Infinite Total Variation of Brownian Paths

Variation and Brownian Increments

A continuous path can oscillate on shorter and shorter time scales, accumulating an unbounded amount of movement even while remaining bounded on every compact interval. For Brownian motion, the size of an increment over a time interval of length \(h\) is typically proportional to \(\sqrt{h}\). Subdividing a fixed interval into \(n\) equal pieces therefore gives a sum of absolute increments of order \(n\sqrt{1/n}=\sqrt n\). We will make this scaling precise and prove that the sums diverge almost surely. Since total variation is at least every partition sum, this implies infinite total variation.

Let \((\Omega,\mathcal F,\mathbb P)\) be a probability space, and let \((B_t)_{t\ge0}\) be standard Brownian motion with state space \(\mathbb R\), equipped, when needed, with its natural filtration \(\mathcal F_t=\sigma(B_s:0\le s\le t)\). We use a version with continuous sample paths, an event of probability one. For a continuous function \(f:[a,b]\to\mathbb R\), its total variation is

The supremum is over all finite partitions of the interval; it is allowed to be infinite. In particular, each individual partition sum is no larger than the total variation. We will use the uniform partition with \(N\) pieces of \([a,b]\), where \(T=b-a>0\), and write its sum as

For each fixed \(N\), the increments in this sum are independent and each has the normal distribution with mean \(0\) and variance \(T/N\). If \(Z\) is standard normal, then \(\mathbb E|Z|=\sqrt{2/\pi}\) and \(\operatorname{Var}(|Z|)=1-2/\pi\): indeed, \(\mathbb E|Z|=2(2\pi)^{-1/2}\int_0^\infty x e^{-x^2/2}\,dx=\sqrt{2/\pi}\), while \(\mathbb E|Z|^2=\mathbb E Z^2=1\). Set \(c=\sqrt{2/\pi}\) and \(v=1-2/\pi\). Scaling a normal increment by \(\sqrt{T/N}\) gives

The expectation grows like \(\sqrt N\). To establish an almost-sure conclusion, we need more than this expectation calculation: we will show that the normalized sums concentrate around their mean along dyadic values of \(N\).

A Dyadic Almost-Sure Scaling Result

Proof. For \(N=2^m\), define \(Y_{m,k}\) to be the absolute value of the \(k\)th increment divided by \(\sqrt{T/N}\), for \(1\le k\le N\). For each fixed \(m\), the \(Y_{m,k}\) are independent and each has the distribution of \(|Z|\). No independence between different values of \(m\) is needed. Their average satisfies $$ \frac{A_N(a,b)}{\sqrt{TN}}=\frac1N\sum_{k=1}^N Y_{m,k}, \qquad \mathbb E\left[\frac1N\sum_{k=1}^N Y_{m,k}\right]=c, \qquad \operatorname{Var}\left(\frac1N\sum_{k=1}^N Y_{m,k}\right)=\frac vN. $$ For any \(\varepsilon>0\), Chebyshev’s inequality therefore gives $$ \mathbb P\left(\left|\frac{A_N(a,b)}{\sqrt{TN}}-c\right|\ge\varepsilon\right) \le \frac{v}{N\varepsilon^2} =\frac{v}{2^m\varepsilon^2}. $$ For fixed \(\varepsilon\), these bounds sum to a finite number over \(m\). The first Borel–Cantelli implication follows directly from the union bound: the probability that any of these events occurs for some \(m\ge M\) is at most the sum of their probabilities over \(m\ge M\), and that tail sum tends to zero as \(M\to\infty\). Thus, with probability one, only finitely many of the events occur. Apply this argument to \(\varepsilon=1/r\) for each positive integer \(r\), and intersect the resulting countably many probability-one events. The normalized sums then converge to \(c\) almost surely.

Since \(c>0\), on this probability-one event there is an index after which \(A_{2^m}(a,b)/\sqrt{T2^m}\ge c/2\). Hence \(A_{2^m}(a,b)\ge(c/2)\sqrt{T2^m}\) for all sufficiently large \(m\), and the right-hand side tends to infinity. Every one of these sums is a partition sum for \([a,b]\). Their divergence forces the supremum defining total variation to be infinite. \(\square\)

The result identifies more than divergence: along these dyadic partitions, the sum is asymptotic to \(c\sqrt{T2^m}\). The proof uses independent increments within each partition and summable concentration bounds across the dyadic sequence; it does not require independence between partitions at different resolutions.

One Probability-One Event for Every Interval

Proof. For each pair of rational numbers \(r,s\) with \(0\le r<s\), the fixed-interval theorem gives an event of probability one on which the variation on \([r,s]\) is infinite. There are only countably many such pairs, so the intersection of these events, together with the event of continuous paths, still has probability one. Fix a path in this intersection and any real \(0\le a<b<\infty\). The density of the rationals gives rational \(r,s\) with \(a\le r<s\le b\). Every partition of \([r,s]\) can be extended to a partition of \([a,b]\) by adjoining \(a\) and \(b\) when necessary. The added terms are nonnegative, so the variation on \([a,b]\) is at least the variation on \([r,s]\). The latter is infinite, hence the former is infinite as well. Since the argument applies to every such \([a,b]\) on the same event, the conclusion is simultaneous over all nontrivial compact intervals. \(\square\)

Why Diverging Expectations Are Not Enough

A common mistake is to argue that because \(\mathbb E A_N(a,b)=c\sqrt{TN}\) tends to infinity, \(A_N(a,b)\) must tend to infinity almost surely. Diverging expectations alone do not justify that conclusion. For a concrete counterexample, let \(U\) be uniform on \((0,1)\) and define \(Y_m=2^{2m}\mathbf 1_{\{U\le2^{-m}\}}\). Then \(\mathbb E Y_m=2^{2m}2^{-m}=2^m\), which diverges. But for every outcome with \(U>0\), the inequality \(U\le2^{-m}\) fails for all sufficiently large \(m\), so \(Y_m=0\) eventually. Thus \(Y_m\to0\) almost surely even though its expectations diverge.

For Brownian partition sums, the correction is the concentration argument: after dividing by \(\sqrt{TN}\), the sums have a common mean \(c\), and Chebyshev’s bounds for deviations at \(N=2^m\) are summable. Borel–Cantelli turns those bounds into almost-sure convergence of the normalized sums. It is that additional step—not the growth of the expectations by itself—that establishes infinite variation.

The conclusion concerns total variation, which takes the supremum over all finite partitions. It is enough to find one sequence of partitions whose sums diverge; there is no need to calculate the supremum directly. The later study of quadratic variation asks about sums of squared increments, a different quantity with a different limiting behavior. Here the absolute increments and their square-root scaling are the essential ingredients.