Questions tagged [stochastic-processes]

A stochastic process describes 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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How to derive transition matrix in this stochastic process?

I am new to stochastic processes and trying to solve a question related to finding a transition matrix of some experiment. The question is a A sequence of experiments is performed, in each of which ...
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Show that $\{(X_{n},X_{n+1})\}_{n\geq 0}$ is a Markov Chain where $\{X_{n}\}_{n\geq 0}$ is a Markov Chain

Show that $\{(X_{n},X_{n+1})\}_{n\geq 0}$ is a Markov Chain where $\{X_{n}\}_{n\geq 0}$ is a Markov Chain. Remark: We know that $\mathbb{P}(A|\emptyset)$ is undefined, I am right? This fact is ...
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Poisson process - time of arrival of a client, given that one client arrived at first interval

I have the following situation: I have a Poisson Process with $λ=7$ (seven customers / hour). This process describes the arrival of customers in a store. The store is open from 9:00 AM to 19:00 PM. My ...
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Strict Sense Cyclostationary and Shifting the X with $\theta$

Fellow stackexchangers, I did my best to put a topic that describes the question that I am going to ask. I am reading Probability, Random Variables, and Stochastic Processes by Papoulis and I am ...
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Why choose a Dirichlet distribution in Latent Dirichlet Allocation (LDA)?

I am studying Latent Dirichlet Allocation (LDA), but I don't have much knowledge of stochastic processes. From Wikipedia: LDA assumes the following generative process for a corpus $D$ consisting of $...
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Probability of service in a queue theory problem with exponential random variable

I have one queue with two servers $S_1$ and $S_2$.The serving times are modeled $\sim exp(\mu_1)$ and $\sim exp(\mu_2)$ respectively. The first server is free while the second has two clients, $A$ ...
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Implementing multivariate stochastic volatility on PyMC3

I am trying to implement a multivariate stochastic volatility model for electric activity on the brain on PyMC3. The data are composed of: Y = temporal series with 3 channels and 700000 samples (...
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Time distribution of a Poisson Process

My problem is the following: I have a Poisson Process with $\lambda = 7$. This process describes the arrival of customers in a store. Now, I must find: The time distribution when the second customer ...
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For a Zipf distribution(n,$\alpha$) how does alpha affect the entropy? [closed]

How does alpha affect the entropy of a zipf distribution, or does it even change it at all? I reason that it does not change much because in a zipf distribution, the entropy is changed more by the ...
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Limiting distribution of $M/M/2/5$ queue with two heterogenous servers

A queue with a total capacity of $5$ customers has $2$ servers who serve at rates $\mu_1 = 1$ customer/hour and $\mu_2 = 2$ customers/hour respectively. Service times are exponentially distributed. ...
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How to get this analytical results for probability of wait times

I'm working with a continious-time stochastic process, where a particular event may happen at some time t with an unkown underlying distribution. One "run" of a simulation of this process ...
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Finding not Gaussian vector

I know from the answer to another question that if $X,Y$ are independent standard Gaussians and $Z=\text{sign}(X)\cdot |Y|$ then $Z+X$ is not Gaussian. So, $(X, Z)$ is not Gaussian vector. How to ...
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Expected value of random sum of random variables

In example 3.2.2 of the book "An Introduction to Stochastic Processes" by Kao, the following statement is presented. Why are $X_i$ and $N(t)$ dependent? I would be thankful if you can ...
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Probability after n steps

I have been studying markov chains for my Introductory Stochastic Processes exam, but i am struggling with the following problem: Question: Consider a matrix with state space $S=\{1,2,3\}$ and the ...
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Could you estimate the probability of arrivals of a poisson process?

Let $T_i$ ~ $exp(\lambda)$ be i.i.d exponential random variables, with unknown $\lambda$. These are the time intervals of the poisson process. And $X_n=\sum_{i=1}^nT_i$, are the arriving time of the ...
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How to understand the prior and posterior of Gaussian process in the function space view?

