Iterated-logarithm laws for convex hulls of random walks with drift (Q6597545)

Let \(Z,Z_1,Z_2,\ldots\) be independent and identically distributed random variables in \(\mathbb{R}^d\) with finite second moments, and \(S_n=Z_1+\cdots+Z_n\) the corresponding random walk started from \(S_0=\mathbf{0}\). Let \(\mathcal{H}_n\) denote the convex hull of this random walk at time \(n\), the smallest convex set in \(\mathbb{R}^d\) that contains \(S_0,\ldots,S_n\). The authors establish laws of the iterated logarithm for geometric functionals of \(\mathcal{H}_n\) in the case where the increment \(Z\) has non-zero mean. For the intrinsic volumes \(V_k(\mathcal{H}_d)\), \(k=1,\ldots,d\), associated with such a random walk, they show the existence of a constant \(\Lambda(d,k,\mathcal{L}_Z)\), depending only on \(d\), \(k\) and the law of \(Z\), such that\N\[\N\limsup_{n\to\infty}\frac{V_k(\mathcal{H}_n)}{\sqrt{2^{k-1}n^{k+1}(\log\log n)^{k-1}}}=\Lambda(d,k,\mathcal{L}_Z)\text{ almost surely.}\N\]\NIn the case \(k=d\) a Strassen-type variational characterization is given for these constants, with an explicit solution derived for \(d=2\). Thus, for example, in the two-dimensional case the authors show that if the increment \(Z\) has non-zero mean \(\mu\) and identity covariance matrix then the area \(A(\mathcal{H}_n)\) of \(\mathcal{H}_n\) satisfies\N\[\N\limsup_{n\to\infty}\frac{A(\mathcal{H}_n)}{n^{3/2}\sqrt{\log\log n}}=\frac{\lVert\mu\rVert}{\sqrt{6}}\text{ almost surely,}\N\]\Nwhere \(\lVert\cdot\rVert\) is the two-dimensional Euclidean norm. A further application is given to the centre of mass (running average) process associated with the random walk. The proofs make use of Strassen's functional law of the iterated logarithm for random walks with drift, and a zero-one law for functionals of convex hulls of random walks with drift.