Machi Millaray Valdivia, marzo de

We only show the results for the walks with drift here. The equivalent result for the zero drift case is the subject of Chapter 4.

First, we have the Normal unit drift, for which we show the result with the vertical axis scaled to see the whole range of possible values the ratio can take.

0e+00 2e+04 4e+04 6e+04 8e+04 1e+05

2.0 2.2 2.4 2.6 2.8 3.0

Figure 2.12: A demonstration of the law of large numbers for the ratio of the perimeter length and diameter applied to the random walk with drift and all coordinates Normally distributed, unit mean. The ratio is plotted for the first 105 _steps.

Clearly, this scaling, although informative in some sense, is too zoomed out to see anything interesting, so we provide the same plot with the vertical axis only showing values near to 2.

Even on this scaling, the convergence seems to be fairly fast. We provide similarly scaled (but note, not exactly) plots for the other two walks with drift.

2.3. Application of results to our examples 58

0e+00 2e+04 4e+04 6e+04 8e+04 1e+05

2.00 2.01 2.02 2.03 2.04 2.05

Figure 2.13: A demonstration of the law of large numbers for the ratio of the perimeter length and diameter applied to the random walk with drift and all coordinates Normally distributed, unit mean. The ratio is plotted for the first 105 _{steps, with the vertical}

2.3. Application of results to our examples 59

0e+00 2e+04 4e+04 6e+04 8e+04 1e+05

2.0000 2.0002 2.0004 2.0006 2.0008 2.0010

Figure 2.14: A demonstration of the law of large numbers for the ratio of the perimeter length and diameter applied to the random walk with drift and all coordinates Normally distributed, mean of length 5. The ratio is plotted for the first 105 _{steps, with the}

2.3. Application of results to our examples 60

0e+00 2e+04 4e+04 6e+04 8e+04 1e+05

2.000 2.005 2.010 2.015 2.020 2.025 2.030

Figure 2.15: A demonstration of the law of large numbers for the ratio of the perimeter length and diameter applied to the random walk with drift and no variance in the first coordinate, mean of length 5. The ratio is plotted for the first 105 _{steps, with the}

Chapter 3

Functional limit approach

The results on the perimeter length and diameter in Chapter 2 can be interpreted as an indication that the convex hull exhibits particular shapes after the random walk takes a large number of steps. In this chapter, we consider this question of the shape of the random walk and in turn the convex hull, and how this can be extended to be informative about functionals of the convex hull, not least, returning to the perimeter length and diameter1_{. Here we do not restrict ourselves to 2 dimensions as before, so}

our functionals become more general too.

The heuristic idea underlying the functional limit theorems starts with the story of law of large numbers and central limit theorem, see Theorem 1.3.14 and Theorem 1.3.15. These early results refer to the sums of random variables, or the endpoint of our random walks, which were the first real quantities of interest in this area due to their application in the contexts of long run profit in gambling games and errors when sampling large amounts of data. The functional limit theorem extension of these results, presented in Billingsley [Bil99], considers the paths of the random walks not just the endpoints confirming the intuitive idea that the path moves linearly towards its endpoint, at least on the law of large numbers scaling.

As the pictures of our examples in Section 2.3 suggest, the case with drift and the zero drift case will, unsurprisingly, be seen to behave differently. Under the law of large

1_{All sections in this chapter except Section 3.4 are based on work published in [LMW18]. The}

theory and the maximum functional example were joint work with the other authors, but the arc-sine law and the convex hull material was written independently.

Chapter 3. Functional limit approach 62

numbers scaling, the walk with drift converges to the unit vector in the direction of the mean whilst the zero-drift case is degenerate converging to the point at the origin (which itself could be considered as the degenerate unit vector in the direction of the origin). The zero-drift case attains a non-trivial limiting distribution under the central limit theorem scaling, for which it converges to Brownian motion in the sense of weak convergence of functions, to be formally defined later. Our contribution to the theory for the random walks is to show that these results extend to higher dimensions in the natural way.

In order to see the theory in action, we present the example of the maximum functional which also serves to demonstrate how the continuous mapping theorem can be used to determine information about functionals of the random walk. We then give another example, this time an original generalisation of the arcsine law to higher dimensions which states that the walk’s direction has no limiting direction or subset of directions. This example is nicely coherent with our shape result in Chapter 4 which, loosely speaking, says that the random walk with zero-drift approximates any shape with unit diameter infinitely often, after appropriate scaling. If the random walk’s direction had a limiting subset of the sphere as our arc-sine law rules out, then it would not be surprising if some shapes could not be well-approximated infinitely often because this would require increasingly large, and unlikely, jumps. Conversely, if the walk had a limiting shape, it would not be surprising to find that the walk’s directions had a limiting set, or at least could not infinitely often spend almost every time point in a subset which would contradict the directions of points in the limiting shape. So although the arc-sine law relates to the walk and not the convex hull, they should not be seen as isolated results.

With the strategy understood and preliminary examples presented, we consider the point set of the random walk. Without even taking the convex hull of these points, we will already be in a position to study the diameter of the convex hull, because this coincides with the diameter of the point set. Next, using either this convergence of sets, or the trajectory convergence, we will extend the results to the convergence of the convex hulls and consider the convergence of the mean width, volume and surface area, all defined later on. Again, the results are quite different in the case with drift, which