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

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106 views

Recurrence of a certain class of $2$-$d$ random walks

As is well known, a symmetric random walk on $\mathbb{Z}^d$ (the lattice of $d$ dimensional vectors with integer components) is recurrent if and only if $d=1,2$. In particular it is transient for ...
2
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0answers
24 views

Sums of independent random variables

I was unsuccessful in deriving a good estimate of the distance below. Let $(X_{n})_{n \geqslant 1}$ be a sequence of i.i.d. random variables, and let $(\varepsilon_{n})_{n\geqslant 1}$ be a sequence ...
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0answers
42 views

Random Walk in confined region and loop configurations

Suppose I take a random walk on a 2 dimensional square lattice, but this lattice plane has a finite size, e.g. Dx*Dy. I can not cross the boundary, my step length is the lattice cell size, I either go ...
0
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1answer
81 views

Solving a differential equation (Lattice Laplacian)

Suppose that $ p_n(t) $ is the probability of finding n particle at a time t. And the dynamics of the particle is described by this equation : $$ \frac{d}{dt} p_n(t) = \lambda \Delta p_n(t) $$ ...
3
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3answers
85 views

A fly on a triangle?

A fly is on the vertex of a triangle. It can move left with probability $\frac 12$ and right with probability $\frac 12$. What is the expected number of moves till it reaches its starting point? ...
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2answers
189 views

Calculating expected value of random walk with one stop value.

I know that for a random walk with two stop values, the expected value of the number of steps needed is $ab$ where the stop values are $-a$ and $b$ and the initial position is at 0. What about for ...
0
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0answers
30 views

How to perform a stochastic search of the locality of a node in a network?

In a graph that may be a random graph (ER graph), scale free network, etc. I would like to obtain a distribution of the locality of the nodes surrounding a ...
1
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2answers
42 views

Random Walk with Edges

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 ...
2
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1answer
254 views

Expected time for winning in biased Gambler's Ruin

Consider the random walk $X_0, X_1, X_2, \ldots$ on state space $S=\{0,1,\ldots,n\}$ with absorbing states $A=\{0,n\}$, and with $P(i,i+1)=p$ and $P(i,i-1)=q$ for all $i \in S \setminus A$, where ...
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0answers
86 views

Bayesian random walk

Suppose that, at first, I am trying to estimate the mean and standard deviation of some data that I assume to be normally distributed. My prior is gaussian with mean $\mu_0$ and variance $\sigma^2_0$. ...
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3answers
119 views

Probability of having X more heads than tails for N tosses.

Giving a a fair coin, and tossing it N times, in how many possible outcomes would there be a point wherein there were more heads than tails tossed, ie, net heads.
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1answer
45 views

Concerning the distribution of a random variable of a random walk that doesn't make any sense to me

Let $\Omega = \{w = (x_1, \dots, x_N) | \; x_i \in \{-1, 1\}\}, \;X_k(w) = x_k, \;S_n(w) = \sum_{k=1}^n X_k(w), \; S_0(w) = 0.$ After having proven a few theorems about $S_n$, in our lecture about ...
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0answers
130 views

Brownian motion, rate of large events

Given the most simple brownian motion: $$ \dot x(t) = \sigma \eta(t)$$ where $\langle \eta(t)\eta(t')\rangle=\delta(t-t')$, I define as large event in a time-frame $\tau$ a portion of the trace ...
1
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0answers
87 views

Stopping time and martingale for random walks

Let $X_0=0, X_1, X_2,\dots, X_N$ be i.i.d. random variables, with Gaussian distribution $\cal N (0,1)$. For $k=0,\dots, N, S_k=\sum_{i=1}^k X_i$ and $\tau=\min\{k:S_k^2\geq N-k\}$. So $\tau$ is a ...
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0answers
40 views

Random walk - proving limits exist

Consider the random walk $\{X_k\}_{k\geq0}$ on $\mathbb{Z}$ with transition probabilities $$\begin{cases} p_{i,i-1} = p_{-1} &> 0 \\ p_{i,i+1} = p_{1} &> 0 \\ p_{i,i+2} = p_{2} ...
2
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1answer
71 views

What is the probability that the robot steps on the bomb?

