# Questions tagged [random-walk]

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

1,428 questions
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### Sample autocorrelation of random walk with drift

I would like to calculate the autocorrelation of a sample whose data generating process is a random walk with drift. I generated the movement over 250 time points of a fictious stock price with ...
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### What is the probability a random walk hits x before it hits y?

This problem was motivated by my bitcoin trading and recalling some of my math education back in the day. I thought I'd ask people who know this much better than I... Suppose there is a continuous, ...
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### Non-markovian random walks and their applications in machine learning

I'm searching applications of random walks in machine learning. In particular, applications of random walks with long memory. An example of this kind of processes is the so called ELEPHANT RANDOM WALK....
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### Conditional expectation for a simple random walk

Suppose that $S_n$ is a simple random walk started at $0$, so that $S_n = X_1 + \dots + X_n$ where $X_j$'s are iid random variables taking values $1$ and $-1$ with probability $p = 3/4$ and $q=1/4$ ...
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### Escape time probability distribution

I have a system where a random walker is moving on $\mathbb{Z}$. However, at each point in $\mathbb{Z}$, there is a probability $q$ that an escape route exists along which the walker can escape. I ...
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### Estimating the discrete random walk probability by error function

I am trying to work out the asymptotic large $t$ behavior of following function f(t ) = \sum_{x = 0}^{2t} { 2t \choose t + x} p^{ t+x } (1 - p)^{ t - x} = \sum_{x = 0}^{2t} { 2t \...
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### Prove a general random walk is not stationary

Here is my attempt. $X_t$ is a general random walk is when we have a sequence of independent and identically distributed random variables $Y_1,Y_2,...$ such that $X_0=0,$ $X_1=Y_1$, $X_2=Y_1+Y_2$ and ...
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### Does the specific sequence of random variables converge almost surely to a given constant?

Suppose, $\{X_n\}_{n = 1}^{\infty}$ is a sequence of i.i.d random variables, such that $P(X_i = 1) = P(X_i = -1) = \frac{1}{2}$. Now, suppose $\{S_n\}_{n = 1}^\infty$ is a sequence of random variables ...
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We consider a Markov chain on a subset of positive integers $S =$ {$0, 1, 2, 3, .......N$}, with transition probabilities defined as follows: The chain jumps only one unit to the left or right. $p(i,... 1answer 48 views ### Simplifying summation of binomials I am working on a proof and think that the following equality holds but am unable to prove it: $$\sum_{k_1=0}^{m} \sum_{k_2=0}^{m-k_1} \dots \sum_{k_{d^2/2}=m-(k_1+k_2+\dots+k_{d^2/2-1})}^{m-(k_1+k_2+... 1answer 34 views ### Symmetric random walk calculate In basic, symmetric random walk with P(Y_{n}=1)=\frac{1}{2}, S_{0}=0, calculate:$$P(S_{1}>0,...,S_{2n-1}>0, S_{2n}=0)$$1answer 206 views ### How can I prove this bijection between random walks? Let R_n be the set of simple random walk paths such that S_n=0. P_n be the set of simple random walk paths such that \forall i \in \{1,2,...,n\}, S_i > 0. N_n be the set of paths ... 0answers 53 views ### Who gets the broccoli stick? Random walk in a circle. [duplicate] Suppose we have n+1 people in a circle {0,1,2,3....n}, we pass around a broccoli stick. The person k has probability p to pass the stick to k+1 and probability q = 1-p to pass it to the k-... 3answers 79 views ### 2D Random Walk: Average distance after 2 steps A simulation of 50,000 iterations gives the average distance after a 2-step (unit step) random walk on a 2 dimensional plane, which is around 1.27. But how can one mathematically prove this? Any ... 0answers 19 views ### Stochastic Processes Formula I am trying to solve this question on stochastic processes which is to show that for$$I_0=0\\\\ I_n=\sum_{j=0}^{n-1}M_j(M_{j+1}-M_j), \quad n=0,1,2,... $$the equation can be writen as;$$I_n=\frac{... 0answers 24 views ### Probability of a stochastic process crossing a boundary in time interval Suppose that we have a stochastic process$X(t)$: $$X(t) = \frac{1}{t}\int_{0}^{t} W(\tau) d\tau$$ where$W(\tau)$is a Wiener process. What is the probability of$X(t)$crossing a barrier$\alpha$($\...
Let $L_{m,m}$ be a $2D$ lattice. Also, suppose that there is a random walker located in position $(0,0)$. The random walker goes right, left, up, or down randomly in each step and cannot get out of \$...