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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Stick-breaking: break sticks of decreasing lengths

The stick-breaking construction used for Dirichlet Processes can create an infinite sequence of probabilities $ \boldsymbol{\pi} $ (stick lengths) that sum to 1 via the following formulae: $\nu_i \sim ...
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Show the ergodicity of a random sum of ergodic processes

We say that a mean stationary stochastic process $(X_t)_{t \in \mathbb N}$ - i.e. $E[X_t]= \mu_X$ for all $t$ - is ergodic mean if \begin{equation}\tag{I} \frac 1 T \sum_{t=1}^T X_t \overset {pr} \...
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Distribution of class sizes above a certain threshold in a stick-breaking process

Given a stick-breaking process, that is to say a sequence of random variables $(Y_n)_n$ such that $$Y_n = X_n \prod_{i=1}^{n-1}(1 - X_i)$$ where the $X_n$ are independent random variables distributed ...
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A question related to uniform asymptotic negligibility (UAN) assumption

The uniform asymptotic negligibility (UAN) assumption is well know in probability theory. In my case, I have a definition of (UAN) for MA processes. Let $(X_n)_{n\geq 1}$ a sequence of MA processes: $$...
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Why do sampled functions from a Gaussian Process perfectly interpolate the data?

I've been studying GPs with Rasmussens book as well as a few of my old probability favorites. I am confused by the results we're getting (doing also a lot of practice) when sampling from our Gaussian ...
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differential entropy rate for stationary processes

the differential entropy rate of a stochastic process $X = (X_i)_{i \in \mathbb{N}}$ is defined as $$h(X) = \lim_{n \to \infty} \frac{1}{n} h(X_1,\ldots,X_n)$$, where $h(X_1,\ldots,X_n)$ is the joint ...
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Statistical test to assess if two stochastic processes have the same law

Suppose that we have two finite, distinct, groups of sample paths. How can we statistically test that they were actually sampled from the same stochastic process? In the case of solutions of rough ...
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Understanding deterministic processes from a variable/feature

I have been trying to understand the statistics behind a deterministic process. I'm currently doing this exercise from this guide to understand better seasonality and time series modeling. This code ...
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Range of $a$ such that $w \leftarrow w-a x \langle w, x \rangle$ converges almost surely?

Edit Sep 19 this answer on Mathoverflow matches simulation results Suppose $x_i$ come from 2-d standard Normal centered at 0. What is the range of $a$ for which the following iteration converges ...
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I want to calculate $\int f(X_t) dB_t$ where $B(t)$ is Brownian motion and $X_t$ satisfies $d X_t = \mu dt + \sigma dB_t$

Let $B_t$ be Brownian motion, and $X_t$ satisfies the following Ito SDE: $$ d X_t = \mu\, dt + \sigma\, d B_t, $$ and $f$ is a function over $X_T$. I want to calculate $\mathbb{E}[f(X_t)dB_t]$. It ...
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Where does the random come in for conditional expectations $\mathbb{E}[X | \mathcal{F}]$?

For continuous random variables $X, Y$ the conditional expectation $\mathbb{E}[X | Y]$ is itself a random variable. I understood this in the sense that for a realisation of $Y$ we can say $$ \mathbb{E}...
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Show an asymptotic property of a realization of an AR(1) [closed]

Let $$X_t = 0.9 \, X_{t-1} + \eta_t, \quad \eta_t \,\, \hbox{i.i.d.} \sim N(0,1)$$ Indeed, I want to show that a fixed realization $(\bar{X}_t)$ satisfies \begin{equation}\tag{I} \lim_{n \to \...
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Control variables cofounding the effect of decisions in linear regression model used in sequential decision optimization

How can we be sure that confounding variables/control variables don’t pickup the effect our decisions w.r.t decision variables had on the actual control variable? Since the term control variable ...
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Convergence of a function having a big summation at each sample

I have the following function. $$ x(k) = \sum_{m} e^{i (U_m k + \beta_m)} $$ Here, $U_m$ samples are random numbers coming from a Gaussian distribution $$U_m \sim \mathcal{N}(\mu_u, \sigma_u)$$ and ...
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Is stationarity of data necesarry in order to do any statistics?

If we assume we have a stochastic process $X_t$ for which we have, $$\mathbb{E}[X_t] = \mu(t) $$ $$\operatorname{Cov}(X_t,X_s) = \gamma(s,t) $$ where the dependence of the functions on $s,t$ are non-...
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What is the maximum Q-value that any state can obtain in DQN?

The q value of a specific state,$s$ and action, $a$ is given by the following equation, as per Sutton and Barto's equation 3.13 - $$q_{\pi}(s,a) = \mathbb{E}_{\pi}[\sum_{k=0}^{\infty}\gamma^{k}R_{t+k+...
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Unconditional variance of AR(2) + GARCH(1, 1)

I am being asked to derive the unconditional variance for stochastic process $\{Y_t\}$, where: $$Y_t = \phi_1 Y_{t-1} + \phi_2 Y_{t-2} + \varepsilon_t$$ $$\varepsilon_t = V_t \sigma_t$$ $$\sigma_t^2 = ...
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Stochastic parametric stability of structures

I am investigating an engineering problem that relates to the dynamic stability of a beam-like structure under the action of a stochastic excitation (say, e.g., a marine structure acted by waves or a ...
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What is the distribution of the Frobenius distance between two covariance matrices?

I am computing the Frobenius norm of the difference between two covariance matrices, \begin{align} |\mathbf{C}-\mathbf{C}'|_F=\sqrt{\sum_{i,j}\left(c_{ij}-c'_{ij}\right)^2}. \end{align} Each of these ...
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Second order weakly stationary process

I was wondering, whether a stochastic process with a covariance, that is not depending on time, imply constant variances. Since I read a textbook, where they call a process, which is second order ...
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Reinforcement Learning MDP stochastic policies

I am struggling to get an understanding behind stochastic policy in a MDP I was reading this paper and in section 3.2 they say At each time step $t$, corresponding to extracting sentence number $t$, ...
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How to verify the correctness of forecast?

I would like to forecast the car rental (count time series). Given hourly integer valued car rentals for a month's period from 24th september to 24th October. I need to forecast car rental demand ...
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How to build a Poisson random measure according a Levy Process

Let $(\Omega, \mathcal{F}, P)$ and $(\Theta, \mathcal{B}, \rho)$. A Poisson random measure (PRM) with intensity $\rho$ is a kernel $\mathcal{N}: \Omega \times \mathcal{B} \to \mathbb{R}$ such that: $\...
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How to calibrate a multi-dimensional Ornstein-Uhlenbeck process?

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,\...
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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 ...
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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 $p=0$ I read this to mean \begin{align*} \mu + \epsilon_t &...
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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 "...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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_{...
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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, ...
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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) ...
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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 ...
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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:...
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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 ...
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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, ...
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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 ...
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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 ...
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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"...
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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 ...
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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" ...
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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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What's the transition probability according to this PDE?

I'm trying to figure out how I can simulate markov chains based on an ODE: dN/dt = alpha N (1 - N / K) - beta N Thus N denotes ...
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