  Consider a discrete-time martingale, and let $V^2$ be its normalized
quadratic variation. As $V^2$ approaches 1, and provided that some Lindeberg
condition is satisfied, the distribution of the rescaled martingale approaches
the Gaussian distribution. For any $p\geq 1$, (Ann. Probab. 16 (1988) 275-299)
gave a bound on the rate of convergence in this central limit theorem that is
the sum of two terms, say $A_p+B_p$, where up to a constant,
$A_p={\|V^2-1\|}_p^{p/(2p+1)}$. Here we discuss the optimality of this term,
focusing on the restricted class of martingales with bounded increments. In
this context, (Ann. Probab. 10 (1982) 672-688) sketched a strategy to prove
optimality for $p=1$. Here we extend this strategy to any $p\geq 1$, thereby
justifying the optimality of the term $A_p$. As a necessary step, we also
provide a new bound on the rate of convergence in the central limit theorem for
martingales with bounded increments that improves on the term $B_p$,
generalizing another result of (Ann. Probab. 10 (1982) 672-688).
