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Local asymptotics for the area of random walk excursions. (English) Zbl 1316.60063

Let \(\xi_1\), \(\xi_2,\ldots\) be independent copies of a random variable \(\xi\) with zero mean and finite variance \(\sigma^2\). Suppose further that the law of \(\xi\) is concentrated on the arithmetic progression \((\rho+dk)_{k\in\mathbb{Z}}\), where \(d>0\) is maximal possible and \(\rho\in [0,d)\). For \(n\in\mathbb{N}\), set \(S_n:=\sum_{i=1}^n \xi_i\), \(A_n:=\sum_{k=1}^n S_k\) and then \(\tau:=\inf\{i\in\mathbb{N}:S_i\leq 0\}\).
The authors show that \[ \lim_{n\to\infty}\sup|n^{3/2}\mathbb{P}\{A_n=a|\tau=n+1\}-\sigma^{-1}df(\sigma^{-1}n^{-3/2}a)|=0, \] where the supremum is taken over appropriate \(a\), \(f\) is the density of \(\int_0^1 e(t)dt\), and \((e(t))_{t\in [0,1]}\) is the standard Brownian excursion. A similar result is also established for \(\mathbb{P}\{A_n=a|\tau=n+1,S_n=x\}\). The authors demonstrate that their findings are relevant to the problem of enumeration of Dyck paths below a line of rational slope. An important intermediate step of the proof of the main results consists in showing that, for every \(z>0\), \[ \lim_{n\to\infty}\sup|n^2\mathbb{P}_z\{A_n=a, S_n=x|\tau>n\}-\sigma^{-2}d^2h(\sigma^{-1}n^{-3/2}a, \sigma^{-1}n^{-1/2}x)|=0, \] where \(\mathbb{P}_z\) is the distribution of the random walk with \(S_0=z\); the supremum is taken over appropriate \((a,x)\), \(h(u,v)\) is the density of \((\int_0^1M(t)dt, M(1))\) with \((M(s))_{s\geq 0}\) being the Brownian meander. Furthermore, it is checked in two different ways that the density \(h(u,v)\) is continuous.

MSC:

60G50 Sums of independent random variables; random walks
60F99 Limit theorems in probability theory
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