A central limit theorem for weighted averages of spins in the high temperature region of the Sherrington-Kirkpatrick model

In this paper we prove that in the high temperature region of the Sherrington-Kirkpatrick model for a typical realization of the disorder the weighted average of spins $\sum_{i\leq N} t_i \sigma_i$ will be approximately Gaussian provided that $\max_{i\leq N}|t_i|/\sum_{i\leq N} t_i^2$ is small.