Math 6040 • Probability

Let \(\{X_{n}\}_{n \ge 1}\) be i.i.d. with mean 0 and variance 1. Let \(\{a_{n}\}_{n \ge 1}\) be a sequence of nonzero real numbers and \[ A_{n} = \left (\sum _{k=1}^{n} a_{k}^{2}\right )^{1/2}. \] Assume that \[ A_{n}^{-1}\left (\max _{1 \le k \le n} |a_{k}|\right ) \to 0 \qquad \text {as } n \to \infty . \] Find (and prove) a distributional limit for \[ A_{n}^{-1}\sum _{k=1}^{n} a_{k}X_{k} \qquad \text {as } n \to \infty . \]