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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aperiodic property and the existence of limiting or stationary distribution for Markov chain

Given a Markov chain, what are the relationship between the property of aperiodic and the existence of stationary distribution or limiting distribution? Moreover, if a Markov chain is claimed to be ...
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Absorption Time of Continuous Markov Chains

I have the following question about the Absorption Times of Markov Chains in Continuous State-Space. I was reading the following article on Absorption Times of Markov Chains (https://en.wikipedia.org/...
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Maximum Likelihood estimation of Double Poisson GAS models in R

I am trying to fitting a Bivariate Poisson generalized autoregressive score (GAS) model in R. The model is developed in this paper: https://papers.tinbergen.nl/17062.pdf . I have found a working code ...
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How to identify a random process as a measure probability on the space of functions?

Given $(\Omega, \mathcal{F}, \mathbb{P})$ a measure space and $(\mathbb{R}, \mathcal{B}(\mathbb{R}) )$. Let $X_t = [X_t: t \in \mathbb{Z}]$ - $X_t: \Omega \to \mathbb{R}$, $\forall t \in \mathbb{Z}$ -...
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Different notion of stationarity for distribution shift

Consider a stochastic process $(X_t)_{t\in T}$. Usually this will be a time series, so for simplicity consider $T=\mathbb{N}$ (or $T=\mathbb{Z}$). For many applications this process will be ...
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Applications of Functional Data Analysis (FDA) [duplicate]

Are there any problems that are particularly well suited for models from the field of Functional Data Analysis (FDA)? I was reading the Wikipedia page for Functional Data Analysis (https://en....
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Scoring rules for Dirichlet process mixtures

Let's say I do a sample-based fitting of a Dirichlet process model: $$ \begin{aligned} X_i &\sim f(x_i\mid \theta_i)\\ \theta_i &\sim G\\ G &\sim \text{DP}(\alpha, G_0) \end{aligned} $$ ...
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Importance of the "Chinese Restaurant Process" in Statistics [closed]

Recently, I came across a concept in statistics called the "Chinese Restaurant Process" (similar to the Poya Urn): In the Polya Urn, there is a vase (an urn) with different colored balls. A ...
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Stochastical process regarding to net profit and cost question

There are three companies $X,Y,Z,O$ they sell $240000,180000,120000,60000$ boxes respectively. The company $Z$ is thinking to release a new product which takes $L$ (money) to spread it. Take $P_{ij}$ ...
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Finding the maximum likely estimate of the parameters of a Linear Birth Death Immigration process given samples from the stationary distribution

So, I am working on a statistical modeling problem where I would need to estimate the maximum likely parameters of a Linear Birth Death Immigration model. So, here we would have the birth rate at ...
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Do stochastic chaotic systems decorrelate with time?

Assume I have a dynamical system with additive process noise of the form $$\mathbf{x}_{t} = \mathbf{F}\left(\mathbf{x_{t-1}}\right) + \mathbf{\epsilon}$$ where $\mathbf{x}_{t}$ is the state at time $t$...
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Taking Samples From a Mixture Model

I am interested in knowing if Monte Carlo algorithms can be used to take samples from a Gaussian Mixture Model. Suppose I have the following Gaussian Mixture Model (2 Components, 4 Dimensional - I ...
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Where does "Uniform" in "Uniform central limit theorem" come from?

We may all know about the CLT. Today I have seen two articles where the use a new term (to me), that is "Uniform central limit theorem". A uniform central limit theorem and efficiency for ...
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Is the mode probability of IMM filter in a multi-sensor problem formulated by pseudo measurement or just a probability product?

I would like to use IMM filter to solve a multi-sensor problem, but it seems that there are two ways of calculating mode probability $\mu_k$. The first way can be written as follows, \begin{aligned} \...
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Calculating V(Xi) and E(Xi) from Markov chain, given a probability matrix and a intitial distribution

*If you have a given probability matrix (P) for a Markov chain, and have calculated the initial distribution (π). How do you calculate E(X10) and V(X10)? {Xt, t= 0,1,...} Sx={1,2,3} P= (1/3, 1/4 ,5/12 ...
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Can I simulate a random field with nested variogram by a summation of independently simulated random field for each componnet of the nested variogram?

I want to simulate a random field $Z(u)$ that has a nested variogram, say $\gamma(h)=\gamma_1(h) + \gamma_2(h) + \gamma_3(h)$, assuming the variogram is isotropic. Whether can I simuate independently ...
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Whether the correlation structure of random field $Z(u) + Y(u)$ is equal to the correlation structure of $Z(u)$ plus that of $Y(u)$?

