Doob martingale (nonfiction)

A Doob martingale (named after Joseph L. Doob, also known as a Levy martingale) is a mathematical construction of a stochastic process which approximates a given random variable and has the martingale property with respect to the given filtration. It may be thought of as the evolving sequence of best approximations to the random variable based on information accumulated up to a certain time.

When analyzing sums, random walks, or other additive functions of independent random variables, one can often apply the central limit theorem, law of large numbers, Chernoff's inequality, Chebyshev's inequality, or similar tools. When analyzing similar objects where the differences are not independent, the main tools are martingales and Azuma's inequality.

Fiction cross-reference

• Crimes against mathematical constants
• Doob Martingale
• Gnomon algorithm
• Gnomon Chronicles
• Mathematics

Nonfiction cross-reference

• Azuma's inequality (nonfiction) - gives a concentration result for the values of martingales that have bounded differences.
• Central limit theorem (nonfiction) - establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.
• Chebyshev's inequality (nonfiction) - guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean.
• Chernoff bound (nonfiction) - gives exponentially decreasing bounds on tail distributions of sums of independent random variables.
• Joseph L. Doob (nonfiction) - Joseph Leo "Joe" Doob (February 27, 1910 – June 7, 2004) was an American mathematician, specializing in analysis and probability theory. The theory of martingales was developed by Doob.
• Filtration (probability theory) (nonfiction) - model of information that is available at a given point in a stochastic process.
• Independence (probability theory) (nonfiction) - a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent[1] if the occurrence of one does not affect the probability of occurrence of the other (equivalently, does not affect the odds). Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.
• Law of large numbers (nonfiction) - a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer to the expected value as more trials are performed.
• Martingale (probability theory) (nonfiction) - a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence, given all prior values, is equal to the present value.
• Mathematics (nonfiction)
• Random variable (nonfiction) - a variable whose values depend on outcomes of a random phenomenon. The formal mathematical treatment of random variables is a topic in probability theory.
• Random walk (nonfiction) - a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.
• Stochastic process (nonfiction) - a mathematical object usually defined as a family of random variables.

External links:

• Doob martingale @ Wikipedia