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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.

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