A Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0
Let \(b\) be a Brownian bridge, extended to a process with period \(1\), indexed by \(t\in {\mathbb R}\), \(e\) be the positive normalized Brownian excursion, \(X_{a}\) be \(|b|\) conditioned to have local time at \(0\) equal to \(a\), \(Y_{a}=\Psi_{a} b\), \(Z_{a}=\Psi_{a} e\), where \((\Psi_{a}f)(t)= \sup_{s\leq t}(f(t)-f(s)-a(t-s))\). For a process \(X\), \(T(X)\) is the inverse of the local time \(L(X)\) at \(0\) of \(X\). Let \(U\) be uniform on \([0,1]\). Results: \(Z_{a}(U+\cdot)\) is identical in law with \(Y_{a}\) if \(U\) is independent of \(Z_{a}\) and the same replacing \(Z_{a}\) by \(X_{a}\). For \(a>0\), \(X_{a}\) is identical in law to \(Z_{a}(\cdot +T_{aU}(Z_{a}))\) if \(U\) is independent of \(Z_{a}\) and the same replacing (everywhere) \(Z_{a}\) by \(Y_{a}\). If \(a>0\), then \(L_{t}(X_{a})\) reaches its maximum at a unique \(V\in [0,1)\) and \(X_{a}(V+\cdot)\) is identical in law with \(Z_{a}\) and the same replacing (everywhere) \(X_{a}\) by \(Y_{a}\); the \(V\) for \(Y_{a}\) is uniform on \([0,1]\) and independent of \(Y_{a}(V+\cdot)\). We have \(b(t)= Y_{a}(t)-Y_{a}(0)-L_{t}(Y_{a})+at\) and \(e(t)=Z_{a}(t)-L_{t}(Z_{a})+at\). The Vervaat results are the first two for \(a=0\). Proofs are given in the paper; they rely upon a concept of ``parking scheme'': Consider \(1,\dots ,n\) written in a circular order and \(\omega_{1},\dots ,\omega_{m}\) among them, the \(i\)th car parks in the first place in \(\omega_{i},\omega_{i}+1,\dots \) which is empty after the previous parked. Among the results involving such schemes we mention those related to the uniform distribution, convergence in law to \(Y_{a}\) of a process constructed from random schemes using the numbers of cars which tried to park in \(k\). In the final section the possibility of other ``shifts'' relating \(X_{a}\), \(Y_{a}\), \(Z_{a}\) is discussed and the following corollary is mentioned: let \(\varrho\) be uniform on \([0,1]\), independent of \(X_{a}\), let \(d\),\(g\) denote the first \(0\) after \(\varrho\) and the last \(0\) before \(\varrho\), respectively. Then \(d-g\) is distributed as \(N^{2}(a^{2}+N^{2})^{-1}\) with \(N\) Gaussian standard, \(X_{a}\) restricted to the interval \((d,g)\) and ``normalized'' gives a normalized Brownian excursion \(q\), independent of \(d\),\(g\) and, conditioned on \((d-g,q)\), \(X_{a}\) restricted to the interval \((d,g+1)\) is identical in law to \(X_{a(1-(d-g))^{-1/2}}\); the same for \(Y_{a}\) and \(Z_{a}\).
scientific article; zbMATH DE number 1906039
scientific article