Tagged Questions

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Intuition in Random walk

Suppose $X_i$ are i.i.d. r.v. $S_n=X_1+\cdots+X_n$ is random walk. Why $\mathcal{F}_n =\sigma(X_1,\cdots,X_n)$ are called the information known at time n? I think We only know the measurability of ...
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Prove that a random walk on $\mathbb{Z}_+\cup \{0\}$ is transient

Prove that a random walk on $\mathbb{Z}_+ \cup \{0\}$ is transient with $p_{i,i+1}=\frac{i^2+2i+1}{2i^2+2i+1}$ and $p_{i,i-1}=\frac{i^2}{2i^2+2i+1}$. So since this Markov chain has only a single ...
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The variance of a simple random walk/process

I've been trying to wrap my head around this for the past day. Please help! Let $\epsilon_i = \pm 1$ with equal probabilities independently for $i=1,...,N$. Then $Z_i = \epsilon_1 + ... + \epsilon_i$ ...
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The random walk $S_n=a+\sum_{i=1}^nX_i$

Consider a variant of random walk defined as $$S_n=a+\sum_{i=1}^nX_i,$$ where $X_i$ takes either value $2$ with prob= $p$ or value $-1$ with prob =$1-p$. What is $P(S_n=b)$?
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Inequality between two Random Walks

Let's consider two Random Walks, $$x^{(1)}_t = x_0 + \sum_{i=1}^{t}\xi^{(1)}_i,$$ $$x^{(2)}_t = x_0 + \sum_{i=1}^{t}\xi^{(2)}_i.$$ The random variables $\xi^{(1)}_i$ are i. i. d. They take values on ...
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A Coupled Random Walk on the xy-Plane

Consider a point on the $xy$-plane whose position is updated in iterations. In each iteration, the point undergoes, with equal probability, either an $A$- or a $B$-update, defined as follows: ...
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Expected hitting time of one of two barriers

In the webpage "hitting time of one of two barriers", the probability that a non symmetric random walk hits one of two barriers is computed. The walker starts from $x=0$ and the barriers are located ...
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hitting time of one of two barriers

Let's consider a one-dimensional Random Walk. At each time the walker moves of one step to the right with probability $p$ and to the left with probability $q$, with $p+q=1$. The walk is not symmetric, ...
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Extinction probability of a simple birth death process

X is a simple birth death process with birth rate $\lambda n$ and death rate $\mu n$ Embedded within a simple birth death process is a simple random walk. Let $Y_n$ be the value of X at the time of ...
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Finding the exact solution of a difference equation

We know a particle moves two units to the right with probability $p$, or $1$ unit to the left with probability $q$, hence $(p+q=1)$. $$q_k=P\left(S_n=0\mid S_0=k\right)$$ We are asked to find the ...
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130 views

Using random walks to predict behavior rather than matrix decomposition

I want to create a model that tries to predict a user's behavior based on the random walks of similar users. The problem is similar to Netflix's recommendation challenge. One of the popular solutions ...
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 ...
I need a set of stochastic processes $x_i(t)$ with the following characteristics: At each time $t$, the jump of each variable can be just $+s$ or $-s$; The processes have to be mean reverting, so ...