I want to simulate a Gaussian random field (RF) with correlation structure (represented by the geostatistic tool 'semivariogram' $\gamma (h) \: +\: pure \: nugget \: effect$). I want to know whether ...
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Wold Decomposition -- summation to infinity

In Wold's decomposition we have $ Y_t = \sum_{j=0}^\infty b_j \varepsilon_{t-j} + \eta_t $, where the variables have definition as in the Wikipedia page. I'm confused about why the summation goes to ...
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ARIMA Process vs Cox Process

In the context of queues, I am very interested in this idea - as we know, arrival rates can be non-linear and can also display seasonal-temporal patterns (e.g. fewer passengers may arrive at the ...
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Why is the probability of ever making a transition to state $j$ starting from $i$ equals to $\sum_{n=1}^{\infty}f_{ij}^n$?

This trouble comes from: I did not understand why this probability would not be: $f_{ij}=\displaystyle\sum_{n=1}^{\infty}P_{ij}^{n}$ Where: $P^{n}_{ij}=P(X_n=j|X_0=i)$ I see no problem using $P^{n}_{...
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Prove that $z_t = \delta + w_t + y_t - y_{t-1}$ is stationary, where $w_t$ is a white noise, $y_t$ is a stationary time serie and $\delta$ is const

Prove that $X_t - X_{t-1} = \delta + w_t + y_t - y_{t-1}$ is stationary, where $w_t$ is a white noise, $y_t$ is a stationary time series and $\delta$ is an scalar. My attempt: Let $Z_t=X_t - X_{t-1}$...
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What is the difference between a Simple Random Walk and a Random Walk and why is one stationary, while the other is not?

To clarify, by a Simple Random Walk I mean $$ Y_i = \begin{cases} -1 & prob = 1/2\\ 1 & prob = 1/2 \end{cases} $$ $$ X_t = \sum_{i=1}^t{Y_i} \quad \textrm{,}\,X_0 = 0 $$ and by Random Walk I ...
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Insurance Claims: Proving a Process is a Poisson Process and Finding its Rate

Let $X(t)$ denote the number of claims received by an insurance company in the time interval $[0,t]$. We will assume that ${X(t) : t ≥ 0}$ can be modelled as a Poisson process, where $t$ is measured ...
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Homework question about Poisson Processes with conditional expectation

I think I have the first two questions figured out, but I'm stuck on conditional expectation. Consider a Poisson process, X = {X(t) : t ≥ 0} with rate λ = 2 per hour. 1) Covariance(X(5) - X(2), X(4) - ...
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Why are there two forms of cubature kalman filters?

When I read cubature Kalman filter (CKF) related papers, I have seen two forms of corvariance $P_{xz}$. I don’t know what is the difference, or are they equivalent? The first form is form the orignal ...
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What is a Self-Normalized Process?

I came across a vast literature by Victor H. de la Peña (you can find the book here) that concern self-normalized processes, but I am struggling to understand the definition of said processes! What ...
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Maximum value on a set of die rolls --- how to prove that this is a Markov chain?

Suppose we have a sequence of rolls of a fair die. Suppose we let $X_n$ be the $n$th outcome and let $Z_n=\max(X_1,...,X_n)$ be the maximum outcome in the first $n$ rolls. For example, if I roll a ...
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Existence of the solution for SDE with Gaussian Process

I'm interested in the existence of the solution for a non-Ito SDE. Sloppy notation but assume a SDE given by $\dot{x}=f(x),\quad f(x)∼GP(0,k(x,x′)),$ where $f$ is a Gaussian Process with kernel $k$. ...
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measure if some binary events happen in random or are correlated

I have thousands of binary processes of uneven length like those below: ...
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How to compare time series with different time stamps

My data consists of several time series of the form $A_i = [ (t^i_0,x^i_0), (t^i_1, x^i_1), \ldots, (t^i_{n_i}, x^i_{n_i})]$ for $i \in \{1, 2, \ldots 20 \}$ $B_j = [ (t^j_0,x^j_0), (t^j_1, x^j_1), \...
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What’s a textbook covering similar content to “Introduction to Probability Models” by Sheldon Ross?

