Almost Sure

17 November 19

Algebraic Probability (continued)

Filed under: Probability Theory — George Lowther @ 8:19 PM
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Continuing on from the previous post, I look at cases where the abstract concept of states on algebras correspond to classical probability measures. Up until now, we have considered commutative real algebras but, before going further, it will help to look instead at algebras over the complex numbers ${{\mathbb C}}$. In the commutative case, we will see that this is equivalent to using real algebras, but can be more convenient, and in the non-commutative case it is essential. When using complex algebras, we will require the existence of an involution, which can be thought of as a generalisation of complex conjugation.

Recall that, by an algebra ${\mathcal A}$ over a field ${K}$, we mean that ${\mathcal A}$ is a ${K}$-vector space together with a binary product operation satisfying associativity, distributivity over addition, compatibility with scalars, and which has a multiplicative identity.

Definition 1 A *-algebra ${\mathcal A}$ is an algebra over ${{\mathbb C}}$ together with an involution, which is a unary operator ${\mathcal A\rightarrow\mathcal A}$, ${a\mapsto a^*}$, satisfying,

1. Anti-linearity: ${(\lambda a+\mu b)^*=\bar\lambda a^*+\bar\mu b^*}$.
2. ${(ab)^*=b^*a^*}$.
3. ${a^{**}=a}$

for all ${a,b\in\mathcal A}$ and ${\lambda,\mu\in{\mathbb C}}$.

10 November 19

Algebraic Probability

Filed under: Probability Theory — George Lowther @ 2:16 PM
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The aim of this post is to motivate the idea of representing probability spaces as states on a commutative algebra. We will consider how this abstract construction relates directly to classical probabilities.

In the standard axiomatization of probability theory, due to Kolmogorov, the central construct is a probability space ${(\Omega,\mathcal F,{\mathbb P})}$. This consists of a state space ${\Omega}$, an event space ${\mathcal F}$, which is a sigma-algebra of subsets of ${\Omega}$, and a probability measure ${{\mathbb P}}$. The measure ${{\mathbb P}}$ is defined as a map ${{\mathbb P}\colon\mathcal F\rightarrow{\mathbb R}^+}$ satisfying countable additivity and normalised as ${{\mathbb P}(\Omega)=1}$.

A measure space allows us to define integrals of real-valued measurable functions or, in the language of probability, expectations of random variables. We construct the set ${L^\infty(\Omega,\mathcal F)}$ of all bounded measurable functions ${X\colon\Omega\rightarrow{\mathbb R}}$. This is a real vector space and, as it is closed under multiplication, is an algebra. Expectation, by definition, is the unique linear map ${L^\infty\rightarrow{\mathbb R}}$, ${X\mapsto{\mathbb E}[X]}$ satisfying ${{\mathbb E}[1_A]={\mathbb P}(A)}$ for ${A\in\mathcal F}$ and monotone convergence: if ${X_n\in L^\infty}$ is a nonnegative sequence increasing to a bounded limit ${X}$, then ${{\mathbb E}[X_n]}$ tends to ${{\mathbb E}[X]}$.

In the opposite direction, any nonnegative linear map ${p\colon L^\infty(\Omega,\mathcal F)\rightarrow{\mathbb R}}$ satisfying monotone convergence and ${p(1)=1}$ defines a probability measure by ${{\mathbb P}(A)=p(1_A)}$. This is the unique measure with respect to which expectation agrees with the linear map, ${{\mathbb E}=p}$. So, probability measures are in one-to-one correspondence with such linear maps, and they can be viewed as one and the same thing. The Kolmogorov definition of a probability space can be thought of as representing the expectation on the subset of ${L^\infty}$ consisting of indicator functions ${1_A}$. In practise, it is often more convenient to start with a different subset of ${L^\infty}$. For example, probability measures on ${{\mathbb R}^+}$ can be defined via their Laplace transform, ${\mathcal L_{{\mathbb P}}(a)=\int e^{-ax}d{\mathbb P}(x)}$, which represents the expectation on exponential functions ${x\mapsto e^{-ax}}$. Generalising to complex-valued random variables, probability measures on ${{\mathbb R}}$ are often represented by their characteristic function ${\varphi(a)=\int e^{iax}d{\mathbb P}(x)}$, which is just the expectation of the complex exponentials ${x\mapsto e^{iax}}$. In fact, by the monotone class theorem, we can uniquely represent probability measures on ${(\Omega,\mathcal F)}$ by the expectations on any subset ${\mathcal K\subseteq L^\infty}$ which is closed under taking products and generates the sigma-algebra ${\mathcal F}$. (more…)

