Ton Coolen, Alessia Annibale, and Ekaterina Roberts

Print publication date: 2017

Print ISBN-13: 9780198709893

Published to Oxford Scholarship Online: May 2017

DOI: 10.1093/oso/9780198709893.001.0001

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(p.229) Appendix A The delta distribution

Source:
Generating Random Networks and Graphs
Publisher:
Oxford University Press

Definition. We define the $δ$-distribution as the probability distribution $δ(x)$ corresponding to a zero-average random variable x in the limit where the randomness in the variable vanishes. So

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By the same token, the expression $δ(x−a)$ will then represent the distribution for a random variable x with average a, in the limit where the randomness vanishes, since

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Formulae for the $δ$-distribution. A problem arises when we want to write down a formula for $δ(x)$. Intuitively one could propose taking a zero-average normal distribution and setting its width to zero:

(A.1)
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This is not a true function in a mathematical sense: $δ(x)$ is zero for $x≠0$ and $δ(0)=∞$. However, we realize that $δ(x)$ only serves to calculate averages; it only has a meaning inside an integration. If we adopt the convention that one should set $σ→0$ in (A.1) only after performing the integration, we can use (A.1) to derive the following properties (for sufficiently well-behaved functions f):

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The following relation links the $δ$-distribution to the step function:

(A.2)
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(p.230) This is proved by showing that both sides of the equation have the same effect inside an integration:

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Finally one can use the definitions of Fourier transforms and inverse Fourier transforms to obtain the following integral representation of the $δ$-distribution:

(A.3)
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