Suppose a robot is initially placed at $0$ on the number line, and is programmed to take steps of integer length in the positive direction between $1$ and $k$, inclusive, where $k$ is a positive ...
2
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3answers
179 views

Random walk returning probability

Consider a two-dimensional random walk, but this time the probabilities are not $1/4$, but some values $p_1, p_2, p_3, p_4$ with $\sum p_i=1$. For example, from $(0,0)$, it goes to $(1,0)$ with $p_1$, ...
1
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1answer
171 views

Probability of being at a certain point after $N$ steps in Random Walk with a single absorbing barrier

A random walker in $1$ dimension starts walking from a point $k>0$ with an absorbing barrier at point $0$. What is the probability that he will reach a point $m>0$ in $N$ steps? How should I ...
2
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0answers
77 views

random walk with possibility to freeze

Consider a Random Walk on a one-dimensional lattice. The walker starts moving at time $0$ from $x=0$. At every step, the walker moves to the right with probability $p$, to the left with probability ...
2
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1answer
74 views

law of large number modified statement

The weak law of large number states that, given $Y_n = \sum_{k=1}^{n} X_k$, where $X_k$ are random variables independent and identically distributed with finite expectation $\mu$, $$ \forall ...
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0answers
54 views

Random walk with non-integer steps

Let $X_k\sim f(x)$ be a random variable taking values on $(-\infty, +a)$. The sum $S(n)=\sum_{k=0}^n X_k$ can be seen as a random walk with general distributed steps. Suppose $c_{neg}<0$ and ...
33
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1answer
874 views

Identity for simple 1D random walk

The question is to find a purely probabilistic proof of the following identity, valid for every integer $n\geqslant1$, where $(S_n)_{n\geqslant0}$ denotes a standard simple random walk: $$ ...
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0answers
79 views

On the second derivative of Wiener process

As we all know, continuous white noise is the derivative, with respect to time, of a Wiener process. My question is that does the second derivative of Wiener process exists? If so, what is it and how ...
2
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1answer
96 views

Computation of a mean (random sum)

Let $X_1$, $X_2$, ... be independent and identially distributed positive random variables and define the sum $S_n = X_1 + X_2 + ... + X_n$. Consider the first time $N$ where $S_N \ge b$ with a given ...
2
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1answer
71 views

A continuous random walk of length 1

Suppose one starts at origo in in the plane and takes $N$ steps of length $1/N$ in a random direction, what is the distribution of the resulting distance from origo as $N$ approaches infinity? For one ...
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1answer
61 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. ...
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2answers
54 views

Random walk confusion

If a ransom walk is binomial (1/2 probability of going forward, 1/2 backward) why isn;t the variance a) $\sigma=(\frac{n}{4})^.5$ b) instead of $\sigma=(n)^.5$ these sources seem to give ...
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0answers
28 views

What random process is this?

This is a rather basic question, is just that I don't know a thing about this subject. Let's say $x$ is an integer. At $t=0$, the value of $x$ is 1. Then, at each time step, one of the following ...
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1answer
65 views

Number non self avoiding closed walks surrounding some point

While studying some Peierls-like arguments in statistical physics I thought about the following problem: We have some 2d-integer lattice like this, for simplicity infinite in all directions. Now fix ...
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0answers
34 views

Optimal stopping for random walks

Let $X_0=0, X_1, X_2,\dots, X_N$ be i.i.d. random variables, with Gaussian distribution $\cal N (0,1)$. For $k=0,\dots, N, S_k=\sum_{i=1}^k X_i$ and $Z_k= (N+1-k)S_k^2$. The goal is to get ...
5
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0answers
129 views

Intuition for the optimality of bold play

There is a standard result (I think originally by Dubins and Savage) that if one wants to maximise the probability of winning a certain amount in an unfair game of chance then an optimal strategy is ...
4
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2answers
133 views

Speculating on the stock exchange

Imagine you model each stock as a random walk (fractal) and also that you can buy and sell at any price. Suppose also that it 'walks' with the pace of 1. If you buy, for example, 1000 shares of ...
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1answer
80 views

When is an infinite sequence of integers purely deterministic with no randomness involved?

