How Long does it Take to Train an Elephant Random Walk

Abstract We study how conditioning on the first k steps, which we think of as training, affects the long-term behavior of the Elephant Random Walk. When the elephant is conditioned to be at position k at time k , the first return time to the origin scales as $$k^{(4-4p)/(3-4p)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> in the diffusive regime, and grows exponentially in the critical regime. We loosely interpret this as a measurement of the rate at which the elephant forgets its training.