# 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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### Can anyone provide me with reference to some lecture notes or an online lecture on Multiplicative Error Models?

As the title says, I am looking for some lecture notes or an online class going over Multiplicative Error Models. I have found a number of academic papers on the topic, but I am having trouble ...
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### Cannot obtain empirical Laplace distribution for increments of a laplace motion

Consider the Laplace motion (a special type of Levy process where the stationary and indepedent increments are Laplace distributed). One representation of the Laplace Motion is through Brownian ...
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### Are Polya urn (CRP) and stick-breaking clusters interchangeable in posterior analysis?

In sampling a DP model, it's more space-efficient to only keep data related to active clusters (clusters with data). Under a CRP model, if I want to do posterior predictive sampling, it's ...
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### Pitman Yor Density, hyperpriors and Parallel tempering

I’m trying to understand the pitman Yor process. For the Dirichlet, I can give the concentration parameter a Gamma prior, and Escobar/West gives a posterior sampling strategy for it. (Actually I’m ...
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There is abundance of literature out there on methods for calibrating a one-dimensional OU process, namely: \begin{equation} dy_t=\kappa(\theta-y_t)dt+\sigma dW_t \end{equation} where $(y_t,\kappa,\... • 1,072 0 votes 0 answers 18 views ### Does a martingale difference sequence$X_{t,i}$imply$E[(\frac{1}{M}\sum_{i=1}^{M}X_{t,i})^2|\mathcal{F}_t]\leq C\times \frac{1}{M}$? Let$X_{t,i}$denote a martingale difference sequence in the$i$th time at day$t$, where$i=1,\cdots,M$. If$X_{t,i}$is independent for$i=1,\cdots,M$, the result is straightforward. And I know that ... 0 votes 0 answers 23 views ### Does$p=0 \implies \sum_{i=1}^{p} \phi_i L^i = 0$? Let us take this$\operatorname{AR}(p)$equation $$\left(1 - \sum_{i=1}^{p} \phi_i L^i \right)X_t = \mu + \epsilon_t$$ as an example. When$p=0I read this to mean \begin{align*} \mu + \epsilon_t &... • 5,062 6 votes 2 answers 570 views ### Can any Models be "Bagged"? I have been learning about "bagging" (bootstrap aggregation) - supposedly, there are many types of statistical models can be bagged together. For example, CART Decision Trees can be "... • 6,068 1 vote 0 answers 27 views ### A lower bound on the probability of an event? Consider a normal distribution\mathcal N(\mu, \sigma^2)$($\sigma>0$). Imagine that one is playing the following "stopping game": she keeps acquiring independent signals from this ... • 111 0 votes 1 answer 39 views ### What do we achieve by imposing ARMA structure on a stationary stochastic process? Suppose we have some set of data$\{x_t\}_{t=1}^T$, which we model as a part of realization of stationary stochastic process$\{X_t|t\in\mathbb{Z}\}$. Now, as I understand, by a virtue of The Wold ... • 160 0 votes 0 answers 45 views ### Gaussian Process Regression Without Kernel I am working on a sequential estimation problem that involves a Bayesian update to a multivariate Gaussian prior from a measurement with Gaussian noise. Specifically, I have a mean vector of length n ... 0 votes 0 answers 42 views ### How to calculate the probability distribution of the sum of two dependent random variables? I need to calculate the PDF and CDF of the polynomial$c_1+c_2 x^2+c_3 x^4$, where$c_1>0$,$c_2<0$and$c_3>0$or$c_3<0$and$x$is a random variable following the truncated half nornal ... 0 votes 0 answers 14 views ### Differentiating a stochastic process i'd like to understand something about the differential of a stochast process: In some exercise I have some stochastic proces$X(t)$which I have to differentiate and I do it using the ito-doeblin ... 0 votes 0 answers 12 views ### Can't find ito differential from infinitesimal generator with partial derivative I have to compute the Ito differential of a diffusion X whose infinitesimal generator coincides in$C^2_0$with$$Lf(x1, x2) = f_{x_1}(x1, x2) + x_2f_{x_2} (x1, x2) + 2x_1f_{x_1x_1}(x1, x2) + 2x_2^2f_{... 1 vote 0 answers 24 views ### Deriving a Stochastic Equation Edit: I'm currently reading a paper (The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing, American Journal of Operations Research, ... 0 votes 0 answers 16 views ### Simulating Iterated Brownian Motion I was going through an interesting article (https://arxiv.org/pdf/1112.3776.pdf) while I was trying to read about subordinated processes. I wanted to simulate subordinated processes (in R or python) ... 1 vote 0 answers 28 views ### Random Walks Question [closed] I am trying to solve this question by using the reflection principles. Let a>c>0 and b>0. A is the set of all paths of a random walk which end at c in their final n’th step. B is the set of ... 2 votes 2 answers 84 views ### Expected Number of Transitions for a Markov Chain to Reach a Certain State I am trying to find out the number of times a die needs to be rolled before observing a 4 followed by a 6. I would like to model this problem using a discrete time Markov chain with 3 states: State 1:... • 6,068 1 vote 0 answers 31 views ### Does the arrival of vehicle on a specified point of a road follow a poisson process? I am asking about a poisson with the same rate but different serving time, because a point in a road have a maximum capacity of letting a number of cars pass through it, but in the same time the speed ... 0 votes 1 answer 46 views ### Balancing "Delayed Entry Bias" and "Survivorship Bias"? This is a question I have always struggled with - suppose you have medical data on patients over a period of time. This includes information on how long they spent in different states: Admission, ... • 6,068 1 vote 0 answers 16 views ### Product of kernels vs Composition of kernels According to Wikipedia there are two main operations between two kernels: product and composition. They look almost identical to me and I cannot figure out what's the intuition between these different ... • 1,560 0 votes 0 answers 32 views ### Poisson processes: Joint probability in overlapping intervals Two teams, A and B, play a soccer match. The number of goals scored by Team A is modelled by a Poisson process$X_t$with rate$\lambda = 0.03$goals per minute. The number of goals scored by Team B ... • 21 0 votes 1 answer 49 views ### Can some Survival Models "Dominate" other Survival Models? I recently heard an interesting interpretation of Survival Models : A "standard" Survival Analysis problem (e.g. where at the end of the study, observations can either be "Censored"... • 6,068 0 votes 0 answers 7 views ### What method can be used to reformulate this kind of Chance Constraint In the above chance constraint, x and y are binary decision variables, is the integer random variable with the range of [4,20], and the right side of the constraint is the probability level. And I ... 2 votes 1 answer 59 views ### Why do we need to Define "Valid" State Transitions in a Multi-State Model? I was watching this video (https://www.youtube.com/watch?v=Wy-WmY6x4tg) and the presenter mentions (@ 8:10) that in a Multi-State Model, the user is required to specify number of "States" ... • 6,068 1 vote 1 answer 45 views ### What is the cubic expectation (third-order moment) of a complex gaussian vector (say, E[$aa^{T}a\$])?

Note: I also posted this question on MATHEMATICS. For a real gaussian vector, an explicit formula for the cubic expectation can be found in Matrix Reference Manual (search 'Cubic Expectations' in this ...
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