# Questions tagged [random-walk]

For questions on random walks, a mathematical formalization of a path that consists of a succession of random steps.

1,384 questions
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### The conductance of a random walk on an undirected graph

Consider a random walk on an undirected graph consisting of an n-vertex path with self-loops at the both ends. With the self loops, we have $p_{xy}=\frac{1}{2}$ on all edges (x,y), and so the ...
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### Markov chain: Relation between absorbent states and its eigenvalue.

I'm a computer science student, and I'm really stuck in this lemma since I cant find out anything about the relation between a absorbent state (in a Markov chain $M$, a absorbent state is when the ...
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### Stopped random walk is not uniformly integrable

I know that in general Doob's Optional Stopping Theorem doesn't hold for unbounded stopping times, but that it does when the system up to the stopping time is uniformly integrable. One counter ...
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### Devise a strategy to get out of finite spaced random walk

Can there be a strategy to come out of 2-d finite spaced random walk in finite steps? (i.e. with probability 1)
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### Mathematical concept of time series modeling

I am trying to learn about time series (self learner), and I read about the mathematical (or statistical) concept behind analysing time series, however I am still confused about some notions. The ...
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### 2D Random Walk: Creating Closed Loops

I was wondering if there are any papers out there that study the following 2D random walk situation: 1) Start a random walk at the origin 2) Once the walk forms a loop, stop the walk and "remove the ...
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### Using the upper bound lemma, show that $\|Q-U\|\geq\frac{1}{2|G|}\sum^*d_{\rho}Tr(\hat{Q}(\rho)\hat{Q}(\rho)^*)$

Using the upper bound lemma: Let $Q$ be a probability on the finite group $G$. Then, $\|Q-U\|^2\leq\frac{1}{4}\sum^*d_{\rho}Tr(\hat{Q}(\rho)\hat{Q}(\rho)^*)$ where the sum is over all non-trivial ...
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### comparison of first hitting times re: discrete, symmetric random walk on $\mathbb{Z}$

Let $c, d \in \mathbb{Z}$ be such that $c < 0 < d$. Let $\tau_k = \inf_{n \geq 0} \left\{n: S_n = k\right\}$ be the first hitting time of state $k$, where $S_n$ denotes the position of a ...
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### Is an exponential function of a stochastic process a smooth function?

Let's say I have an Ito process $X_t$, and another process $Y_t=e^{\int_t^{t+\delta} X_s ds}$. I want to know that quadratic variation of $Y_t$ and another process. I know that the quadratic ...
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### Random walk on thin ice?

My Question: Is the stochastic process which is described below a (special case of a) well-studied model? What kind of properties are known under which assumptions? Let's think of a random walker on ...
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### Expected distance squared of random walk on an infinite hexagonal grid

I had a probability test and that was one of the questions: We have an infinite grid of hexagons like this: Each edge have length 1 and all the degrees are 120°. There's a particle in one of the ...
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### Conditional expectation of a bounded harmonic function

Let $G$ be a discrete group. Consider $G^{\mathbb{N}}$, the space of all sequences in $G$ equipped with product $\sigma$-algebra. Let $m:G^{\mathbb{N}} \to G^{\mathbb{N}}$ be the multiplication map ...
I wonder if someone can please check if I am on the right lines. I want to show that a random walk with a probability of a right step of $\frac{1}{4}$ is recurrent. I think I might have done ...
Let a random walf with a certain stopping time $T$. Say that step $i$ costs $i$. If we can compute the average value of $T$, what is the correct (non biaised) way to compute the average cost? Le \$E(...