Stability of Stochastic Systems

Gokul B Alex
2 min readMar 27, 2024
Photo by Paul Wong on Unsplash

The stochasitc systems demonstrate random probability distributions. At the same time they are amenable to the statical analysis. We categorize 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 result in countless possibilties and promises.

As per Wikipedia, a stochasitc process can be defined as the following.

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 apprent sequence of random variables underscores the polymorphic and polynomial behaviors of randomness in a stochastic setting. This sequence could be a pointer to the possible approaches to analyse the stability of stochasitc systems. The roots of the sequential behavior could be the convergence of concave and convex nature in the mutltitude of randomness that coexist in stochastic processes.

Sequences are never empirical events in time. Sequences are the manifestations of convergence of symmetric and asymptotic tendancies. Asymptotic tendencies gets truncated and and conjugated in a symmetrical systems. Symmetric tendancies gets modulated and marginalized in asymptotic systems. This results in sequences. Sequences of harmonic and impulsive kinds. Without sequences, symmetries will only result in singular occurances. Without sequences, asymptotes will only results in spiralling occurances.

Thus sequences are the results of convergences of different kind. Sequences could be stable or unstable. By analyzing the sequences and their symmtric and asymptotic tendancies 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.