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Origin-seeking random walk

Biased random walks in 2dBiased alternating random walk on a lattice in 1DRandom walk, Cat and mouseRandom walk problem in the planeRandom walk - expected distance not from originNumber of steps in a 2D random walk return to origindistance from the origin in a simple random walk on $mathbb Z^2$1-dimensional biaised random walk with chosen walk's pace and two absorbing boundariesWill a 2 dimensional random walk with random orientations almost certainly return near the origin infinitely often?(2-D Random Walk) Probability of Returning to Origin

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A random walker can choose the distance of each step but not the direction, which is continuous in q dimensions. The walker, who can initially be anywhere in q-space, wants to get ever-closer to the origin, in the sense that by taking enough steps he can be arbitrarily certain of being arbitrarily close. The walker travels, at each step t, a distance d_t = f(O_t-1,q) where O_t-1 is the distance from the origin at the end of step t-1.

I postulate that choosing d_t = O_t-1 will result in a path which converges on the origin for all q, and that no other choice will converge faster. Either prove this, or find the minimum value of q for which the path will not converge.

Is there a value of q for which (d_t = O_t-1) converges, but no other choice of d_t = f(O_t-1,q) converges?

random-walk

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0

$begingroup$

A random walker can choose the distance of each step but not the direction, which is continuous in q dimensions. The walker, who can initially be anywhere in q-space, wants to get ever-closer to the origin, in the sense that by taking enough steps he can be arbitrarily certain of being arbitrarily close. The walker travels, at each step t, a distance d_t = f(O_t-1,q) where O_t-1 is the distance from the origin at the end of step t-1.

I postulate that choosing d_t = O_t-1 will result in a path which converges on the origin for all q, and that no other choice will converge faster. Either prove this, or find the minimum value of q for which the path will not converge.

Is there a value of q for which (d_t = O_t-1) converges, but no other choice of d_t = f(O_t-1,q) converges?

random-walk

share|cite|improve this question

edited 17 hours ago

Justin

asked yesterday

JustinJustin

92

New contributor

Justin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

A random walker can choose the distance of each step but not the direction, which is continuous in q dimensions. The walker, who can initially be anywhere in q-space, wants to get ever-closer to the origin, in the sense that by taking enough steps he can be arbitrarily certain of being arbitrarily close. The walker travels, at each step t, a distance d_t = f(O_t-1,q) where O_t-1 is the distance from the origin at the end of step t-1.

I postulate that choosing d_t = O_t-1 will result in a path which converges on the origin for all q, and that no other choice will converge faster. Either prove this, or find the minimum value of q for which the path will not converge.

Is there a value of q for which (d_t = O_t-1) converges, but no other choice of d_t = f(O_t-1,q) converges?

random-walk

share|cite|improve this question

edited 17 hours ago

Justin

asked yesterday

JustinJustin

92

New contributor

Justin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited 17 hours ago

Justin

asked yesterday

JustinJustin

92

New contributor

Justin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited 17 hours ago

Justin

asked yesterday

JustinJustin

92

New contributor

Justin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked yesterday

JustinJustin

92

New contributor

Justin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked yesterday

JustinJustin

92