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.
KeywordsRandom Walk Characteristic Function Transition Function Simple Random Walk Renewal Theorem
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