Lyapunov central limit theorem

Fix $\epsilon >0$. On $\mathrm{\mid }{X}_{n,k}\mathrm{\mid }>\epsilon s_n>0$, monotonicity of the positive real power gives $\mathrm{\mid }{X}_{n,k}{\mathrm{\mid }}^{\delta }>(\epsilon s_n{)}^{\delta }$. Multiplying by $X_{n,k}^2\mathrm{/}(\epsilon s_n{)}^{\delta }$ gives $X_{n,k}^2\le \mathrm{\mid }{X}_{n,k}{\mathrm{\mid }}^{2+\delta }\mathrm{/}(\epsilon s_n{)}^{\delta }$ there. Off that event the truncated square is zero and the right side is nonnegative, including X=0. [F3] therefore bounds $L_n(\epsilon )\le {\epsilon }^{-\delta }{s}_n^{-(2+\delta )}{\sum }_k\mathbb{E}\mathrm{\mid }{X}_{n,k}{\mathrm{\mid }}^{2+\delta }$. The exponent simplification uses [F4]. For positive delta, monotonicity of log and exp in the defining formula [F5] gives the asserted power monotonicity.

The right side tends to zero for this fixed positive epsilon, so Lindeberg holds for every epsilon. All hypotheses of [F2] are now satisfied: finite second moments, centered independent rows and positive total standard deviations were given. Apply it to obtain the stated limit. The positive delta is fixed across all rows; delta=0 is excluded because the displayed normalized second-moment sum would be one. AC is inherited from [F2].