I’m taking a class with a instructor using said textbook, and I find the explanations in it lacking. It’d be great if anyone can offer an alternative book covering similar content (i.e. conditioning ...
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how to calculate the number of events N(t) (counting process) at a certain time point

I have a simulated survival data of failure time, censoring time, observed time and delta like the following pic (a part of the data. There are 1000 rows in total. The first column is just index ...
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Markov Chains for Hospital Stays

Suppose I have the following problem: Suppose you access to the hospital records: you have the history about how different patients passed through the different "stages" of the hospital (...
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Stochastic simulation, what to do after generate the initial random sample

I don't have a background in statistics but currently learning the basics. I want to do a stochastic simulation, which I assume here I should iterate my simulation multiple times. And I am stuck now ...
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Measurement of regularity of events

I am dealing with events generated by different processes. My data includes time stamps when the events occur, so I am trying to differentiate the processes in regular or irregular categories or ...
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Can a time dependent lagged function be used as a rate parameter?

Question: Can the rate parameter in a mm1 queue or a poisson process be a lagged function of itself? https://en.m.wikipedia.org/wiki/Poisson_point_process https://en.m.wikipedia.org/wiki/M/M/1_queue ...
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Can any discrete stochastic be considered as a "point process"?

"A point process is a stochastic model underlying the occurrence of events in time and/or space. In this blog, we will emphasis on purely temporal aspects of point process i.e., the space in ...
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Check if real data follows the memoryless property

Are there any methods to test if some real world data follows the memoryless property? For example, are there any statistical hypothesis tests that can be used to check if the data is truly "...
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Correlation in weak stationary process

I read this chapter about weakly stationary process from the book "Introduction to Probability, Statistics, and Random Processes" by Hossein Pishro-Nik. Here is the defition of it: $$E[X(t_1)...
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Geometric Brownian motion drift estimation problem

Given: $$ d \ln{S_t} = \left( \mu - \frac{\sigma^2}{2} \right)dt+ \sigma dW_t $$ We have that: $$ \mathbb{E}_t[d \ln(S)]=\left( \mu - \frac{\sigma^2}{2} \right)dt $$ $$ \mathbb{V}_t[d \ln(S)]=\sigma^2 ...
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Generating a 2 dimensional vector Random Process with Brownian Motion and Ornstein-Uhlenback Process as components

I want to generated a Random Process $(X_1(t), X_2(t))$ with prescribed covariance structure such that $$ X_1(t) \sim \text{Brownian Motion}(\mu_1, \sigma_1, x_0) $$ $$ X_2(t) \sim \text{Ornstien-...
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Randomly Sampling from one Random Signal

The Question I was given a set of data from a local commodity market trader. The trader had employed two different trading algorithms on the same commodity. The price of the commodity, $X_t$, is ...
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Simple Symmetric Random Walk on $\mathbb{Z}$ is null recurrent

Question: Consider a simple symmetric random walk on integers, where from every state $i$ you move to states $i-1$ and $ i+1$ with probability half each. Show that this random walk is is null ...
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Infimum of a random function

Suppose that we have a random variable $X$ and this is a function of an index $t\in T$. Here, what is the meaning of the infimum of this random function? $$\inf_{t\in T}X(t)$$ How can we interpret ...
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The meaning of a limiting process

In some papers, the authors use the word "limiting process". I think, this is the limit of a stochastic process. But, I am not sure because the limit of a stochastic process is just a random ...
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What are the differences between different volatility models?

I would like to understand the differences between different volatility models like in simple terms and what are pros and cons over the other models Local volatility Model(Dupire) Heston Model SABR ...
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Computation of Posterior Mixture Weights

My question concerns the below example, where the author analyzes rainfall occurrences via a first order Markov chain. The transition probabilities are such that $p_{11} + p_{12} = 1$ and $p_{21} + p_{...
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Problem understanding the intuition behind Slepian's inequality

Slepian's inequality is defined as follows: Let $X\in\mathbb{R}^n$ and $Y\in\mathbb{R}^n$ be centered Gaussian random vectors such that \begin{align} \mathbb{E}X_iX_j&\geq \mathbb{E}Y_iY_j,\quad \...
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How to forecast simulated possible futures of a time series, rather than just a deterministic forecast with prediction intervals?

Time series forecasting packages typically provide the ability to see a deterministic projection of the future of the time series, often with prediction intervals. But suppose I want to show many ...
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Trying to find a distribution for time dependent drought

I would like to seek your support on a modelling issue for which I could not find relevant past postings or published literature resolve it. I am running a cost benefit model to assess the impact of ...

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