# Tagged Questions

13 views

### Simple random walk conditioning on non-return

Consider a simple symmetric random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0=0$. Let $P_{k,j}$ be the probability that the walker hits the point $k$ without returning to the origin in ...
34 views

### Exact probability distribution for hitting time of simple random walk

Consider simple random walk on the line starting from the site $y \in \mathbb{N}$. With probability $p$ the walker moves to the right and with probability $1-p$ to the left. Call $\tau$ the first time ...
31 views

### Writing down the transition matrix of a discrete Markov chain

Please consider the following scenario: One person is walking along a discrete circle induced by $\mathbb{Z}/n\mathbb{Z}$ In each round we roll a dice with $w\in\left\{2,\ldots n\right\}$ sides If ...
70 views

### proving null recurrence of random walk (Markov chain)

How would I prove that the zero state of a random walk with a positive probability of staying in the same state is null recurrent. (sorry if this isn't a random walk and just a Markov chain.) eg. ...
70 views

26 views

### Prove equilibrium theorem without irreducibility and aperiodicity

I have to solve the following question: Consider a random walk Markov chain on $S = \{1, 2, \ldots, 100\}$. If the chain is between 2 and 99, it selects one of the adjacent states with equal ...
26 views

### Why must a stochastic process be at least second order in terms of differential equations?

A first order differential equation in $q(t)$ has a unique path through each possible value of $q(0)$. This is opposed to a stochastic process (e.g. random walk), where any place might be "hopped ...
64 views

### Absorbing state for a collection of random walks

Further to this question; having learned some stuff since I posed it. Consider a collection of random walks $X_i$ which take finite integer values. These evolve as time-inhomogeneous Markov Chains. ...
129 views

### Conditional probability and integrating out part of a random walk

Suppose that I have a random walk process defined by $\alpha_{t+1}$ ~ N$(\alpha_t, \omega^2)$. Given $\alpha_t$ and $\alpha_{t+2}$, I understand why the conditional formula for ...
135 views

### geometric sum - weighted random walk

I am trying to model the following sum: $\sum_{i=0}^{n}{W_i \alpha^{i}}$ where $\alpha \in[0, 1)$ and $W_n$ takes values 0 or 1 and may be modeled as a markow chain or for simplicity as a binary ...
356 views

### Covariance of Brownian-motion-like processes

We know that $\operatorname{Cov}(B_s,B_t)=\min(s,t)$ if $B_t$ is Brownian motion. What is $\operatorname{Cov}(B_{f(s)},B_{f(t)})$ for some injective $f$? How can I write $B_{f(t)}$ in an Ito ...