Martingale (probability theory) - Wikipedia
In probability theory, a martingale is 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 is equal to the present value, regardless of all prior values.
17.1: Introduction to Martingalges - Statistics LibreTexts
Apr 24, 2022 · Here are two martingales associated with the branching process: Each of the following is a martingale with respect to \( \bs X \). \( \bs{Y} = \{Y_n: n \in \N\} \) where \( Y_n = X_n / m^n \) for \( n \in \N \).
The theory of martingales plays a very important ans ueful role in the study of stochastic processes. A formal definition is given below. Definition 5.1. Let (Ω, F, P ) be a probability space. A martingale se-quence of length n is a chain X 1, X 2, , Xn of random variables and corre- · · ·.
17: Martingales - Statistics LibreTexts
The importance of martingales extends far beyond gambling, and indeed these random processes are among the most important in probability theory, with an incredible number and diversity of applications.
Informally a martingale is simply a stochastic process Mt defined on some probability space (Ω, F, P) and indexed by some ordered set T that is “con-ditionally constant,” i.e., whose predicted value at any future time s > t is the same as its present value at the time t of prediction.
3. Martingales - Yale University
Interpretation: A random process that evolves over time. Common examples are martingales (described below), and Markov processes, where the distribution of X i+1 depends only on X i and not on any previous states. 2. Sigma-algebras.
Martingales by D. Cox December 2, 2009 1 Stochastic Processes. Definition 1.1 Let T be an arbitrary index set. A stochastic process indexed by T is a family of random variables (Xt: t ∈ T) defined on a common probability space (Ω,F,P). If T is clear from context, we will write (Xt). If T is one of ZZ, IN, or
CONDITIONAL EXPECTATION AND MARTINGALES 1. INTRODUCTION Martingales play a role in stochastic processes roughly similar to that played by conserved quantities in dynamical systems. Unlike a conserved quantity in dynamics, which remains constant in time, a martingale’s value can change; however, its expectation remains constant in time.
The above examples illustrate two important kinds of martingales: those obtained as sums of independent random variables (each with mean zero) and those obtained products of independent random variables (each with mean one).
Martingales: examples Example. Galton-Watson (or branching) process: Start with one subject in generation 0. Every subject in every generation produces an independent number of offspring in the next generation. This number has offspring distribution given by pk = P(no. of children = k), k = 0,1,2,... Let µ= P ∞ k=0 kpk. 8