Finite moments control increments over short moving windows

First assume independence and fix positive integers $A$ and $d$ with $d\ge 0$. Choose $a$ strictly between $p$ and $q$, put $M_q=\mathbb{E}\mathrm{\mid }{Z}_1{\mathrm{\mid }}^q$, and, for dyadic integers $N=2^m\ge 2$, put $b_N=N^{1\mathrm{/}a}$, ${\mathrm{\ell }}_N=\log N$, and $r_N=\lceil A{N}^{1\mathrm{/}p}{\mathrm{\ell }}_N^d\rceil$. The second moment $v=\mathbb{E}{Z}_1^2$ is finite since $\mathrm{\mid }z{\mathrm{\mid }}^2\le 1+\mathrm{\mid }z{\mathrm{\mid }}^q$. If $v=0$, the nonnegative variable $Z_1^2$ has zero expectation and is zero almost surely: for each integer $h\ge 1$, [F2] bounds $\mathbb{P}(Z_1^2\ge 1\mathrm{/}h)$ by zero, and the union of these events is $\{Z_1\mathrm{\ne }0\}$. Equality of marginal laws and countable subadditivity make every $Z_k$ zero simultaneously almost surely. This proves the assertion in this case. Henceforth assume $v>0$.

The union bound and [F2] give $\mathbb{P}({\max }_{k\le N}\mathrm{\mid }{Z}_k\mathrm{\mid }>b_N)\le M_q{N}^{1-q\mathrm{/}a}$. Since $1-q\mathrm{/}a<0$, these probabilities are summable over dyadic $N$. By [F3], eventually no one of the first $N$ variables is truncated at level $b_N$, almost surely along the dyadic horizons. Define $m_N=\mathbb{E}[Z_1{\mathbf{1}}_{\{\mathrm{\mid }{Z}_1\mathrm{\mid }\le b_N\}}]$ and $V_{N,k}=Z_k{\mathbf{1}}_{\{\mathrm{\mid }{Z}_k\mathrm{\mid }\le b_N\}}-m_N$. For each fixed $N$, these variables are independent and centered, $\mathrm{\mid }{V}_{N,k}\mathrm{\mid }\le 2b_N$, and $\mathrm{Var}(V_{N,k})\le v$. Moreover centering of $Z_1$ gives $\mathrm{\mid }{m}_N\mathrm{\mid }\le \mathbb{E}[\mathrm{\mid }{Z}_1\mathrm{\mid }{\mathbf{1}}_{\{\mathrm{\mid }{Z}_1\mathrm{\mid }>b_N\}}]\le M_q{b}_N^{1-q}$.

For an interval of at most $r_N$ indices, [F1] bounds the tail of its $V_{N,k}$ sum by $2\exp (-x^2\mathrm{/}[2(v{r}_N+2b_{N}x\mathrm{/}3)])$; if its actual variance is zero the bound still holds because the tail is zero. Set $x_N=C(\sqrt{r_N{\mathrm{\ell }}_N}+b_N{\mathrm{\ell }}_N)$, where $C\ge \max \{\sqrt{20v},40\mathrm{/}3,1\}$. Then $2v{r}_N\le x_N^2\mathrm{/}(10{\mathrm{\ell }}_N)$ and $4b_N{x}_N\mathrm{/}3\le x_N^2\mathrm{/}(10{\mathrm{\ell }}_N)$, so the bound is at most $2e^{-5{\mathrm{\ell }}_N}=2N^{-5}$. There are at most $N^2$ intervals with integer endpoints in $[0,N]$. The probability that any interval of length at most $r_N$ violates the bound $x_N$ is therefore at most $2N^{-3}$, summable over dyadic $N$. By [F3], all such interval sums are bounded by $x_N$ eventually almost surely.

On the eventual no-truncation event from step 2.1, every original interval sum differs from its centered truncated sum by at most $r_N\mathrm{\mid }{m}_N\mathrm{\mid }$. Now $\sqrt{r_N{\mathrm{\ell }}_N}=o(N^{1\mathrm{/}p})$ because $1\mathrm{/}(2p)<1\mathrm{/}p$, and $b_N{\mathrm{\ell }}_N=o(N^{1\mathrm{/}p})$ because $1\mathrm{/}a<1\mathrm{/}p$. Also $r_N\mathrm{\mid }{m}_N\mathrm{\mid }\le M_q{r}_N{N}^{(1-q)\mathrm{/}a}=o(N^{1\mathrm{/}p})$. Thus step 3.1 proves the claimed little-o bound at dyadic horizons. These comparisons follow by dividing by $N^{1\mathrm{/}p}$: in each expression a strictly negative power of $N$ multiplies a fixed power of $\log N$, which tends to zero.

For $2^{m-1}<n\le 2^m=N$, monotonicity of $n^{1\mathrm{/}p}(\log n{)}^d$ gives $r_n\le r_N$, so every admissible interval for $n$ is included among those for $N$. Since $N^{1\mathrm{/}p}\le 2^{1\mathrm{/}p}{n}^{1\mathrm{/}p}$, step 4.1 proves the assertion at every horizon $n$. Intersect the probability-one events obtained for the countably many integer pairs $A\ge 1,d\ge 0$. For arbitrary real $A>0,d\ge 0$, choose integers $A^{\mathrm{\prime }}\ge A$ and $d^{\mathrm{\prime }}\ge d$; when $n\ge 3$, $\log n\ge 1$ makes the real-parameter window no longer than the integer-parameter window. The conclusion therefore holds on one event for every $A,d$ in the Statement.

Under one-dependence, each parity subsequence is independent. Indeed, the sigma-algebra of any finite initial collection of even variables is contained in $\sigma (Z_1,\dots ,Z_{2h})$, which is independent of $Z_{2h+2}$ by the given separation; induction on the number of factors proves the finite rectangle factorization for all even variables. The identical argument applies to the odd variables. Both subsequences retain the common centered marginal and its $q$th moment. Apply step 5.1 to each subsequence and intersect their two probability-one events. An interval of at most $r_n$ original indices splits into one contiguous interval of each parity, each with at most $r_n$ terms and with its subsequence endpoints at most $n$. Each component is $o(n^{1\mathrm{/}p})$ by the independent result, uniformly over the original intervals; their sum has the same bound. No independence between the parity subsequences is asserted or needed. All windows are nonempty by $i<j$; empty component windows contribute zero. This completes both dependence cases, including zero variance and every permitted window parameter.