Contents
What is independent stochastic processes?
Independence of two stochastic processes is a property between two stochastic processes and that are defined on the same probability space . Formally, two stochastic processes and are said to be independent if for all and for all , the random vectors and are independent, i.e. if.
Is probability a stochastic process?
Stochastic processes are probabilistic models for random quantities evolving in time or space. The evolution is governed by some dependence relationship between the random quantities at different times or locations.
How are Stochastic Processes classified in different ways?
A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space.
How is a stochastic process represented in probability theory?
In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Many stochastic processes can be represented by time series. However, a stochastic process is by nature continuous while a time series is a set of observations indexed by integers.
How is a sample function formed in a stochastic process?
Sample function. A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if is a stochastic process, then for any point , the mapping is called a sample function, a realization, or,…
When does a stochastic process have continuous state space?
Classifications. If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process. If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space.