McLeish's master inequality for the characteristic function of a bounded martingale difference sum

Let $(\mathcal{F}_k)_{k\in\mathbb{N}}$ be a filtration on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and let $(Z_k)_{k\in\mathbb{N}}$ be an adapted, integrable, uniformly bounded martingale difference sequence: $\mathbb{E}[Z_{k+1}\mid\mathcal{F}_k]=0$ a.s., $\mathbb{E}[Z_0]=0$, and $|Z_k|\le C$ everywhere. Fix $\theta, M\in\mathbb{R}$ and $n\in\mathbb{N}$, and suppose that along every path

• the squared variation is bounded: $\sum_{k<n} Z_k^2 \le M$, and
• the increments are small at scale $\theta$: $|\theta Z_k| \le 1$ for every $k<n$.

Then for every complex constant $c$, writing $S_n=\sum_{k<n}Z_k$,

What it does. This is the single inequality that carries McLeish's argument from the algebraic decomposition to the central limit theorem. It bounds the distance between the characteristic function of $S_n$ and an arbitrary target constant $c$ by two explicitly controllable quantities: a third-moment term and the distance from the random Gaussian factor $\exp(-\tfrac{\theta^2}{2}\sum_{k<n}Z_k^2)$ to $c$.

Everything after this point is limit-taking with no further structure. Choosing $c = e^{-\theta^2\sigma^2/2}$:

• the third-moment term is dominated by $\bigl(\max_{k<n}|\theta Z_k|\bigr)\,\theta^2\sum_{k<n}Z_k^2 \le |\theta|^3 M \max_{k<n}|Z_k|$, which vanishes under the negligibility hypothesis;
• the second term vanishes whenever $\sum_{k<n}Z_k^2\to\sigma^2$, by continuity of $t\mapsto e^{-\theta^2 t/2}$ and bounded convergence.

Hence $\mathbb{E}[e^{i\theta S_n}]\to e^{-\theta^2\sigma^2/2}$ for every $\theta$, and Lévy's continuity theorem gives $S_n \Rightarrow \mathcal{N}(0,\sigma^2)$.

Why an arbitrary constant $c$ is the right formulation. The martingale property enters only through the exact identity $\mathbb{E}\bigl[\prod_{k<n}(1+i\theta Z_k)\bigr]=1$. Because the expectation is exactly $1$, subtracting a constant $c$ is the same as subtracting $c\prod_{k<n}(1+i\theta Z_k)$, and the difference factors as

This step fails for a random $c$: one would be left with the extra term $\mathbb{E}[c\,(J^{(1)}_n-1)]$, which need not vanish. That is exactly why the random Gaussian factor cannot be used directly as the comparison object and must itself be compared to the constant $e^{-\theta^2\sigma^2/2}$ — the second term on the right-hand side.

The role of $M$. The prefactor $e^{\theta^2 M/2}$ is the uniform bound on $|J^{(1)}_n|$, coming from $|J^{(1)}_n|^2=\prod_{k<n}(1+\theta^2Z_k^2)\le e^{\theta^2\sum_{k<n}Z_k^2}\le e^{\theta^2 M}$. In McLeish's general theorem this pointwise bound is replaced by uniform integrability of $\{|J^{(1)}_n|\}$; the pathwise bound assumed here is the form in which the hypothesis is available in the applications (bounded or truncated arrays), and it keeps the estimate completely explicit.