for a sample function of a stochastic process include trajectory, path function or path. Increment An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if is a stochastic process with state space and index set , then for any two non-negative numbers and such that , the difference is a -valued random variable known as an increment. When interested in the increments, often the state space is the real line or the natural numbers, but it can be -dimensional Euclidean space or more abstract spaces such as Banach spaces. Further definitions Law For a stochastic process defined on the probability space , the law of stochastic process is defined as the image measure: where is a probability measure, the symbol denotes function composition and is the pre-image of the measurable function or, equivalently, the -valued random variable , where is the space of all the possible -valued functions of , so the law of a stochastic process is a probability measure. For a measurable subset of , the pre-image of gives so the law of a can be written as: The law of a stochastic process or a random variable is also called the probability law, probability distribution, or the distribution. Finite-dimensional probability distributions For a stochastic process with law , its finite-dimensional distribution for is defined as: This measure is the joint distribution of the random vector ; it can be viewed as a "projection" of the law onto a finite subset of . For any measurable subset of the -fold Cartesian power , the finite-dimensional distributions of a stochastic process can be written as: The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions. Stationarity Stationarity is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. In other words, if is a stationary stochastic process, then for any the random variable has the same distribution, which means that for any set of index set values , the corresponding random variables all have the same probability distribution. The index set of a stationary stochastic process is usually interpreted as time, so it can be