I've been reading Gaussian process for machine learning by Rasmussen and Williams and I'm confused by the prior and posterior in the weight space view and function space view. The prior and posteior ...
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Densities of Arrival Times of Poisson Process

People arrive at a store as a non-homogeneous Poisson process with rate t, where t is the time measured in hours between noon and 6pm. If we know that precisely one person arrived in the first hour, ...
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Intuitive/Practical meaning of non-stationarity of GDP Data

As i just read in a time series book that a particular GDP data under consideration is non-stationary verified through various tests. From stationarity definition this means that the process has ...
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understanding definition of stochastic process

I am trying to understand the definition of a stochastic process and related terminology by myself. I found this intuitive: http://www.eco.uc3m.es/~jgonzalo/teaching/PhDTimeSeries/...
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Convergence of a stochastic process

Consider the discrete time random process $X_n,n\in \mathbb N$, with $$X_{n+1}=(1-K)\cdot X_n+K\cdot\frac{G_n}{c}\cdot X_n$$ where $G_n$ is a random variable with expectation $\mathbb E[G_n\mid X_n]=\...
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Can a stochastic process be decomposed as follows?

So for instance an AR(1) process \begin{equation} x_t=\rho x_{t-1}+u_t \end{equation} where, say, $u_t\sim IID(0,1)$ for $t\in \mathbb{N}$, can be expressed using backward iteration as follows: \begin{...
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Renewal process vs inhomogeneous Poisson process?

I am analyzing a dataset with recurrent events, and considering two candidate models: A model based on a renewal process (time is measured from the previous event) A model based on an inhomogeneous ...
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Strategic Multi Armed Bandit

As a part of my project, I have been tasked with formulating a multi-armed bandit problem with strategic arms. What I have found out is a Gittin's index approach to the problem provides a solution ...
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Interpolation using Gaussian processes

This is about Gaussian process interpolations, where the given data are f(0) = 1, f(0.4) = 3 and f(1) = 2. Assume that the covariance function used is the exponential covariance, where the expectation ...
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Estimate partially observed Poisson process

I try to estimate the intensity of a Poisson process $P_1$, but it is not fully observable. There are some "obervers" coming to the system which follow another Poisson process $P_2$. In ...
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Time series - Stationarity and invertibility?

Sometimes when I take material from time series to study, it appears out of nowhere "for a process to be stationary it is necessary for the roots of the characteristic polynomial to fall outside ...
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Stochastic Differential Equation with drift multiplied by everywhere discontinuous random process

(WARNING: this has been crossposted on physics.stackexchange (questions/588606). It has been suggested to post also here, let me know if it is against the rules). Let the stochastic process $\{X_t\}$ ...
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Key elements of a Poisson Process? [duplicate]

I'm new to Stochastic Processes, in general. However, I see that they come in varying forms of complexity. A gaussian process (and gaussian process regression) are quite complicated and can easily fit ...
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Autoregressive model with observable noise

The classical autoregressive model is a linear model for the dynamic variable $x$, where the added noise $\epsilon$ is directly affecting the dynamics of the model $$x_{t} = \sum_i \alpha_i x_{t-i} + \...
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Frequentist vs Bayesian and deterministic vs stochastic [closed]

So this is sort of a general/basic, likely dumb question. I'm hoping to get a general idea, to better guide what I search/read. How do these terms relate to each other. I know with Bayesian theory, ...
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In a Galton–Watson process, obtain the observation report and simulation table, estimate the $E(Z_n)$ and $Var(Z_n)$ by using Monte Carlo method

The following is mathematical formulation of Galton–Watson process, which is from the wikipedia, I have two questions about this branching process. 1. How to obtain the observation report and ...
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A Skip Free Negative Random Walk

Suppose $\{X_{n}|n\geq 1\}$ is independent, identically distributed distribuited. Define $S_{0}=X_{0}=1$ and for $n\geq 1$ $$S_{n}=X_{0}+X_{1}+\cdots+X_{n}.$$ For $n\geq 1$ the distribution of $X_{n}$ ...
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Covariance Matrix and Gaussian Process

In a paper i'm reading they use gaussian processes but i'm a little bit confused about their use of the covariance matrix. The setup is as follows: the inputs are $x_i \in \mathbb{R}^Q$ and there are $...
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In this ball-and-bag scenario (with several twists), how does one determine probability that an event occurred within some time frame?

We have a bag into which we add red balls one at a time at some known but varying rate $x$ throughout the course of the experiment. Balls are never removed at any time during the experiment. In the ...
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How do I obtain the standard errors for Ornstein Uhlenbeck parameter estimates?

