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An invariance principle for random walk bridges conditioned to stay positive. (English) Zbl 1291.60090

Let \((X_{n})_{n\geq 1}\), be a sequence of i.i.d. random variables with its associated random walk \(S_{n}=X_1+\dots+X_{n}\), \(n\geq 1\). Assume there exists a positive sequence \(({a_{n}})_{n\geq 1}\), regularly varying with index \(1/\alpha\), \(0<\alpha\geq 2\), such that \(S_{n}/a_{n}\) converges in distribution to \(Y_1\) as \(n\rightarrow \infty\), where \(({Y_{t}})_{t\geq 0}\) is a stable Lévy process with index \(\alpha\) and parameter \(\rho\in ]0,1[\). Further, assume that either the law of \(X_1\) is supported by the lattice \(c+hZ\), where \(\operatorname{span} h\) is maximal and \(c\in [0,h[\), or the law of \(Y_1\) is absolutely continuous with respect to the Lebesgue measure, and the density \(\operatorname{P}(S_{n}\in dx)/dx\in L^{\infty}\) for some \(n\geq 1\). For \(N\geq 2\) and \(x,y\in [0,\infty[\), the random walk bridge of length \(N\), conditioned to stay positive, starting at \(x\) and ending at \(y\), is defined by the law \(\operatorname{P}_{x}(\cdot|S_1 \geq 0,\dots,S_{N-1}\geq 0,S_{N}=y)\). Analogously, for \(T>0\) and \(a,b\in [0,\infty[\), the bridge of \({Y_{t}}\), \(t\geq 0\) of length \(T\), conditioned to stay positive, starting at \(a\) and ending at \(b\), is defined by the law \(\operatorname{P}_{a}(\cdot|\inf_{0\leq t\leq T}Y_{t}\geq 0,Y_{T}=b)\). The authors show that the bridge of \(({S_{n}})_{n\geq 1}\), conditioned to stay positive, suitable rescaled, converges in distribution to the bridge of \(({Y_{t}})_{t\geq 0}\), conditioned to stay positive.

MSC:

60G50 Sums of independent random variables; random walks
60G51 Processes with independent increments; Lévy processes
60F17 Functional limit theorems; invariance principles
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