Fourth-Moment Bounds for Random-Walk Tightness

From Increment Moments to Path Control

To compare random walks with continuous paths, it is not enough to understand their values at a fixed finite set of times. A path could make large excursions between those times. Tightness in the space of continuous functions rules out that possibility by controlling how rapidly paths can vary on short time intervals. For polygonally interpolated, diffusively rescaled walks, a fourth-moment estimate gives exactly the increment control needed for this purpose.

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space, and let \((\xi_k)_{k\ge1}\) be independent, identically distributed real random variables with \(\mathbb{E}\xi_1=0\), \(\mathbb{E}\xi_1^2=\sigma^2<\infty\), and \(\mathbb{E}\xi_1^4=\beta<\infty\). Write \(S_0=0\) and \(S_k=\sum_{r=1}^k\xi_r\) for \(k\ge1\). For each positive integer \(n\), define a continuous, linearly interpolated path on \([0,\infty)\) by \(\widetilde S_n(t)=S_k+(nt-k)\xi_{k+1}\) when \(k=\lfloor nt\rfloor\), and set \(W_n(t)=n^{-1/2}\widetilde S_n(t)\). Thus \(W_n\) has state space \(\mathbb{R}\), continuous time parameter \(t\ge0\), and \(W_n(0)=0\). The goal is a bound uniform in \(n\) and in the time points.

The Fourth-Moment Bound

The key calculation is the fourth moment of a weighted sum. The weights arise because an interval can begin or end partway through an interpolation segment. For independent centered variables, terms involving an unpaired first power vanish, but terms involving two squared increments do not vanish; those cross terms are essential.

Proof. It is enough to consider \(s\le t\), since the absolute increment is unchanged when the endpoints are exchanged. Put \(h=t-s\). The unscaled interpolated increment \(\widetilde S_n(t)-\widetilde S_n(s)\) can be written as \(\sum_r w_r\xi_r\), where \(w_r\) is the length of the intersection of \([ns,nt]\) with \([r-1,r]\). Only finitely many weights are nonzero, each lies in \([0,1]\), and their sum is the length of \([ns,nt]\): \[ \sum_r w_r=nh. \] For any finite set of deterministic weights, independence and centering give \[ \mathbb{E}\left(\sum_r w_r\xi_r\right)^4 =\beta\sum_r w_r^4+6\sigma^4\sum_{r<q}w_r^2w_q^2. \] Indeed, in the expansion of the fourth power, a term with an index appearing just once has expectation zero by independence and \(\mathbb{E}\xi_r=0\). The remaining terms either use one index four times, contributing \(\beta w_r^4\), or two indices twice each; the latter have six orderings and contribute \(6\sigma^4w_r^2w_q^2\). Now \[ \beta\sum_r w_r^4+6\sigma^4\sum_{r<q}w_r^2w_q^2 \le C_4\left(\sum_r w_r^2\right)^2, \] because the square on the right expands as \(\sum_r w_r^4+2\sum_{r<q}w_r^2w_q^2\), and \(C_4\ge\beta\), \(2C_4\ge6\sigma^4\). Since \(0\le w_r\le1\), \(w_r^2\le w_r\), so \(\sum_r w_r^2\le nh\). Dividing by \(n^2\) for the rescaling \(W_n=n^{-1/2}\widetilde S_n\) gives \[ \mathbb{E}|W_n(t)-W_n(s)|^4 \le n^{-2}C_4(nh)^2=C_4h^2. \] If \(s=t\), both sides are zero, so the bound holds in that case too. \(\square\)

The proof covers intervals shorter than a mesh interval as well as intervals spanning many mesh points. In particular, it does not assume that either endpoint lies on the walk’s time grid. The exponent \(2\) on the right is what allows a dyadic argument to control all small increments at once.

A Dyadic Tightness Criterion

Here is the path-space result that turns an increment estimate into tightness. We give the dyadic argument because its summation shows precisely why the moment exponent matters. The result is stated for fourth moments, which is the case needed for the interpolated walks.

Proof. Fix \(0<\gamma<\tfrac14\), a number \(b>0\), and a nonnegative integer \(m_0\). At level \(m\), use the dyadic grid points \(t_{m,k}=kT2^{-m}\), \(k=0,\ldots,2^m\). Call a level-\(m\) adjacent increment bad if \[ |Z_a(t_{m,k+1})-Z_a(t_{m,k})|>b2^{-\gamma m} \] for at least one \(k\). Markov’s inequality gives, for each adjacent pair, \[ \mathbb{P}\bigl(|Z_a(t_{m,k+1})-Z_a(t_{m,k})|>b2^{-\gamma m}\bigr) \le \frac{C(T2^{-m})^2}{b^4 2^{-4\gamma m}} =\frac{CT^2}{b^4}2^{-(2-4\gamma)m}. \] There are \(2^m\) adjacent pairs at level \(m\), so the union bound gives probability at most \(\frac{CT^2}{b^4}2^{-(1-4\gamma)m}\) that level \(m\) has a bad increment. Summing over \(m\ge m_0\), and using \(1-4\gamma>0\), shows that the probability of any bad increment at any such level is at most \[ \frac{CT^2}{b^4}\, \frac{2^{-(1-4\gamma)m_0}}{1-2^{-(1-4\gamma)}}. \] This bound is uniform over \(a\).

