Principles of Random Walk pp 54-104 | Cite as

# Harmonic Analysis

## Abstract

As in the last chapter *R* will denote the *d*-dimensional group of lattice points *x* = (*x* ^{1}, *x* ^{2}, *...* ,*x* ^{ d }) where the *x* ^{ i } are integers. To develop the usual notions of Fourier analysis we must consider still another copy of Euclidean space, which we shall call *E.* It will be the whole of Euclidean space (not just the integers) and of the same dimension *d* as *R.* For convenience we shall use Greek letters to denote elements of *E*,and so, if *R* is α-dimensional, the elements of *E* will be *θ* = (*θ* _{1}, *θ* _{2},..., *θ* _{ d }), where each *θ* _{ i } is a real number for *i* = 1, 2,..., *d*. The following notation will be convenient.

## Keywords

Random Walk Characteristic Function Transition Function Simple Random Walk Renewal Theorem## Preview

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