Skip to main content
MathOverflow

Return to Question

got rid of erroneous comment on Doob-h transform
Source Link
Matthew Junge
Matthew Junge
  • 457
  • 2
  • 9

Consider simple random walk started at $1$ on the path $[0,M]$. Let $\tau_x$ be the time to reach $x$ and condition on the event $\tau_M < \tau_0$ (the walk hits $M$ before reaching $0$). Conditioned on this event, let $V_k$ be the number of visits to $k$ up to time $\tau_M$. What is $\mathbb E V_k$?

All we need is a bound independent of $M$. Something like there exists $C>0$ such that

$$\sup_{M >0 }\Bigl[ \max_{0< k < M} \mathbb E V_k \Bigr]< C.$$

I suspect the Doob-$h$ transform will be useful. This guarantees that the conditioned walk has transition probabilities $P(i,j) = \frac{j}{2i}$ (see Example 1 of http://www.math.harvard.edu/~alexb/rm/Doob.pdf). So essentially the conditioned process is a biased random walk towards $M$ with shrinking bias.

Consider simple random walk started at $1$ on the path $[0,M]$. Let $\tau_x$ be the time to reach $x$ and condition on the event $\tau_M < \tau_0$ (the walk hits $M$ before reaching $0$). Conditioned on this event, let $V_k$ be the number of visits to $k$ up to time $\tau_M$. What is $\mathbb E V_k$?

All we need is a bound independent of $M$. Something like there exists $C>0$ such that

$$\sup_{M >0 }\Bigl[ \max_{0< k < M} \mathbb E V_k \Bigr]< C.$$

I suspect the Doob-$h$ transform will be useful. This guarantees that the conditioned walk has transition probabilities $P(i,j) = \frac{j}{2i}$ (see Example 1 of http://www.math.harvard.edu/~alexb/rm/Doob.pdf). So essentially the conditioned process is a biased random walk towards $M$ with shrinking bias.

Consider simple random walk started at $1$ on the path $[0,M]$. Let $\tau_x$ be the time to reach $x$ and condition on the event $\tau_M < \tau_0$ (the walk hits $M$ before reaching $0$). Conditioned on this event, let $V_k$ be the number of visits to $k$ up to time $\tau_M$. What is $\mathbb E V_k$?

All we need is a bound independent of $M$. Something like there exists $C>0$ such that

$$\sup_{M >0 }\Bigl[ \max_{0< k < M} \mathbb E V_k \Bigr]< C.$$

changed $\tau_1$ to $\tau_0$
Source Link
Matthew Junge
Matthew Junge
  • 457
  • 2
  • 9

Consider simple random walk started at $1$ on the path $[0,M]$. Let $\tau_x$ be the time to reach $x$ and condition on the event $\tau_M < \tau_1$$\tau_M < \tau_0$ (the walk hits $M$ before reaching $0$). Conditioned on this event, let $V_k$ be the number of visits to $k$ up to time $\tau_M$. What is $\mathbb E V_k$?

All we need is a bound independent of $M$. Something like there exists $C>0$ such that

$$\sup_{M >0 }\Bigl[ \max_{0< k < M} \mathbb E V_k \Bigr]< C.$$

I suspect the Doob-$h$ transform will be useful. This guarantees that the conditioned walk has transition probabilities $P(i,j) = \frac{j}{2i}$ (see Example 1 of http://www.math.harvard.edu/~alexb/rm/Doob.pdf). So essentially the conditioned process is a biased random walk towards $M$ with shrinking bias.

Consider simple random walk started at $1$ on the path $[0,M]$. Let $\tau_x$ be the time to reach $x$ and condition on the event $\tau_M < \tau_1$ (the walk hits $M$ before reaching $0$). Conditioned on this event, let $V_k$ be the number of visits to $k$ up to time $\tau_M$. What is $\mathbb E V_k$?

All we need is a bound independent of $M$. Something like there exists $C>0$ such that

$$\sup_{M >0 }\Bigl[ \max_{0< k < M} \mathbb E V_k \Bigr]< C.$$

I suspect the Doob-$h$ transform will be useful. This guarantees that the conditioned walk has transition probabilities $P(i,j) = \frac{j}{2i}$ (see Example 1 of http://www.math.harvard.edu/~alexb/rm/Doob.pdf). So essentially the conditioned process is a biased random walk towards $M$ with shrinking bias.

Consider simple random walk started at $1$ on the path $[0,M]$. Let $\tau_x$ be the time to reach $x$ and condition on the event $\tau_M < \tau_0$ (the walk hits $M$ before reaching $0$). Conditioned on this event, let $V_k$ be the number of visits to $k$ up to time $\tau_M$. What is $\mathbb E V_k$?

All we need is a bound independent of $M$. Something like there exists $C>0$ such that

$$\sup_{M >0 }\Bigl[ \max_{0< k < M} \mathbb E V_k \Bigr]< C.$$

I suspect the Doob-$h$ transform will be useful. This guarantees that the conditioned walk has transition probabilities $P(i,j) = \frac{j}{2i}$ (see Example 1 of http://www.math.harvard.edu/~alexb/rm/Doob.pdf). So essentially the conditioned process is a biased random walk towards $M$ with shrinking bias.

Source Link
Matthew Junge
Matthew Junge
  • 457
  • 2
  • 9

Local time for conditioned simple random walk

Consider simple random walk started at $1$ on the path $[0,M]$. Let $\tau_x$ be the time to reach $x$ and condition on the event $\tau_M < \tau_1$ (the walk hits $M$ before reaching $0$). Conditioned on this event, let $V_k$ be the number of visits to $k$ up to time $\tau_M$. What is $\mathbb E V_k$?

All we need is a bound independent of $M$. Something like there exists $C>0$ such that

$$\sup_{M >0 }\Bigl[ \max_{0< k < M} \mathbb E V_k \Bigr]< C.$$

I suspect the Doob-$h$ transform will be useful. This guarantees that the conditioned walk has transition probabilities $P(i,j) = \frac{j}{2i}$ (see Example 1 of http://www.math.harvard.edu/~alexb/rm/Doob.pdf). So essentially the conditioned process is a biased random walk towards $M$ with shrinking bias.