Feller converse to Lindeberg-Feller

Fix real t and put $v_{n,k}=\mathbb{E}{X}_{n,k}^2$ and $w_{n,k}=\mathbb{E}(e^{it{X}_{n,k}}-1)$. Centering, [F1], [F9] and [F10] give $\mathrm{\mid }{w}_{n,k}\mathrm{\mid }\le t^2{v}_{n,k}$. Thus ${\sum }_k\mathrm{\mid }{w}_{n,k}\mathrm{\mid }\le t^2$, ${\max }_k\mathrm{\mid }{w}_{n,k}\mathrm{\mid }\to 0$, and ${\sum }_k\mathrm{\mid }{w}_{n,k}{\mathrm{\mid }}^2\le t^4{\max }_k{v}_{n,k}\to 0$. By [F2]–[F5] and the assumed normal limit, $\exp ({\sum }_k{w}_{n,k})\to e^{-t^2\mathrm{/}2}$. This inference uses the forward continuity theorem only, not Lindeberg sufficiency.

Define the nonnegative function $d_t(x)=t^2{x}^2\mathrm{/}2-(1-\cos (tx))$. Nonnegativity follows from the cosine Taylor bound in [F1]; also $d_t(x)\le t^2{x}^2\mathrm{/}2$ because $\cos (tx)\le 1$. Hence $D_n(t):={\sum }_k\mathbb{E}{d}_t(X_{n,k})$ is finite and nonnegative. Total variance one and [F9] give $\mathrm{Re}{\sum }_k{w}_{n,k}=-t^2\mathrm{/}2+D_n(t)$. Exponential addition and Euler modulus show $\mathrm{\mid }\exp ({\sum }_k{w}_{n,k})\mathrm{\mid }=e^{-t^2\mathrm{/}2+D_n(t)}$. Step 1.1 therefore implies $e^{D_n(t)}\to 1$. Since the nonnegative series gives $0\le D_n(t)\le e^{D_n(t)}-1$, we obtain $D_n(t)\to 0$. No complex logarithm or subsequence of measures is needed.

Now fix any $\epsilon >0$ and take the single frequency $t=4\mathrm{/}\epsilon$. On $\mathrm{\mid }x\mathrm{\mid }>\epsilon$ we have $t^2{x}^2>16$, and $1-\cos (tx)\le 2$ yields $d_t(x)\ge t^2{x}^2\mathrm{/}2-2\ge t^2{x}^2\mathrm{/}4$. On the complementary set d_t is nonnegative. Integrating and summing gives ${\sum }_k\mathbb{E}[X_{n,k}^2{\mathbf{1}}_{\{\mathrm{\mid }{X}_{n,k}\mathrm{\mid }>\epsilon \}}]\le 4D_n(t)\mathrm{/}{t}^2\to 0$. This is precisely [F11], for every positive epsilon. AC is inherited from the target normal-law construction; the proof uses no Helly selection, uniqueness inversion or backward application of sufficiency. Zero entries and t=0 in the earlier steps are harmless, but the final chosen t is nonzero.