Questions tagged [stochastic-processes]

A stochastic process describes the evolution of random variables/systems over time and/or space and/or any other index set. It has applications in areas such as econometrics, weather, signal processing, etc. Examples: Gaussian process, Markov process, etc.

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106
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5answers
103k views

What does a “closed-form solution” mean?

I have come across the term "closed-form solution" quite often. What does a closed-form solution mean? How does one determine if a close-form solution exists for a given problem? Searching online, I ...
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6answers
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Is a time series the same as a stochastic process?

A stochastic process is a process that evolves over time, so is it really a fancier way of saying "time series"?
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6answers
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Why does the variance of the Random walk increase?

The random walk that is defined as $Y_{t} = Y_{t-1} + e_t$, where $e_t$ is white noise. Denotes that the current position is the sum of the previous position + an unpredicted term. You can prove that ...
29
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5answers
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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 ...
29
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5answers
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What are examples of statistical experiments that allow the calculation of the golden ratio?

There are some very simple experiences that can be done by a kid at home, whose result allows one to statistically approach famous numbers such as $\pi$ or $e$. An example where $\pi$ shows up is ...
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6answers
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How likely am I to be descended from a particular person born in the year 1300?

In other words, based on the following, what is p? In order to make this a math problem rather than anthropology or social science, and to simplify the problem, assume that mates are selected with ...
25
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2answers
64k 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 ...
25
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1answer
2k views

How to create a multivariate Brownian Bridge?

It is known, that a standard multivariate Brownian bridge $ y(\mathbf u) $ is a centered Gaussian process with covariance function $$ \mathbb E(y(\mathbf u) y(\mathbf v)) = \prod_{j=1}^d (u_j \wedge ...
24
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2answers
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What are the main differences between Granger's and Pearl's causality frameworks?

Recently, I ran across several papers and online resources that mention Granger causality. Brief browsing through the corresponding Wikipedia article left me with the impression that this term refers ...
24
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1answer
853 views

Wavelet-domain Gaussian processes: what is the covariance?

I've been reading Maraun et al, "Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significant testing" (2007) which defines a class of non-stationary GPs that ...
22
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2answers
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Intuitive explanation for periodicity in Markov chains

Can someone explain me in a intuitive way what the periodicity of a Markov chain is? It is defined as follows: For all states $i$ in $S$ $d_i$=gcd$\{n \in \mathbb{N} | p_{ii}^{(n)} > 0\} =1$ ...
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2answers
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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 are ...
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7answers
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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 ...
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2answers
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stochastic vs deterministic trend/seasonality in time series forecasting

I have moderate background in time series forecasting. I have looked at several forecasting books, and I don't see the following questions addressed in any of them. I have two questions: How would I ...
18
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2answers
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Whether a AR(P) process is stationary or not?

In practice, how to evaluate whether a AR(P) process is stationary or not? How to determine the order for the AR and MA model?
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5answers
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How do you see a Markov chain is irreducible?

I have some trouble understanding the Markov chain property irreducible. Irreducible is said to mean that the stochastic process can "go from any state to any state". But what defines whether it can ...
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2answers
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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 ...
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4answers
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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 $ ...
18
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2answers
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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 ...
16
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1answer
503 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), \;\...
16
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2answers
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What is the difference between Markov chains and Markov processes?

What is the difference between Markov chains and Markov processes? I'm reading conflicting information: sometimes the definition is based on whether the state space is discrete or continuous, and ...
15
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2answers
3k views

What does it mean to say that an event “happens eventually”?

Consider a 1 dimensional random walk on the integers $\mathbb{Z}$ with initial state $x\in\mathbb{Z}$: \begin{equation} S_n=x+\sum^n_{i=1}\xi_i \end{equation} where the increments $\xi_i$ are I.I.D ...
15
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2answers
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Derivative of a Gaussian Process

I believe that the derivative of a Gaussian process (GP) is a another GP, and so I would like to know if there are closed form equations for the prediction equations of the derivative of a GP? In ...
14
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2answers
2k views

Numeric solvers for stochastic differential equations in R: are there 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 ...
13
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3answers
3k views

Is stationarity preserved under a linear combination?

Imagine we have two time-series processes, which are stationary, producing: $x_t,y_t$. Is $z_t=\alpha x_t +\beta y_t$, $\forall \alpha, \beta \in \mathbb{R}$ also stationary? Any help would be ...
13
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2answers
471 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 (\epsilon^...
13
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1answer
381 views

A question related to Borel-Cantelli Lemma

Note: Borel-Cantelli Lemma says that $$\sum_{n=1}^\infty P(A_n) \lt \infty \Rightarrow P(\lim\sup A_n)=0$$ $$\sum_{n=1}^\infty P(A_n) =\infty \textrm{ and } A_n\textrm{'s ...
12
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5answers
5k views

What are the differences between stochastic and fixed regressors in linear regression model?

If we have stochastic regressors, we are drawing random pairs $(y_i,\vec{x}_i)$ for a bunch of $i$, the so-called random sample, from a fixed but unknown probabilistic distribution $(y,\vec{x})$. ...
12
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2answers
8k views

What is the difference between a distribution and a process (Poisson)?

