If you spot an error on this site, we would be grateful if you could report it to us by using the contact link at the top of this page and we will endeavour to correct it as soon as possible. The word “convergence” here is not being used in the standard manner, instead it is being used in an abstract manner. I have learned about the Intuition on the Kullback-Leibler (KL) Divergence as how much a model distribution convergence metric function differs from the theoretical/true distribution of the data. This article incorporates material from the Citizendium article “Stochastic convergence”, which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL. These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been studied.

It essentially means that “eventually” a sequence of elements get closer and closer to a single value. To formalize this requires a careful specification of the set of functions under consideration and how uniform the convergence should be. Where Ω is the sample space of the underlying probability space over which the random variables are defined. This is the type of stochastic convergence that is most similar to pointwise convergence known from elementary real analysis.

## Metric Convergence

Also, keep in mind that stating that an algorithm converges requires a proof (as we did for our 0.001, 0.0001, …, example). A metric is a number that we give to a given result that the algorithm produces. For instance, in A.I / Machine Learning iterative algorithms it is very common for us to keep track of the “error” that the algorithm is generating based on the input.

That is, there is the potential to make a functionally risk-free profit by purchasing the lower-priced commodity and selling the higher-priced futures contract—assuming the market is in contango. Our algorithm for every set of numbers spits for each of them if they are even or odd. For that, we can define a metric error as being the number of times it got wrong divided by the total number of elements that were given.

## Sure convergence or pointwise convergence

More precisely, no matter how small an error range you choose, if you continue long enough the function will eventually stay within that error range around the final value. The basic idea behind this type of convergence is that the probability of an “unusual” outcome becomes smaller and smaller as the sequence progresses. Convergence simply means that, on the last day that a futures contract can be delivered to fulfill the terms of the contract, the price of the futures and the price of the underlying commodity will be equal.

And what the algorithm tries to do is to minimize that error so it ever gets smaller and smaller. We say that the algorithm converges if it sequence of errors converges. We say Xn converges to a given number L if for every positive error that you think, there is a Xm such that every element Xn that comes after Xm differs from L by less than that error. Otherwise, convergence in measure can refer to either global convergence in measure or local convergence in measure, depending on the author. Convergence in measure is either of two distinct mathematical concepts both of which generalize

the concept of convergence in probability. Again, we will be cheating a little bit and we will use the definite article in front of the word limit before we prove that the limit is unique.

It depends on a topology on the underlying space and thus is not a purely measure theoretic notion. At the same time, the case of a deterministic X cannot, whenever the deterministic value is a discontinuity point (not isolated), be handled by convergence in distribution, where discontinuity points have to be explicitly excluded. Then as n tends to infinity, Xn converges in probability (see below) to the common mean, μ, of the random variables Yi.

However, Egorov’s theorem does guarantee that on a finite measure space, a sequence of functions that converges almost everywhere also converges almost uniformly on the same set. We first define uniform convergence for real-valued functions, although the concept is readily generalized to functions mapping to metric spaces and, more generally, uniform spaces (see below). This is the notion of pointwise convergence of a sequence of functions extended to a sequence of random variables. Using Morera’s Theorem, one can show that if a sequence of analytic functions converges uniformly in a region S of the complex plane, then the limit is analytic in S. This example demonstrates that complex functions are more well-behaved than real functions, since the uniform limit of analytic functions on a real interval need not even be differentiable (see Weierstrass function). In mathematics and statistics, weak convergence is one of many types of convergence relating to the convergence of measures.

- For example, lets say we are writing an algorithm that prints all the digits of PI.
- More precisely, no matter how long you continue, the function value will never settle down within a range of any “final” value.
- Where Ω is the sample space of the underlying probability space over which the random variables are defined.
- Otherwise, convergence in measure can refer to either global convergence in measure or local convergence in measure, depending on the author.

The concept of convergence in probability is used very often in statistics. For example, an estimator is called consistent if it converges in probability to the quantity being estimated. Convergence in probability is also the type of convergence established by the weak law of large numbers. With this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given probability distribution. When we take a closure of a set \(A\), we really throw in precisely those points that are limits of sequences in \(A\).

Convergence in distribution is the weakest form of convergence typically discussed, since it is implied by all other types of convergence mentioned in this article. However, convergence in distribution is very frequently used in practice; most often it arises from application of the central limit theorem. While the above discussion has related to the convergence of a single series to a limiting value, the notion of the convergence of two series towards each other is also important, but this is easily handled by studying the sequence defined as either the difference or the ratio of the two series. A set is closed when it contains the limits of its convergent sequences. The topology, that is, the set of open sets of a space encodes which sequences converge.