Large deviations for independent random walks on the line (Q1902954)

For a system of infinitely many independent symmetric walks on \(\mathbb{Z}\) let \(K_n(x)\) be the number of visits of \(x \in \mathbb{Z}\) from time 0 to \(n -1\). The difference between two distinct sites \(x\) and \(y\) is \(L_n(x,y) = K_n(x) - K_n(y)\) (for simplicity, \(x = 0\) and \(y = 1\)). Let \(\zeta_0\) be the zeroth walk. For a deterministic initial configuration with asymptotic density 1, that is, \[ \lim_{n\to \infty} (2k + 1)^{-1} \sum^k_{x = -k} \zeta_0(x) =1, \] let \(Q\) be the associated probability measure. It was proved by \textit{J. T. Cox} and \textit{R. Durrett} [Probab. Theory Relat. Fields 84, No. 1, 67-82 (1990; Zbl 0692.60028)] that \[ \lim_{n \to \infty} n^{-1/2 - 3\theta/2} \log Q\{K_n > an^{1+\theta}\} = -4(a/3)^{3/2} \] with \(a > 0\), \(\theta \in (0,1)\); \[ \lim_{n \to \infty} n^{-1/2 - 6\theta/5} \log Q \{L_n > an^{3/4 + \theta}\} = -(5/4) (2/3)^{6/5} (b)^{6/5} \] with \(b > 0\), \(\theta \in (0,5/4)\). It can be checked that \(\zeta_n(x)\), \(x \in \mathbb{Z}\), converges in distribution to independent mean 1 Poisson variables as \(n \to \infty\). When \(\zeta_0(x)\), \(x \in \mathbb{Z}\), are independent mean 1 Poisson random variables, let \(P\) be the associate probability measure. Cox and Durrett also proved that \[ \liminf_{n \to \infty} (\log n)^{-1} n^{-1/2 - \theta} \log P\{L_n > bn^{3/4+\theta}\} \geq -(1 + 2\theta),\qquad b > 0. \] The goal of the paper is to determine more precise the behavior of the processes \(P\) and \(Q\) in the above described problems. Let \(\lambda_n\) be a sequence such that \(\lambda_n \to \infty\) and \((\log \lambda_n)/(\log n) \to \theta > 0\). The main result states that, with respect to \(P\), the system \[ ((\lambda^{-1}_n n^{-1} K_n, \lambda^{-1}_n (\log n)^{-1/4} n^{-3/4} L_n), \lambda_n (\log n)^{1/2} n^{1/2}) \] is a large deviation system, and so does the system \[ ((n^{-\theta - 1} K_n, n^{-5\theta/4-3/4} L_n), n^{3\theta/2 + 1/2}) \] with respect to \(Q\). The rate functions are given explicitly.