  We prove the central limit theorem (CLT) for a sequence of independent
zero-mean random variables $\xi_j$, perturbed by predictable multiplicative
factors $\lambda_j$ with values in intervals
$[\underline\lambda_j,\overline\lambda_j]$. It is assumed that the sequences
$\underline\lambda_j$, $\overline\lambda_j$ are bounded and satisfy some
stabilization condition. Under the classical Lindeberg condition we show that
the CLT limit, corresponding to a "worst" sequence $\lambda_j$, is described by
the solution $v$ of one-dimensional $G$-heat equation. The main part of the
proof follows Peng's approach to the CLT under sublinear expectations, and
utilizes H\"{o}lder regularity properties of $v$. Under the lack of such
properties, we use the technique of half-relaxed limits from the theory of
viscosity solutions.
