# 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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### 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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### 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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