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A limit theorem for two-dimensional random walk conditioned to stay in a cone. (English) Zbl 0741.60068
Consider a two-dimensional random walk \({\mathcal Z}=\{Z_ k\}^ \infty_{k=0}\) on a probability space \((\Omega,{\mathcal F},P)\), starting at (0,0), and having stationary independent increments together with a few further properties. By “connecting the dots” and normalizing, one obtains a sequence of random functions \(Z^{(n)}: \Omega\to C[0,1]\), where \(C[0,1]\) denotes the Banach space of continuous \(\mathbb{R}^ 2\)-valued functions on \([0,1]\), and where \[ Z^{(n)}(t)=(Z_ k+(nt-k)(Z_{k+1}- Z_ k))/\sqrt n,\quad k/n\leq t\leq(k+1)/n,\quad k=0,1,\dots,n-1. \] The author studies the probabilistic behavior of \(Z^{(n)}\), conditioned on the event \(R_ n\) that \({\mathcal Z}\) remains inside a closed cone \(F\) up to and including time \(n\). (\(F\) is determined by two rays emanating from (0,0) at an angle \(\alpha<2\pi\), and its interior, \(F^ 0\), is assumed to contain the positive \(y\)-axis.) Thus, \(R_ n=\bigcap^ n_{k=0}(Z_ k\in F)\).
Let \(\tilde W^{(n)}\) denote the conditional distribution of \(Z^{(n)}\) given \(R_ n\). Thus, \(\tilde W^{(n)}(B)=P(Z^{(n)}\in B\mid R_ n)\) for every Borel subset \(B\) of \(C[0,1]\). The main result asserts that the sequence \(\{\tilde W^{(n)}\}\) converges weakly in \(C[0,1]\) to a probability \(\tilde W\) described by the author as the “law of the conditioned Brownian motion which starts at (0,0) and enters at once into \(F^ 0\), then stays there for a unit of time”. Also, the function \(L(n)=n^{\pi/2\alpha}P(R_ n)\) is slowly varying at infinity.

60G50 Sums of independent random variables; random walks
60F17 Functional limit theorems; invariance principles