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RANDOM FIELDS AND THEIR GEOMETRY

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6 1. Random fields<br />

space T is what makes Gaussian fields such a useful model for random processes<br />

on general spaces. To build an appreciation for this, we need to look<br />

at some examples. The following Section looks at entire class of examples<br />

that are generated from something called ‘Gaussian noise’, that we shall<br />

also exploit in Section 1.4 to develop the notion of stationarity of random<br />

fields. Many more examples can be found in Chapter 2, which looks at<br />

Gaussian fields and their properties in far more depth.<br />

1.3 The Brownian family of processes<br />

Perhaps the most basic of all random fields is a collection of independent<br />

Gaussian random variables. While it is simple to construct such random<br />

fields for finite and even countable parameter sets, deep technical difficulties<br />

obstruct the construction for uncountable parameter sets. The path<br />

that we shall take around these difficulties involves the introduction of random<br />

measures which, at least in the Gaussian case, are straightforward to<br />

formulate.<br />

Let (T, T ,ν) be a σ-finite measure space and denote by Tν the collection<br />

of sets of T of finite ν measure. A Gaussian noise based on ν, or ‘Gaussian<br />

ν-noise’ is a random field W : Tν → R such that, for all A, B ∈ Tν,<br />

(1.3.1)<br />

(1.3.2)<br />

(1.3.3)<br />

W (A) ∼ N(0, ν(A)).<br />

A ∩ B = ∅ ⇒ W (A ∪ B) = W (A) + W (B) a.s.<br />

A ∩ B = ∅ ⇒ W (A) and W (B) are independent.<br />

Property (1.3.2) encourages one to think of W 10 as a random (signed)<br />

measure, although it is not generally σ-finite. We describe (1.3.3) by saying<br />

that W has independent increments.<br />

Theorem 1.3.1 If (T, T , ν) is a measure space, then there exists a real<br />

valued Gaussian noise, defined for all A ∈ Tν, satisfying (1.3.1)–(1.3.3).<br />

Proof. In view of the closing remarks of the preceeding Section all we<br />

need do is provide an appropriate covariance function on Tν × Tν. Try<br />

(1.3.4)<br />

Cν(A, B) ∆ = ν(A ∩ B).<br />

10 While the notation ‘W ’ is inconsistent our determination to use lower case Latin<br />

characters for random functions, we retain it as a tribute to Norbert Wiener, who is the<br />

mathematical father of these processes.

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