Shift invariance of the Brownian bridge process

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In this note the distribution for the occuptation time of a one-dimensional Brownian bridge process on any Lebesgue measurable set between the initial and final states of the bridge is shown to be invariant under translation and reflection, so long as the translation or reflection also lies between the initial and final states of the bridge. The proof employs only the strong Markov property and elementary symmetry properties of the Brownian bridge process.

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