Martingale central limit theorem

Write $v_{n,k}=\mathbb{E}[Z_{n,k}^2\mid {\mathcal{F}}_{n,k-1}]$. For every $\epsilon >0$, $v_{n,k}\le {\epsilon }^2+\mathbb{E}[Z_{n,k}^2{1}_{\{\mathrm{\mid }{Z}_{n,k}\mathrm{\mid }>\epsilon \}}\mid {\mathcal{F}}_{n,k-1}]\mathrm{.}$ Taking the supremum in $k$, bounding it by the sum of the nonnegative tail terms, and taking expectations gives $\mathbb{E}{\sup }_k{v}_{n,k}\le {\epsilon }^2+L_n(\epsilon )\mathrm{.}$ Consequently $\mathbb{E}{\sup }_k{v}_{n,k}\to 0$ after first taking $n\to \mathrm{\infty }$ and then $\epsilon \downarrow 0$.

First suppose ${\mathrm{\Gamma }}_{n,\mathrm{\infty }}\le c$ almost surely for one deterministic $c$. Fix $t\in \mathbb{R}$ and put $Q_{n,m}=\exp (it{M}_{n,m})\exp (t^2{\mathrm{\Gamma }}_{n,m}\mathrm{/}2)\mathrm{.}$ The exact telescoping identity is $Q_{n,m}-Q_{n,m-1}=e^{it{M}_{n,m-1}}{e}^{t^2{\mathrm{\Gamma }}_{n,m}\mathrm{/}2}(e^{it{Z}_{n,m}}-e^{-t^2{v}_{n,m}\mathrm{/}2})\mathrm{.}$ Here the prefactor before the parentheses is ${\mathcal{F}}_{n,m-1}$-measurable and has modulus at most $e^{t^{2}c\mathrm{/}2}$.

For any bounded ${\mathcal{F}}_{n,m-1}$-measurable complex $H$ and integrable complex $Y$, the defining conditional-expectation event integrals in F3 give $\mathbb{E}[HY]=\mathbb{E}[H\mathbb{E}[Y\mid {\mathcal{F}}_{n,m-1}]]$: prove it first for simple real $H$, approximate bounded real and imaginary parts by bounded simple functions, and apply F7 to each integrable product. The prefactor in step 1.2 is such an $H$, so this pull-out identity licenses conditioning the telescoping increment. Conditional centering, F2, and a split at $\mathrm{\mid }{Z}_{n,m}\mathrm{\mid }=\epsilon$ give $\mid \mathbb{E}[e^{it{Z}_{n,m}}\mid {\mathcal{F}}_{n,m-1}]-(1-t^2{v}_{n,m}\mathrm{/}2)\mid \le C_t\epsilon v_{n,m}+t^2\mathbb{E}[Z_{n,m}^2{1}_{\{\mathrm{\mid }{Z}_{n,m}\mathrm{\mid }>\epsilon \}}\mid {\mathcal{F}}_{n,m-1}],$ where $C_t$ is deterministic. The elementary exponential remainder also gives $\mathrm{\mid }{e}^{-t^2{v}_{n,m}\mathrm{/}2}-(1-t^2{v}_{n,m}\mathrm{/}2)\mathrm{\mid }\le C_{t,c}{v}_{n,m}{\sup }_j{v}_{n,j}\mathrm{.}$ Sum the expected telescoping errors from step 1.2. Since ${\sum }_m{v}_{n,m}={\mathrm{\Gamma }}_{n,\mathrm{\infty }}\le c$, step 1.1 and the Lindeberg hypothesis yield $\mathrm{\mid }\mathbb{E}{Q}_{n,\mathrm{\infty }}-1\mathrm{\mid }\le C_{t,c}(\epsilon +L_n(\epsilon )+\mathbb{E}{\sup }_j{v}_{n,j})⟶0$ after $n\to \mathrm{\infty }$ and then $\epsilon \downarrow 0$. The infinite telescoping limit is legitimate because $M_{n,m},{\mathrm{\Gamma }}_{n,m}$ converge almost surely and $\mathrm{\mid }{Q}_{n,m}\mathrm{\mid }\le e^{t^{2}c\mathrm{/}2}$; the displayed summable error bound controls passage of expectation through the partial telescopes.

Still under the bounded clock assumption, $\begin{array}{rl}{\displaystyle \mathrm{\mid }\mathbb{E}{e}^{it{M}_{n,\mathrm{\infty }}}-e^{-t^2\mathrm{/}2}\mathrm{\mid }}& {\displaystyle \le \mathbb{E}\mid 1-e^{t^2({\mathrm{\Gamma }}_{n,\mathrm{\infty }}-1)\mathrm{/}2}\mid }\\ {\displaystyle }& {\displaystyle +e^{-t^2\mathrm{/}2}\mathrm{\mid }\mathbb{E}{Q}_{n,\mathrm{\infty }}-1\mathrm{\mid }\mathrm{.}}\end{array}$ The first term tends to zero by bounded convergence from ${\mathrm{\Gamma }}_{n,\mathrm{\infty }}\to 1$ in probability (every subsequence has an almost-surely convergent subsubsequence), and the second tends to zero by step 2.1. Thus the characteristic functions converge to $e^{-t^2\mathrm{/}2}$. F4 proves $M_{n,\mathrm{\infty }}\Rightarrow N(0,1)$ in the bounded-clock case.

For the general case fix $c>1$ and define predictable truncated differences ${\tilde{Z}}_{n,k}=Z_{n,k}{1}_{\{{\mathrm{\Gamma }}_{n,k}\le c\}}\mathrm{.}$ The event is ${\mathcal{F}}_{n,k-1}$-measurable because ${\mathrm{\Gamma }}_{n,k}={\mathrm{\Gamma }}_{n,k-1}+v_{n,k}$. Their variance clock is bounded by $c$, their Lindeberg sums do not increase, and on $\{{\mathrm{\Gamma }}_{n,\mathrm{\infty }}\le c\}$ their terminal sum equals $M_{n,\mathrm{\infty }}$. Moreover their clock equals ${\mathrm{\Gamma }}_{n,\mathrm{\infty }}$ on that event, so it still converges in probability to $1$. step 3.1 gives ${\tilde{M}}_{n,\mathrm{\infty }}\Rightarrow N(0,1)$, while $\mathbb{P}({\tilde{M}}_{n,\mathrm{\infty }}\mathrm{\ne }{M}_{n,\mathrm{\infty }})\le \mathbb{P}({\mathrm{\Gamma }}_{n,\mathrm{\infty }}>c)⟶0.$ F5 transfers the weak limit to $M_{n,\mathrm{\infty }}$. No independence was used: the clock hypothesis and the unconditional Lindeberg hypothesis entered separately. AC has precisely the role in F6.