The stochastic-processes tag has no wiki summary.
13
votes
2answers
262 views
Random walk with momentum
Consider an integer random walk starting at 0 with the following conditions:
The first step is plus or minus 1, with equal probability.
Every future step is: 60% likely to be in the same direction ...
12
votes
2answers
262 views
Exploratory analysis of spatio-temporal forecast errors
The data: I have worked recently on analysing the stochastic properties of a spatio-temporal field of wind power production forecast errors. Formally, it can be said to be a process $$ \left ...
11
votes
2answers
744 views
Why does the supremum of the Brownian bridge have the Kolmogorov–Smirnov distribution?
The Kolmogorov–Smirnov distribution is known from the Kolmogorov–Smirnov test. However, it is also the distribution of the supremum of the Brownian bridge.
Since this is far from obvious (to me), I ...
11
votes
1answer
604 views
What are some techniques for sampling two correlated random variables?
What are some techniques for sampling two correlated random variables:
if their probability
distributions are parameterized
(e.g., log-normal)
if they have non-parametric
distributions.
The data ...
10
votes
1answer
459 views
Modifying linear ballistic accumulator (LBA) simulation in R
The "Linear Ballistic Accumulator" model (LBA) is a rather successful model for human behaviour in speeded simple decision tasks. Donkin et al (2009, PDF) provide code that permits estimating the ...
9
votes
5answers
963 views
Time taken to hit a pattern of heads and tails in a series of coin-tosses
Inspired by Peter Donnelly's talk at TED, in which he discusses how long it would take for a certain pattern to appear in a series of coin tosses, I created the following script in R. Given two ...
9
votes
3answers
2k views
Expected number of coin tosses to get N consecutive, given M consecutive
Interviewstreet had their second CodeSprint in January that included the question below. The programmatic answer is posted but doesn't include a statistical explanation.
(You can see the original ...
9
votes
1answer
222 views
Closed form expression for the quantiles of $\alpha_1\sin(x)+\alpha_2\cos(x)$
I have two random variables, $\alpha_i\sim \text{iid }U(0,1),\;\;i=1,2$ where $U(0,1)$ is the
uniform 0-1 distribution.
Then, these yield a process, say:
$$P(x)=\alpha_1\sin(x)+\alpha_2\cos(x), ...
9
votes
2answers
210 views
Numeric solvers for stochastic differential equations in R: are they any?
I'm looking for a general, clean and fast (i.e. using C++ routines) R package for simulating paths from a non-homogeneous nonlinear diffusion like (1) using the Euler-Maruyama scheme, the Milstein ...
8
votes
7answers
1k views
How will studying “stochastic processes” help me as a statistician?
I wish to decide if I should take a course called "INTRODUCTION TO STOCHASTIC PROCESSES" which will be held next semester in my University.
I asked the lecturer how studying such a course would help ...
8
votes
1answer
128 views
Density of robots doing random walk in an infinite random geometric graph
Consider an infinite random geometric graph in which the node locations follow a Poisson point process with density $\rho$ and edges are placed between the nodes that are closer than $d$. Therefore, ...
7
votes
1answer
145 views
Generalization of Brownian motion to $\alpha$-stable distributions
Brownian motion is constructed as a limit of the sum i.i.d. Gaussian increments. Can one use a non-Gaussian $\alpha$-stable distribution (e.g. the Cauchy distribution) instead, and still construct a ...
6
votes
4answers
350 views
Is winning a soccer match independent of previous wins\losses?
$\quad$ I have a friend of mine who is a bit of a gambler ask me this question. He is of poor mathematical background, but has a sense of logic and will probably accept a logical answer in the natural ...
6
votes
1answer
285 views
Origin of strange formula for equilibrium standard deviation
In the paper
M. Avellaneda and J. H. Lee, Statistical arbitrage in the U.S. equities market, July 2008,
in the Appendix on page 46, how does he get equilibrium standard deviation as following:
...
6
votes
2answers
175 views
Textbooks on the numerical solution of stochastic differential equations
Is there a canonical textbook or textbooks on the use of numerical methods in solving stochastic differential equations that you would recommend?
6
votes
3answers
774 views
The expected value of random variable on tosses of a coin
Came across an interesting problem today. You are given a coin and x money, you double money if you get heads and lose half if tails on any toss.
What is the expected value of your money in n tries
...