27 October 19

The Functional Monotone Class Theorem

Filed under: Probability Theory — George Lowther @ 8:29 PM
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The monotone class theorem is a very helpful and frequently used tool in measure theory. As measurable functions are a rather general construct, and can be difficult to describe explicitly, it is common to prove results by initially considering just a very simple class of functions. For example, we would start by looking at continuous or piecewise constant functions. Then, the monotone class theorem is used to extend to arbitrary measurable functions. There are different, but related, monotone class theorems’ which apply, respectively, to sets and to functions. As the theorem for sets was covered in a previous post, this entry will be concerned with the functional version. In fact, even for the functional version, there are various similar, but slightly different, statements of the monotone class theorem. In practice, it is beneficial to use the version which most directly applies to the specific application. So, I will state and prove several different versions in this post. (more…)

6 October 19

The Monotone Class Theorem

Filed under: Probability Theory — George Lowther @ 11:00 AM
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The monotone class theorem, and closely related ${\pi}$-system lemma, are simple but fundamental theorems in measure theory, and form an essential step in the proofs of many results. General measurable sets are difficult to describe explicitly so, when proving results in measure theory, it is often necessary to start by considering much simpler sets. The monotone class theorem is then used to extend to arbitrary measurable sets. For example, when proving a result about Borel subsets of ${{\mathbb R}}$, we may start by considering compact intervals and then apply the monotone class theorem. I include this post on the monotone class theorem for reference. (more…)

6 January 19

Essential Suprema

Filed under: Probability Theory — George Lowther @ 2:00 PM
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Given a sequence ${X_1,X_2,\ldots}$ of real-valued random variables defined on a probability space ${(\Omega,\mathcal F,{\mathbb P})}$, it is a standard result that the supremum

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle X\colon\Omega\rightarrow{\mathbb R}\cup\{\infty\},\smallskip\\ &\displaystyle X(\omega)=\sup_nX_n(\omega). \end{array}$

is measurable. To ensure that this is well-defined, we need to allow X to have values in ${{\mathbb R}\cup\{\infty\}}$, so that ${X(\omega)=\infty}$ whenever the sequence ${X_n(\omega)}$ is unbounded above. The proof of this fact is simple. We just need to show that ${X^{-1}((-\infty,a])}$ is in ${\mathcal F}$ for all ${a\in{\mathbb R}}$. Writing,

$\displaystyle X^{-1}((-\infty,a])=\bigcap_nX_n^{-1}((-\infty,a]),$

the properties that ${X_n}$ are measurable and that the sigma-algebra ${\mathcal F}$ is closed under countable intersections gives the result.

The measurability of the suprema of sequences of random variables is a vital property, used throughout probability theory. However, once we start looking at uncountable collections of random variables things get more complicated. Given a, possibly uncountable, collection of random variables ${\mathcal S}$, the supremum ${S=\sup\mathcal S}$ is,

 $\displaystyle S(\omega)=\sup\left\{X(\omega)\colon X\in\mathcal S\right\}.$ (1)

However, there are a couple of reasons why this is often not a useful construction:

• The supremum need not be measurable. For example, consider the probability space ${\Omega=[0,1]}$ with ${\mathcal F}$ the collection of Borel or Lebesgue subsets of ${\Omega}$, and ${{\mathbb P}}$ the standard Lebesgue measure. For any ${a\in[0,1]}$ define the random variable ${X_a(\omega)=1_{\{\omega=a\}}}$ and, for a subset A of ${[0,1]}$, consider the collection of random variables ${\mathcal S=\{X_a\colon a\in A\}}$. Its supremum is

$\displaystyle S(\omega)=1_{\{\omega\in A\}}$

which is not measurable if A is a non-measurable set (e.g., a Vitali set).

• Even if the supremum is measurable, it might not be a useful quantity. Letting ${X_a}$ be the random variables on ${(\Omega,\mathcal F,{\mathbb P})}$ constructed above, consider ${\mathcal S=\{X_a\colon a\in[0,1]\}}$. Its supremum is the constant function ${S=1}$. As every ${X\in\mathcal S}$ is almost surely equal to 0, it is almost surely bounded above by the constant function ${Y=0}$. So, the supremum ${S=1}$ is larger than we may expect, and is not what we want in many cases.