I have used least squares estimation to obtain estimates for parameters to be used in Ornstein Uhlenbeck process. Now, I would like to compute the standard errors of estimates. $𝑑𝑆𝑡=𝜆(𝜇−𝑆𝑡)𝑑𝑡+...
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Geometric Brownian motion with target skewness and kurtosis

The Cholesky inversion method can be adopted to set a target correlation matrix when artificially generating a multivariate geometric Brownian motion dataset Can the moments of a univariate GBM be ...
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Why are departure rates of $M/M/c$ queues not equal to service rate $\mu$?

I think this issue should have an answer somewhere but I could not find in any materials. In every textbook I read about $M/M/c$ queueing systems, it is always acknowledged from the beginning that the ...
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Does it make sense to make a conditional Markovianity assumption?

Please do excuse if the title is somewhat vague. As we know, a discrete-time Markov chain is a sequence of random variables $S_1,S_2,\cdots$ with the Markov property - i.e. \begin{equation} P[S_{t+1}=...
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How can I test if a time-series data satisfies Markov's property and it is a martingale?

My question is about investigating some properties of time-series. How can I test if my time-series data satisfies Markov's property? How can I test if my time-series data is a martingale? I wonder ...
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Yule (preferential attachment process vs. birth&death process

What is the distinction between the Yule (preferential attachment) process and the birth&death process? Are they the same thing, called differently in different contexts, or is the birth&death ...
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How to calculate Markov Chain parameters knowing AR(1) coefficient $\rho$?

Given that a discrete-time two-states (low and high) Markov chain $X_t$ can take two values: \begin{pmatrix} X_l\\ X_h \end{pmatrix} with its transition probability matrix $P^{T}$: \begin{pmatrix} p_1 ...
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Bayesian inference of stochastically evolving model parameters

I have a question related to self-calibration in radio interferometry, but I will try to phrase it as generic as possible. I have a set of data points $$D = \{ d_{0, t_0}, d_{1, t_0}, \cdots, d_{M, ...
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Converge of Scaled Bernoulli Random Process

Suppose a random sequence is defined by $X_n := n B_n$, where $B_n$ is a Bernoulli sequence such that $\mathbb{P}(B_n = 1) = 1/n$. I am interested in the convergence properties of this random process ...
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What correlation structure is necessary to ensure a random walk is almost surely bounded?

Say I have a stochastic process $\{X_t\}_{t \in \mathbb{N}}$ such that their cumulative sum $\{S_t\}_{t \in \mathbb{N}}$ is a random walk process: $$ S_t = \sum_{i = 1}^t X_i $$ If each $X_t$ is i.i.d ...
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Expected number of running heads in coin toss

How to find the expected number of running heads of a specific length (say 'k' exactly) in 'n' tosses of a coin (fair/biased). For example, consider the output of a coin toss as follows "...
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Are the following model assumptions on a data stream too restrictive?

Suppose that you were to model a "generic" continuous-time real-world data signal $X$ taking values in a bounded continuum $K\subset\mathbb{R}^d$ (e.g. the body temperature of a patient or ...
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Is the MA($\infty$) process with i.i.d. noise strictly stationary?

I have a MA($\infty$) process defined by $$ X_t = \sum_{k=0}^\infty \alpha_{k} \epsilon_{t-k}, \qquad t\in\mathbb{Z} $$ where the sums converge a.s. and the $\epsilon_t$ are i.i.d. centered noise ...
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Is a weakly stationary AR(p) process also strictly stationary if the noise is i.i.d.?

An AR($p$) process is any causal and weakly stationary solution to the equations $$ X_t = \beta_1 X_{t-1} + \dotsc + \beta_p X_{t-p} + \epsilon_t, \qquad t \in \mathbb{Z} $$ where the polynomial $B(z)...
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Difference between Infinite HMM and Latent Variable Model with Sequential Chinese Restaurant Prior?

What is the difference between an infinite HMM (http://mlg.eng.cam.ac.uk/zoubin/papers/ihmm.pdf) and a latent variable model where the latent states are given by a distance-dependent Chinese ...
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Test for correlated vs uncorrelated increments in random walks

Is there a test that can distinguish the strictest form of the random walk, $$P_{t}=P_{t-1}+\varepsilon_{t}, \varepsilon_{t} \sim \mathrm{IID}\left(0, \sigma^{2}\right)$$ where each step is assumed to ...

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