real values. This term is also used when the index sets are mathematical spaces other than the real line, while the terms stochastic process and random process are usually used when the index set is interpreted as time, and other terms are used such as random field when the index set is -dimensional Euclidean space or a manifold. Notation A stochastic process can be denoted, among other ways, by , , or simply as or , although is regarded as an abuse of function notation. For example, or are used to refer to the random variable with the index , and not the entire stochastic process. If the index set is , then one can write, for example, to denote the stochastic process. Examples Bernoulli process One of the simplest stochastic processes is the Bernoulli process, which is a sequence of independent and identically distributed (iid) random variables, where each random variable takes either the value one or zero, say one with probability and zero with probability . This process can be linked to repeatedly flipping a coin, where the probability of obtaining a head is and its value is one, while the value of a tail is zero. In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, where each coin flip is an example of a Bernoulli trial. Random walk Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. But some also use the term to refer to processes that change in continuous time, particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism. There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines. A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, , or decreases by one