I execute a random walk on complete graph, the L2 norm between the old and the new vectors (p(t+1)-p(t)) still decreases till the iteration x and after they go to increases, then again decreases an so one? DOES this means something? how can I interpret that? When can I Stop the random walk in this case...Read more

The setup for the specific problem that led to this question is as follows: You are playing a game at a casino and have \$10,000; The bank has \$2,000. You are making \$1,000 bets, with a equal payout. You play until either you or the bank run out. What is the probability that you will be the one to run out, versus the bank, and what is the distribution for the path lengths in either case?The way I interpret it is that this is a random walk that is biased in one direction, with the added factor that if it reaches either end it will stay there...Read more

Suppose $\{ X_t \}$ is a sequence of i.i.d. random variables, with support $\{-1,1\}$ and distribution $P(1)=P(-1)=1/2$. Thus, $S_t = \sum_{s=1}^{t} X_s$ is a zero mean random walk. Also, $S_t$ is a martingale, but the conditions for Doob's martingale convergence theorem do not apply. What is it possible to say about the limiting behavior of $S_t$?...Read more

Consider simple random walk: $X_n=\pm1$ with equal probabilities. $S_n =\sum_{i=1}^nX_i$. For finite $n$ we can write$$S_n=\sum_{i=1}^nX_i=\sum_{i=1}^nX_i^+ -\sum_{i=1}^nX_i^-$$So that$$E[S_n]=\sum_{i=1}^nE[X_i^+] -\sum_{i=1}^nE[X_i^-]$$However, $\lim_{n\rightarrow\infty}\sum_{i=1}^nE[X_i^+]=\infty$ and $\lim_{n\rightarrow\infty}\sum_{i=1}^nE[X_i^-]=\infty$.So, what happens to $E[S_n]$ in the limit?...Read more

I am trying to prove the following asymptotic formula for $a_n$, the number of closed paths of length $n$ on the hexagonal lattice (paths starting and ending at the same hexagon):$$a_n\sim \frac{6^n \sqrt{3}}{2\pi n}$$I have already proven an analogous formula for a square lattice using binomial coefficients. Does anyone know of an elementary/intuitive way to prove the above?Even better, if anyone knows of a way to prove the exact formula$$a_n=\sum_{i=0}^n (-2)^{n-i}\binom{n}{i}\sum_{j=0}^i \binom{i}{j}^3$$I can probably figure out how to deriv...Read more

I'm writing a random number generator test. The kit it will run on is way less powerful than a desktop PC, and I've come across this wiki article for determining $\pi$. I'm interested in the random walk method, utilising this formula:There is also this explanatory graph with an example 5 walks of 200 steps each:So I perform some number (n) of random walks of length L. If the generator is working correctly, I will calculate Pi to some accuracy approaching the true value. In this example, I will have used 5 * 200 = 1000 random numbers. So let...Read more

I have a random walk process, discrete in time and state, where at each step the probability of $+1$ is $p$ and $−1$ is $q$. $p+q=1$ and $p$ may be different from $q$ (i.e. the random walk is "biased", "asymmetric", has "drift").Starting from $x$, where $a<x<b$, I'd like to find the expected hitting time to hit either $a$ or $b$.There's usually a nonzero probability of the random walk never hitting either boundary, which may imply difficulties getting an expectation. So a median of the hitting time distribution could be a useful alternati...Read more

I want to perform graph isomorphism tests for a very long random walk with fixed windows. That is given a target graph, say a triangle, I want to find how many consecutive 3 nodes in the random walk induce a triangle.Graph isomorphism test is very costly and there may be repetitive graph patterns appearing in the random walk. Thus, it is expensive to do the isomorphism test on-the-fly when the random walk is simulated.Hence, I want to store the random walk first. Afterwards, I want to use some pruning techniques to reduce the number of isomorph...Read more

Consider a classical symmetric random walk that starts at origin $x=0$ and lasts for $N$ steps. If $x=-1$ is reached, a walk is terminated. What is the expected value of this process?I divided a walk into 2 parts.A path hits $x=-1$ at some step $n \le N$. Then, I can compute a part of a total expected value summing the probabilities with weight $-1$ as it was presented in https://math.stackexchange.com/a/182376However, I encounter a problem with a part giving rise from walking only in the nonnegative part of the axis. How should I proceed?...Read more

Let $(X_1,X_2,...)$ be i.i.d random variables, with $P(X_t=1)=P(X_t=-1)=1/2$. Then$S_t= \frac{1}{t}\sum_{i=1}^{t}X_i $is a zero mean random walk. Let $\tau$ be the stopping time corresponding to the first time that $S_t$ hits $M \in (0,1/2)$,\begin{equation}\tau=\inf\{t \geq 1 \mid S_t \geq M \}\end{equation}I am trying to show that $P(\tau = \infty)>0$. For starters, clearly $P(\tau < \infty)>0$ since $P(\tau=1)=1/2$. Also, the probability that $S_t$ exceeds $M$ infinitely often is zero, since $S_t$ converges to $0$. But I would like ...Read more

Can anyone provide the first hitting time distribution for a discrete random walk?Edit: Specifically, a 1D random walk, starting at $k=0$. Each step moves either $-1$ or $+1$ without any boundaries. I require the distribution for the first hitting time at some arbitrary point $m>0$.I cannot find it anywhere. I can only find it for continuous Brownian motion....Read more

enter image description hereIs it possible to prove that for any symmetric Random Walk the absolute value of its partial sum $S_n$ never exceeds $\sqrt{2 \pi n}+\sqrt{\pi / 2}$ ?I run some simulation, clearly for initial value it is not true but for large n it seems true that it is a real absolute bound....Read more

I am reading a book that is talking about continuous random walk. It first starts with defining one dimensional discrete random walk as starting at point 0 and move to either to the right or left at the rate of $1$ per unit of time.Then, it said that if we instead move $\sqrt{h}$ per $h$ units of time, and took the limit as $h$->$0$, we would have continuous random walk. It will then converge into brownian motion.My questions are:1) Why are me moving $\sqrt{h}$ instead of $h$ per $h$ unit of time? Can someone show me the calculation?2) Before t...Read more

Consider a random walk on a infinitely countable connected graph.We assume that each vertex has finitely many neighbors and that we have a uniform bound of the number of neighbors at each vertex.The probability to move from x to a neighbor y of x is equal to the inverse of the number of neighbors of x.Can we prove the existence of a invariant probability measure for the corresponding Markov chain? Under which assumptions?Thanks!...Read more

If $k$ random $n$-step $\pm 1$ walks start at 0, and the $i$th walk ends at position $X_i$, how big is $\text{median}_i \, |X_i|$?Is there a bound along the lines of $\text{P}(\text{median}_i \, |X_i| > \ldots) < \ldots$? When $k = \Theta(\log n)$, Azuma + Chernoff show this median to be no more than $O(\sqrt{n})$ with high probability, but is there a better bound on $\text{median}_i \, |X_i|$ with less-than-high probability?...Read more