Invariance Principle for Lifts of Geodesic Random Walks

We consider a certain class of Riemannian submersions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>π</mml:mi> <mml:mo>:</mml:mo> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>M</mml:mi></mml:mrow> </mml:math> and study lifted geodesic random walks from the base manifold <i>M</i> to the total manifold <i>N</i>. Under appropriate conditions on the distribution of the speed of the geodesic random walks, we prove an invariance principle, i.e., convergence to horizontal