CHAPTER 1 Introduction Let Xi be symmetric i.i.d. random vectors taking values in Z2 with mean 0 and finite covariance matrix Γ, set Sn = ∑n i=1 Xi, and suppose that no proper subgroup of Z2 supports the random walk Sn. For any random variable Y we will use the notation Y = Y − EY. Let (1.1) Rn = #{S1, . . . , Sn} be the range of the random walk up to time n. The purpose of this paper is to obtain moderate deviation results for Rn and −Rn. With two exceptions, throughout this paper we only assume that the random walks have second moments. The two exceptions are Proposition 5.2 and Corollary 1.3, which supposes slightly more than two moments. For moderate deviations of Rn we have the following. Let (1.2) H(n) = n k=0 P0(Sk = 0). Since the Xi have two moments, by (4.23) below, H(n) = n k=0 P0(Sk = 0) ∼ log n 2π √ det Γ and H(n) − H([n/bn]) = n k=[n/bn]+1 P0(Sk = 0) ∼ log bn 2π √ det Γ . Our first main result is the following. Theorem 1.1. Let {bn} be a positive sequence satisfying bn → ∞ and log bn = o((log n)1/2) as n → ∞. There are two constants C1, C2 0 independent of the choice of the sequence {bn} such that −C1 ≤ lim inf n→∞ bn −1 log P Rn ≥ n H(n)2 (H(n) − H([n/bn])) ≤ lim sup n→∞ bn −1 log P Rn ≥ n H(n)2 (H(n) − H([n/bn]) ≤ −C2. (1.3) Remark 1.2. The proof will show that C2 in the statement of Theorem 1.1 is equal to the constant L given in Theorem 1.3 in [2]. We believe that C1 is also equal to L, but we do not have a proof of this fact. 1

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