6
votes
3answers
277 views
Finding the MLE for a univariate exponential Hawkes process
The univariate exponential Hawkes process is a self-exciting point process with an event arrival rate of:
$ \lambda(t) = \mu + \sum\limits_{t_i<t}{\alpha e^{-\beta(t-t_i)}}$
where $ t_1,..t_n $ ...
6
votes
2answers
2k views
How do betting houses determine betting odds for sports?
Let's take football (soccer) for example. There are 3 possible outcomes, home win, draw, away win. I took a random game from bet365
...
6
votes
1answer
54 views
Importance sampling of finite path of stochastic difference equation
Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
5
votes
1answer
182 views
Taking random items out of a container with replacement
I take vitamins in the morning, but one of them I only take a half tablet.
So, I have an initial container with 100 full tablets, and every morning I take out a random tablet. If it's a full ...
5
votes
3answers
740 views
What are the examples for stochastic processes in Electrical Engineering and Computer Science?
I wanted to find out what kind of different usages of stochastic processes theory in EE & CS are out there. For example, I find these kinds of usages interesting:
using stochastic signal as ...
5
votes
2answers
162 views
Modeling a 1D random walk with nonconstant probabilities
I have (what I will term, for lack of a better word) a random walk that has a particular property: it tends to be right of the origin some fraction k of the time and left 1-k of the time (and on the ...
5
votes
1answer
437 views
Why are cumulative residuals from regression on stock and index returns mean reverting
In the paper
M. Avellaneda and J. H. Lee, Statistical arbitrage in the U.S. equities market, July 2008,
in the Appendix on page 44, I have some questions.
First he runs the regression of ...
4
votes
2answers
511 views
Simulating a Gaussian process with an exponentially decaying covariance function
I'm trying to generate many draws (i.e., realizations) of
a Gaussian process $e_i(t)$, $1\leq t \leq T$ with mean 0 and covariance
function $\gamma(s,t)=\exp(-|t-s|)$.
Is there an efficient way ...
4
votes
1answer
368 views
Conditions for Central Limit Theorem for dependent sequences
Cumbersome technical assumptions (e.g., mixing properties) are used in the literature to prove Central Limit Theorems for dependent sequences. I sketched a proof that does not require any of these ...
4
votes
1answer
205 views
Intuitive understanding covariance, cross-covariance, auto-/cross-correliation and power spectrum density
I'm currently studying for my finals in basic statistics for my ECE bachelor.
While I think I have the math mostly down, I lack intuitive understanding what the numbers actually mean.(Preamble: I'll ...
4
votes
1answer
160 views
Meaning of the “existence” proof
After doing some reading on stochastic processes for work I've found that a proof that the specific process exists is often one of the first things presented.
Could someone please explain, in ...
4
votes
1answer
173 views
Distribution of arrival times to server for an M/M/1 queue (what the server experiences)
In an M/M/1 queue, we know that inter-arrival times are exponentially distributed, and that service times are the same. What is the distribution of to-server inter-arrival times (aka service start ...
4
votes
1answer
220 views
Probability of visiting all other states before return
Question (a)
Random walk on a clock. Consider the numbers $1, 2, \dots, 12$ written around a clock. Consider a Markov chain that jumps with equal probability to one of the two adjacent numbers each ...
4
votes
2answers
150 views
Computing limiting transition probabilities with absorbing states
A bank classifies loans as paid in full (F), in good standing (G), in arrears (A), or as a bad debt (B). Loans move between the categories according to the following transition probability:
$$B = ...
4
votes
1answer
998 views
Example of a 2nd order stationary, but not strictly stationary process
Does anybody have a nice example of a stochastic process that is 2nd-order stationary, but is not strictly stationary?
4
votes
2answers
252 views
What is the distribution of $\chi^n_k$?
$\chi^n_k=\sum_{i=1}^kx_i^n$ where $x_i$ are Gaussian variables and $n>2$?
4
votes
1answer
258 views
Why local martingale property is important in Girsanov theorem?
In Girsanov theorem, the change of probability measure variable $Z_t = \frac{dQ}{dP}|_{\mathcal{F}_t}$, why does it need to be a martingale with respect to measure $P$ for the change of measure ...
4
votes
0answers
35 views
Limiting distributions identified as functions of Brownian motion or stochastic integrals
I am teaching a stochastic processes course to MA stat students, and to stay on topic I would like some examples of limiting distributions in stat that are identified as functions of brownian motion ...
