Laws of large numbers and Birkhoff’s ergodic theorem
نویسنده
چکیده
In preparation for the next post on the central limit theorem, it's worth recalling the fundamental results on convergence of the average of a sequence of random variables: the law of large numbers (both weak and strong), and its strengthening to non-IID sequences, the Birkhoff ergodic theorem. 1 Convergence of random variables First we need to recall the different ways in which a sequence of random variables may converge. Let Y n be a sequence of real-valued random variables and Y a single random variable to which we want the sequence Y n to " converge ". There are various ways of formalising this.
منابع مشابه
On Stability Property of Probability Laws with Respect to Small Violations of Algorithmic Randomness
We study a stability property of probability laws with respect to small violations of algorithmic randomness. A sufficient condition of stability is presented in terms of Schnorr tests of algorithmic randomness. Most probability laws, like the strong law of large numbers, the law of iterated logarithm, and even Birkhoff’s pointwise ergodic theorem for ergodic transformations, are stable in this...
متن کاملRecords from Stationary Observations Subject to a Random Trend
We prove strong convergence and asymptotic normality for the record and the weak record rate of observations of the form Yn = Xn + Tn, n ≥ 1, where (Xn)n∈Z is a stationary ergodic sequence of random variables and (Tn)n≥1 is a stochastic trend process, with stationary ergodic increments. The strong convergence result follows from the Dubins-Freedman law of large numbers and Birkhoff’s ergodic th...
متن کاملUniform Ergodic Theorems for Dynamical Systems Under VC Entropy Conditions
The classic limit theorems of Vapnik and Chervonenkis [27,28] show that if a function class F satisfies a random entropy condition, then the strong law of large numbers holds uniformly over F . In this paper we show that an analogous weighted entropy condition implies that Birkhoff’s pointwise ergodic theorem holds uniformly over F . In this way we obtain a variety of uniform ergodic theorems f...
متن کاملAbstracts of the Talks
S OF THE TALKS Hillel Furstenberg, Hebrew U. Qualitative Laws of Large Numbers. Abstract: If X1,X2, . . . ,Xn, . . . is an iid sequence of non-singular matrices, and we form the ”random product” Yn = X1 ∗ X2 ∗ X3 ∗ ⋯ ∗ Xn and let n → ∞ we find that the Yn tend to have a certain form. We analyze this phenomenon. If X1,X2, . . . ,Xn, . . . is an iid sequence of non-singular matrices, and we form ...
متن کاملA PRELUDE TO THE THEORY OF RANDOM WALKS IN RANDOM ENVIRONMENTS
A random walk on a lattice is one of the most fundamental models in probability theory. When the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (RWRE). The basic questions such as the law of large numbers (LLN), the central limit theorem (CLT), and the large deviation principle (LDP) are ...
متن کامل