I'm doing my PhD in geomechanics. I thought we use a Poisson-Weibull distribution (for the variability of a parameter at the rock), but reading more about the subject I think maybe is a Poisson-...
12
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2answers
5k views

If a time series is second order stationary, does this imply it is strictly stationary?

A process $X_t$ is strictly stationary if the joint distribution of $X_{t_1},X_{t_2},...,X_{t_m}$is the same as the joint distribution of $X_{t_1+k},X_{t_2+k},...,X_{t_m+k}$ for all $m$, for all $k$ ...
12
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1answer
287 views

Special probability distribution

If $p(x)$ is a probability distribution with non-zero values on $[0,+\infty)$, for what type(s) of $p(x)$ does there exist a constant $c\gt 0$ such that $\int_0^{\infty}p(x)\log{\frac{ p(x)}{(1+\...
12
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1answer
218 views

Will there ever be an unhappy Tribble in Oz?

Here's an amusing problem brought to me by a student. Although it was originally phrased in terms of mutually annihilating bullets fired at regular intervals by a gun, I thought you might enjoy a ...
11
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3answers
14k views

What is the difference between deterministic and stochastic model?

Simple Linear Model: $x=\alpha t + \epsilon_t$ where $\epsilon_t$ ~iid $N(0,\sigma^2)$ with $E(x) = \alpha t$ and $Var(x)=\sigma^2$ AR(1): $X_t =\alpha X_{t-1} + \epsilon_t$ where $\epsilon_t$ ~...
11
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4answers
3k views

Ergodicity explained in layman terms

I've been told that Ergodicity gives us a practical vision of processes WSS (Wide-sense stationary) and a bunch of integrals. For me, it is not enough to fully understand it. Could someone explain me ...
11
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1answer
12k 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 ...
11
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3answers
585 views

Optimization of stochastic computer models

This is a tough topic for myself to google since having the words optimization and stochastic in a search almost automatically defaults to searches for stochastic optimization. But what I really want ...
11
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1answer
885 views

Why is an unbiased random walk non-ergodic?

Wikipedia says "An unbiased random walk is non-ergodic." Let's look at a simple random walk. It's defined as: take independent random variables $Z_{1},Z_{2}$, where each variable is either $1$ or $−1,...
11
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1answer
499 views

The Fishing Problem

Suppose you want to go fishing at the nearby lake from 8AM-8PM. Due to overfishing, a law has been instated that says you may only catch one fish per day. When you catch a fish, you can choose to ...
11
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1answer
716 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 ...
10
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3answers
19k views

How can I show that a random walk is not covariance stationary?

How can I show that a random walk ($y$ follows a random walk) is not covariance stationary? I tried to work on the formula below (with no results) Could you give me just a hint on how to proceed ...
10
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3answers
7k 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 ...
10
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1answer
5k views

Example of a process that is 2nd order stationary but not strictly stationary

Does anybody have a nice example of a stochastic process that is 2nd-order stationary, but is not strictly stationary?
10
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1answer
278 views

How to test if “previous state” has influence on “subsequent state” in R

Imagine a situation: We have historical records (20 years) of three mines. Does the presence of silver increases the probability of finding gold in next year? How to test such question? Here is ...
10
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1answer
303 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, ...
10
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1answer
1k views

What's the name for a time series with constant mean?

Consider a random process $\{X_t\}$ for which the mean $\mathbb{E}(X_t)$ exists, and is constant, for all times $t$, i.e. $\mathbb{E}(X_t)=\mathbb{E}(X_{t+\tau})$ for all times $t$ and time shifts (or ...
9
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1answer
6k views

Proof of Chapman Kolmogorov equation

In the proof of Chapman Kolmogorov Equation $p_{ij}^{(m+n)}=\sum_{k=0}^{\infty}p_{ik}^{(n)}p_{kj}^{(m)}$ Proof: $p_{ij}^{(m+n)}=P[X_{m+n}=j|X_0=i]$ By the total probability it says $P[X_{m+...
9
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2answers
233 views

Reference Request: Book on Unit Root Theory

In trying to do time series analysis, I almost regularly stumble upon unit root and cointegration tests. The design of most these tests is based on a null of unit root (for both linear and non-linear ...
9
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1answer
149 views

Having a hard time with the law of the iterated logarithm

Let's say you have infinitely many i.i.d. Bernouilli variables $X_1, X_2, \cdots$ of parameter $p=\frac{1}{2}$. For instance, the binary digits of a random number. Let $S_n = X_1 + \cdots X_n$. The ...
9
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1answer
251 views

Plain English explanation of Ito's integral?

I'm looking for a plain English explanation of Ito's integral. I don't need an exhaustive proof, derivation, etc. Just a simple ~this is effectively what it does and why it's better than a Riemann sum ...
9
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1answer
2k views

MLE of a multivariate Hawkes process

I'm struggling with implementing the maximum likelihood estimator for a multivariate Hawkes process (HP). Specifically, while the analytical expression for a log-likelihood function of a univariate HP ...

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