Let \((V_n)_{n\geq1}\) be a real-valued martingale with respect to a filtration \(({\mathcal G}_n)_{n\geq1}\). Suppose that \((V_n)_{n\geq1}\) converges in the mean to a random variable \(V\). For every integer \(n\geq1\), set \(W_n=\sqrt{n}(V_n-V)\) and let \(K_n\) denote a version of the conditional distribution of \(W_n\) given \({\mathcal G}_n\). Sufficient conditions are presented under which the probability measures \(K_n(\omega,\cdot)\) converge weakly to the Gaussian kernel \(N(0,U(\omega))\) for almost all \(\omega\) in the underlying probability space, where \(U\) is an appropriate positive random variable. The result is then applied to a generalized Pólya urn model.