikipedia, a stochastic process can be defined as: In probability theory and related fields, a stochastic process is a mathematical object usually defined as a sequence of random variables in a probability space, where the index of the sequence often has the interpretation of time.
This demonstrates the hidden interconnections between intrinsic randomness and extrinsic order. The fact that there is an apparent sequence of random variables underscores the polymorphic and polynomial behaviours of randomness in a stochastic setting. This sequence could be a pointer to the possible approaches to analyse the stability of stochastic systems. The roots of the sequential behaviour could be the convergence of concave and convex nature in the multitude of randomness that coexist in stochastic processes.
Sequences are never empirical events in time. Sequences are the manifestations of convergence of symmetric and asymptotic tendencies. Asymptotic tendencies get truncated and conjugated in a symmetrical system. Symmetric tendencies get modulated and marginalised in asymptotic systems. This results in sequences. Sequences of harmonic and impulsive kinds. Without sequences, symmetries will only result in singular occurrences. Without sequences, asymptotes will only result in spiralling occurrences.
Thus sequences are the results of convergences of different kinds. Sequences could be stable or unstable. By analysing the sequences and their symmetric and asymptotic tendencies we could arrive at the stability of stochastic systems. Stable stochastic systems could be the confluence of both symmetric and asymptotic processes. However unstable stochastic systems will be monotonic. They will be either purely symmetric or purely asymptotic in nature.