The essential supremum can be used to correct these deficiencies, and has been important in several places in my notes. See, for example, the proof of the debut theorem for right-continuous processes. So, I am posting this to use as a reference. Note that there is an alternative use of the term essential supremum’ to refer to the smallest real number almost surely bounding a specified random variable, which is the one referred to by Wikipedia. This is different from the use here, where we look at a collection of random variables and the essential supremum is itself a random variable.

The essential supremum is really just the supremum taken within the equivalence classes of random variables under the almost sure ordering. Consider the equivalence relation ${X\cong Y}$ if and only if ${X=Y}$ almost surely. Writing ${[X]}$ for the equivalence class of X, we can consider the ordering given by ${[X]\le[Y]}$ if ${X\le Y}$ almost surely. Then, the equivalence class of the essential supremum of a collection ${\mathcal S}$ of random variables is the supremum of the equivalence classes of the elements of ${\mathcal S}$. In order to avoid issues with unbounded sets, we consider random variables taking values in the extended reals ${\bar{\mathbb R}={\mathbb R}\cup\{\pm\infty\}}$.

Definition 1 An essential supremum of a collection ${\mathcal S}$ of ${\bar{\mathbb R}}$-valued random variables,

$\displaystyle S = {\rm ess\,sup\,}\mathcal{S}$

is the least upper bound of ${\mathcal{S}}$, using the almost-sure ordering on random variables. That is, S is an ${\bar{\mathbb R}}$-valued random variable satisfying

• upper bound: ${S\ge X}$ almost surely, for all ${X\in\mathcal S}$.
• minimality: for all ${\bar{\mathbb R}}$-valued random variables Y satisfying ${Y\ge X}$ almost surely for all ${X\in\mathcal S}$, we have ${Y\ge S}$ almost surely.

22 May 17

The Gaussian Correlation Inequality

When I first created this blog, the subject of my initial post was the Gaussian correlation conjecture. Using ${\mu_n}$ to denote the standard n-dimensional Gaussian probability measure, the conjecture states that the inequality

$\displaystyle \mu_n(A\cap B)\ge\mu_n(A)\mu_n(B)$

holds for all symmetric convex subsets A and B of ${{\mathbb R}^n}$. By symmetric, we mean symmetric about the origin, so that ${-x}$ is in A if and only ${x}$ is in A, and similarly for B. The standard Gaussian measure by definition has zero mean and covariance matrix equal to the nxn identity matrix, so that

$\displaystyle d\mu_n(x)=(2\pi)^{-n/2}e^{-\frac12x^Tx}\,dx,$

with ${dx}$ denoting the Lebesgue integral on ${{\mathbb R}^n}$. However, if it holds for the standard Gaussian measure, then the inequality can also be shown to hold for any centered (i.e., zero mean) Gaussian measure.

At the time of my original post, the Gaussian correlation conjecture was an unsolved mathematical problem, originally arising in the 1950s and formulated in its modern form in the 1970s. However, in the period since that post, the conjecture has been solved! A proof was published by Thomas Royen in 2014 [7]. This seems to have taken some time to come to the notice of much of the mathematical community. In December 2015, Rafał Latała, and Dariusz Matlak published a simplified version of Royen’s proof [4]. Although the original proof by Royen was already simple enough, it did consider a generalisation of the conjecture to a kind of multivariate gamma distribution. The exposition by Latała and Matlak ignores this generality and adds in some intermediate lemmas in order to improve readability and accessibility. Since then, the result has become widely known and, recently, has even been reported in the popular press [10,11]. There is an interesting article on Royen’s discovery of his proof at Quanta Magazine [12] including the background information that Royen was a 67 year old German retiree who supposedly came up with the idea while brushing his teeth one morning. Dick Lipton and Ken Regan have recently written about the history and eventual solution of the conjecture on their blog [5]. As it has now been shown to be true, I will stop referring to the result as a `conjecture’ and, instead, use the common alternative name — the Gaussian correlation inequality.

In this post, I will describe some equivalent formulations of the Gaussian correlation inequality, or GCI for short, before describing a general method of attacking this problem which has worked for earlier proofs of special cases. I will then describe Royen’s proof and we will see that it uses the same ideas, but with some key differences. (more…)

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