I see in literature very different descriptions of what is a deterministic system such as: "... a system in which no randomness is involved in the development of future states of the system...>>>" I ...
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0answers
53 views

Random walk type problem with time increments

Imagine you have $\$50$ and every $2$ minutes you either gain or lose $33$ cents. How would you model the evolution of the hypothetical bankroll for the next hour? My approach based on what i've read ...
5
votes
1answer
283 views

Random walk as a martingale?

Let $S_0$, $Z_1$, $Z_2$, $\ldots$ be independent random variables. $S_n=S_0+Z_1+\cdots+Z_n$, $n=0,1,2,\ldots$ $S_n$ is a random walk starting in a random point, $S_0$ I need to find out, when it is a ...
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1answer
78 views

Arguing on stopping time probability

Consider the random walk where $X_t=\sum_{i=1}^t Y_i$, $Y_i$s are iid and take $\pm$1 with probabilities $p$ and $1-p$ respectively, where $0<p<0.5$. Define stopping time ...
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0answers
149 views

Reference Request, Random Walk [duplicate]

The expected distance from origin after a random walk of $N$ steps in a $d$ dimensional space, is close to ...
0
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1answer
211 views

Coin toss probability calculation

A gambler bets on coin flips. With each flip, he wins $1$ dollar with probability $p$, and loses $1$ dollar with probability $1-p$. He starts with $2$ dollar and stops when he reaches either $0$ or ...
0
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0answers
83 views

Fitting 2D random walk to data points given on cartesian plane

I have a collection of data points given with coordinates $(x,y)$ in $\Re^2$. I suspect that these data points can be modeled using a random walk. How can I fit a random walk model to these 2D ...
2
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2answers
63 views

Ties in Matching Pennies

Players A and B match pennies N times. They keep a tally of their gains and losses. After the first toss, what is the chance that at no time during the game will they be even? I have seen a solution ...
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1answer
120 views

Random walk probability non-symmetric steps

I currently have a probability class tutorial question that I have no idea where to begin. At first instinct, I thought it may have been a CTMC question or branching question, but now I have no idea, ...
2
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1answer
60 views

Random walk - Solution verification

A particle moves randomly 2n times. Each time it chooses one direction: north, south, west, east with even probability and without effect from previous choices, it then moves one step in that ...
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1answer
99 views

Flipping an unfair coin

An coin has probability $p<0.5$ of landing heads up. We flip it infinitely many times, after $n$ tosses, let $S_n$ be the number of heads minus the number of tails. What is the expected value of ...
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0answers
122 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 ...
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0answers
48 views

How to prove that the hitting time for a random $(2,k)$-walk is $\mathcal{O}(\frac{k^4}{r})$?

I'm using the following definitions: An $(x, y)$-partial-rectangle is a sequence of x integers $(i_1,i_2,\ldots,i_x)$ such that $0 \leq i_1 \leq i_2 \leq \ldots i_x \leq y$. One ...
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0answers
55 views

Dimension free Concentration bounds for Martingales

Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$: At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...
2
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1answer
77 views

An application of Donsker's theorem.

Let ${X_i}$ be iid with $E[X]=0$ and $Var(X)=\sigma^2$ Let $S_0=0$ and $S_n=X_1+...+X_n$ for all $n \ge 1$. Show that $\lim_{n\rightarrow \infty}P(S_k>0 \space for \space k=n,n+1,\dots,2n)=1/4 $. ...
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1answer
148 views

Random walk serial correlation

Given a model $$Y_t =b_0 + b_1 \cdot X_t + b_2 \cdot Z_t + e_t,$$ where the error term $e_t$ follows a random walk form of serial correlation $e_t = e_{t-1} + u_t$. Further assume $u_t$ has zero mean ...
2
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1answer
187 views

Proof: Mean and Variance of the squared distance of a random walk in n-dimensional space

consider a $x$ step random walk starting from origin in $n$-dimensional space where each step is taken into a random direction and has a distance of 1, i.e., each step is a vector on the ...
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1answer
176 views

A Boundary crossing result for discrete brownian bridge

Let $S_n$ be a random walk with gaussian increments with $S_0=0$, i.e. $S_n-S_{n-1}\sim N(0,1), n\geq 1$. Fix $a>0,b\in \mathbb{R}$ and $c<a+bn$. Define the new process $$ ...