Abstract
Let \(\left( X_{i}\right) _{i\ge 1}\) be a sequence i.i.d. random variables and \(S_{n}=\sum _{i=1}^{n}X_{i}\), \(n\ge 1\). For any starting point \(y>0\) denote by \(\tau _{y}\) the first moment when the random walk \( \left( y+S_{k}\right) _{k\ge 1}\) becomes negative. We give some bounds of order \(n^{-3/2}\) for the expectations \(\mathbb {E}\left( g\left( y+S_{n}\right) ;\tau _{n}>n\right) \), \(y\in \mathbb {R}_{+}^{*}\) which are valid for a large class of bounded measurable function g with constants depending on some norms of the function g.
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Grama, I., Le Page, É. (2017). Bounds in the Local Limit Theorem for a Random Walk Conditioned to Stay Positive . In: Panov, V. (eds) Modern Problems of Stochastic Analysis and Statistics. MPSAS 2016. Springer Proceedings in Mathematics & Statistics, vol 208. Springer, Cham. https://doi.org/10.1007/978-3-319-65313-6_6
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DOI: https://doi.org/10.1007/978-3-319-65313-6_6
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