THE PROBLEM AND OUR RESULTSLet X = (X m ) m∈Z d be a centered, stationary Gaussian process on Z d . This means that for any k ≥ 1 and any m 0 , . . . , m k ∈ Z d , the vector (X mj +m0 ) 1≤j≤k has a multivariate Gaussian distribution with zero mean and a covariance matrix that does not depend on m 0 . For basics on Gaussian processes, consult for example, the book by Adler [1].For a subset A ⊆ Z d , we define the persistence probability (also called gap probability or hole probability) of X in A asIn particular, one may be interested inThis paper is exclusively about getting bounds on the persistence probability under some additional conditions on the Gaussian process.A stationary Gaussian process on Z d is uniquely described by its covariance kernel Cov(X m , X n ) = Cov(X 0 , X m−n ). Further, there exists a unique finite Borel measure µ on T d that is symmetric about the origin (i.e., µ(I) = µ(−I) for any Borel set I ⊆ T d ) such that Cov(X 0 , X m ) =μ(m) whereμ(m) = T d e i m,t dµ(t) with the usual notation for the inner product m, t = m 1 t 1 + . . . + m d t d . The measure µ is called the spectral measure of the process X. Write dµ(t) = b(t)dλ(t) + dµ s (t) where µ s is singular to Lebesgue measure and b ∈ L 1 (T d , λ) is non-negative. In all the results of this paper, it will be assumed that b is not identically zero. In other words, the spectral measure is not singular. Lastly, for a subset A ⊆ Z d , we denote the covariance matrix of (X m ) m∈A by Σ A := (μ(j − k)) j,k∈A and the cardinality of A by |A|.These notations will be maintained throughout the paper without further mention. In addition, there will appear many constants denoted by C, c, γ etc. Unless otherwise mentioned, the constants depend on the given process X (or equivalently, on the spectral measure µ).We now state our results and then give an overview of past results in the literature in Section 2. Our first theorem has already been proved by N. Feldheim and O. Feldheim [8] and but we explain in Section 2 why we include it here nevertheless.