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Timothy Chow
Timothy Chow
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The most commonly considered random walk on the line takes one step left or right with equal probability until a barrier is reached (if there are any barriers).

More generally, suppose we fix any probability distribution on the integers with compact support, and consider the random walk that begins at 0 and, at each step, takes an independent random draw from this probability distribution, and moves by that amount. (The random walk in the previous paragraph corresponds to the probability distribution $P(1)=1/2$ and $P(-1)=1/2$.) Suppose also that there is a positive integer $n$ such that

  1. $P(m)=0$ whenever $|m| \ge n$, and

  2. there are absorbing barriers at $+m$ and at $-m$ whenever $m\ge n$.

Obviously there are absorbing barriers at $+m$ and at $-m$ whenever $m\ge n$. Obviously, if the probability distribution is symmetric about 0 then the random walk is equally likely to terminate at a "positive" barrier as at a "negative" barrier.

Does there exist an asymmetric probability distribution such that, for all sufficiently large $n$, the associated random walk is equally likely to terminate at a positive barrier as at a negative barrier?

The most commonly considered random walk on the line takes one step left or right with equal probability until a barrier is reached (if there are any barriers).

More generally, suppose we fix any probability distribution on the integers, and consider the random walk that begins at 0 and, at each step, takes an independent random draw from this probability distribution, and moves by that amount. (The random walk in the previous paragraph corresponds to the probability distribution $P(1)=1/2$ and $P(-1)=1/2$.) Suppose also that there is a positive integer $n$ such that

  1. $P(m)=0$ whenever $|m| \ge n$, and

  2. there are absorbing barriers at $+m$ and at $-m$ whenever $m\ge n$.

Obviously, if the probability distribution is symmetric about 0 then the random walk is equally likely to terminate at a "positive" barrier as at a "negative" barrier.

Does there exist an asymmetric probability distribution such that, for all sufficiently large $n$, the associated random walk is equally likely to terminate at a positive barrier as at a negative barrier?

The most commonly considered random walk on the line takes one step left or right with equal probability until a barrier is reached (if there are any barriers).

More generally, suppose we fix any probability distribution on the integers with compact support, and consider the random walk that begins at 0 and, at each step, takes an independent random draw from this probability distribution, and moves by that amount. (The random walk in the previous paragraph corresponds to the probability distribution $P(1)=1/2$ and $P(-1)=1/2$.) Suppose also that there is a positive integer $n$ such that there are absorbing barriers at $+m$ and at $-m$ whenever $m\ge n$. Obviously, if the probability distribution is symmetric about 0 then the random walk is equally likely to terminate at a "positive" barrier as at a "negative" barrier.

Does there exist an asymmetric probability distribution such that, for all sufficiently large $n$, the associated random walk is equally likely to terminate at a positive barrier as at a negative barrier?

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Timothy Chow
Timothy Chow
  • 82.7k
  • 26
  • 363
  • 587

Asymmetric random walk on the line with barriers

The most commonly considered random walk on the line takes one step left or right with equal probability until a barrier is reached (if there are any barriers).

More generally, suppose we fix any probability distribution on the integers, and consider the random walk that begins at 0 and, at each step, takes an independent random draw from this probability distribution, and moves by that amount. (The random walk in the previous paragraph corresponds to the probability distribution $P(1)=1/2$ and $P(-1)=1/2$.) Suppose also that there is a positive integer $n$ such that

  1. $P(m)=0$ whenever $|m| \ge n$, and

  2. there are absorbing barriers at $+m$ and at $-m$ whenever $m\ge n$.

Obviously, if the probability distribution is symmetric about 0 then the random walk is equally likely to terminate at a "positive" barrier as at a "negative" barrier.

Does there exist an asymmetric probability distribution such that, for all sufficiently large $n$, the associated random walk is equally likely to terminate at a positive barrier as at a negative barrier?