## 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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