It has been argued that the first definition of a Markov chain, where it has discrete time, now tends to be used, despite the second definition having been used by researchers like Joseph Doob and Kai Lai Chung. Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics. The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as formula_1-dimensional Euclidean space, which results in collections of random variables known as Markov random fields. A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued, but they can also be complex-valued or even more general. A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. For a sequence of independent and identically distributed random variables formula_199 with zero mean, the stochastic process formed from the successive partial sums formula_200 is a discrete-time martingale. In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables. Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the "compensated Poisson process". Martingales can also be built from other martingales. For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales. Martingales mathematically formalize the idea of a fair game, and they were originally developed to show that it is not possible to win a fair game.