4
votes
0answers
200 views
CIR Process-Variance reduction
I'm trying to evaluate a path dependent function, $f(r_t)$, on a Cox-Ingersoll-Ross process:
$dr_t = \theta (\mu - r_t)dt + \sigma \sqrt r_t dW_t$
by Monte Carlo simulation.
Could anyone suggest ...
3
votes
3answers
459 views
Modeling a birth-death process that is not memoryless
How does one approach the problem of modeling a "birth-death process" where the arrivals are dependent on the current state in the following way: if the population is above a certain point, the ...
3
votes
2answers
265 views
How to find the closest distribution of a given data?
I have inter-arrival times of vehicles recorded by a vehicle detection algorithm. I want to find the closest distribution (e.g., Poisson or other) of this data.
How can I do that?
Here is a graph ...
3
votes
2answers
120 views
What techniques are used for empirical, stochastic simulation of a time series?
Suppose you have recorded a set of paths in the $y,t$ plane, with $y = f(t)$, $f$ is a stochastic function (i.e. there is a noise term), and $t$ might be time or some other monotonic increasing ...
3
votes
2answers
327 views
Calculating probabilites of an nth step transition matrix for discrete time markov chains
"Let $\{X_n, n \geq 0\}$ be a DTMC with state space $S = \{1, 2, 3, 4, 5\}$ and the following transition probability matrix:
$$
P = \begin{pmatrix} 0.1 & 0.0 & 0.2 & 0.3 & 0.4 \\ 0.0 ...
3
votes
1answer
148 views
Proving a non-stopping time
Let me begin by first confirming that this is indeed the correct place to post this (other ideas I had were math.SE). That said,
Let $X_n$ be a Markov chain on the state space $\mathcal S$ and for ...
3
votes
2answers
308 views
Trees generated by multi-type branching processes in n steps
I am trying to develop some algorithm to compute probabilities in multi-type branching trees, and I doubt I am doing this right...
Let us consider a multi-type branching process with two types, ...
3
votes
1answer
191 views
Meaning of this expectation equation?
I was actually looking at this problem on slide 12. I will write it here briefly:
Problem:
Unknown number of people arriving in a fixed time period and my goal is to maximize my probability of ...
3
votes
1answer
106 views
Stationary matrix given a transition matrix
I am given the following transition matrix
$$P= \pmatrix{ 1-\alpha & \alpha \\ \beta & 1-\beta}, \ \alpha,\beta \in (0,1)$$
with the states $S=\{1,2\}$.
I want to determine the stationary ...
3
votes
1answer
350 views
Using the sde package in R to simulate a SV model with leverage
Using the sde package in R, I would like to simulate the following model for stock prices $p_t$:
$\mathrm{d}\sigma^2_t = (\theta_1 - \theta_2\sigma^2_t)\mathrm{d}t + ...
3
votes
1answer
102 views
Proof of theorem on recurrent states and its equivalence class
A theorem states the following:
Theorem
if $i \in S$ is a state which is recurrent, then every state in the equivalence class of $i$ $(\ K(i) \ )$ is recurrent.
Additional information on the ...
3
votes
0answers
63 views
Distribution/expected length of the shortest path in infinite random geometric graphs
Consider an infinite random geometric graph $G(\rho,d)$ in which vertices are uniformly and independently scattered over the 2D plane with density $\rho$ and edges connect the vertices that are closer ...
3
votes
0answers
38 views
Can we reconstruct the hidden (latent) variables after executing EM?
The question is in the title. I know that EM algorithm could do maximum likelihood estimation for models that have latent variables. I would like to know can we get the (estimated) value of these ...
3
votes
0answers
26 views
Is a sequence of random variables indexed by a homogeneous Poisson process process strictly stationary?
I'm revising for an exam and have no idea how to approach this question:
Let $\{N_t\}_{t\geq 0}$ be a homogeneous Poisson process of parameter $\lambda > 0$. Let $\{X_k\}_{k\geq 0}$ be a sequence ...
3
votes
0answers
47 views
Statistical properties of a 3D field from spatial averages at different scales
I'm trying to estimate the statistical properties of a 3D vector field (the magnetic field vector in the solar atmosphere) over a 2D field of view from observations. I will neglect for the moment that ...
3
votes
0answers
43 views
Updating a set of estimated forecasts
Suppose I have some stochastic process $X_t$. At each time $t$, I receive an estimated probability distribution for $x_t$, followed by an observation $x_t$. After receiving a set of observations ...