On the event that no level \(m\ge m_0\) has a bad increment, continuity controls increments between grid points too. For \(x\in[0,T]\), let \(x_m\) be the largest level-\(m\) grid point not exceeding \(x\). For \(x<T\), the points \(x_l\) and \(x_{l+1}\) are either equal or adjacent at level \(l+1\), and \(x_l\) tends to \(x\). Continuity and the bounds on adjacent increments therefore imply \[ |Z_a(x)-Z_a(x_m)| \le\sum_{l=m+1}^{\infty}b2^{-\gamma l} =\frac{b2^{-\gamma(m+1)}}{1-2^{-\gamma}}. \] For \(x=T\), the approximation is exact. If \(|x-y|\le T2^{-m}\), the level-\(m\) grid points \(x_m,y_m\) are equal or adjacent, so their values differ by at most \(b2^{-\gamma m}\). Applying the preceding approximation to both endpoints yields \[ \omega_{Z_a}(T2^{-m}) \le b2^{-\gamma m} +\frac{2b2^{-\gamma(m+1)}}{1-2^{-\gamma}} =D_\gamma b2^{-\gamma m}, \qquad D_\gamma=1+\frac{2\cdot2^{-\gamma}}{1-2^{-\gamma}}. \] Consequently, for every \(m\ge m_0\), the probability that this modulus bound fails is at most the displayed uniform probability of a bad dyadic increment.

To conclude tightness, fix \(\varepsilon>0\). For each integer \(r\ge1\), choose \(m_r\) sufficiently large that \(D_\gamma 2^{-\gamma m_r}\le1/r\) and the probability bound above, with \(b=1\) and \(m_0=m_r\), is less than \(\varepsilon 2^{-r}\). Define \[ K=\{f\in C[0,T]: f(0)=0,\ \omega_f(T2^{-m_r})\le1/r\text{ for every }r\ge1\}. \] Each condition defining \(K\) is closed in the uniform distance: changing a path by at most \(\eta\) changes its modulus at any fixed scale by at most \(2\eta\). The conditions also make \(K\) equicontinuous, since \(1/r\) tends to zero. The paths are uniformly bounded as well: using the \(r=1\) condition and \(f(0)=0\), divide \([0,T]\) into finitely many intervals of length at most \(T2^{-m_1}\); each step changes a path value by at most \(1\). Thus the Arzelà–Ascoli compactness theorem applies, and \(K\) is compact. By the union bound, each \(Z_a\) lies in \(K\) with probability at least \(1-\sum_{r\ge1}\varepsilon2^{-r}=1-\varepsilon\). This is tightness. \(\square\)

The only compactness theorem used here is Arzelà–Ascoli, the standard characterization that a closed, uniformly bounded, equicontinuous subset of \(C[0,T]\) is compact. The probability estimate preceding it provides the required uniform equicontinuity; it does not assume that the paths are differentiable or that a bound on a finite grid alone controls their behavior between grid points.

Why the Cross Terms Matter

A common mistake is to compute the fourth moment of a sum of independent centered increments by adding only the individual fourth moments. For two independent Rademacher increments, that incorrect calculation gives \(\mathbb{E}(\xi_1+\xi_2)^4=1+1=2\). The correction follows directly from the expansion: \[ \mathbb{E}(\xi_1+\xi_2)^4 =\mathbb{E}\xi_1^4+\mathbb{E}\xi_2^4 +6\mathbb{E}(\xi_1^2\xi_2^2) =1+1+6(1)(1)=8. \] The same formula also explains why one must identify the increment distribution before quoting a numerical fourth moment. For independent, identically distributed centered increments with variance one and fourth moment \(\beta\), the answer for two increments is \(2\beta+6\), not always \(8\). For example, standard normal increments have \(\beta=3\), giving \(2(3)+6=12\).

There is a second important distinction: a uniform fourth-moment bound controls increments of the full interpolated paths, including times between mesh points. A calculation only at mesh times would not by itself establish the needed modulus control for continuous paths. The weights in the proposition handle partial segments, and the dyadic argument extends grid estimates to every time. These tools provide tightness of the rescaled paths on each finite interval; by themselves, they do not identify a limiting path law or prove convergence to Brownian motion.