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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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Nested CV with Online Learning

I have a time series binary classification dataset. I am implementing an online learning Logistic Regression algorithm in Sklearn and am cross validating with Sklearn's TimeSeriesSplit method. I am ...
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What are some easy to understand books for discrete stochastic process simulation using R?

What are some easy to understand books for discrete stochastic process simulation using R programming language? I mean for the starters?
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Random walks-heavy tailed case

Let $\beta > 0$ and $S_{0}=0$, and let $S_{n}=\xi_{1}+\dots+\xi_{n}$,$n \geq 1$, be a random walk with i.i.d. increments $\{\xi_{n}\}$ having a common distribution $P(\xi_{1}=-1)=1-C_{\beta}$ and $...
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Combining probabilities to find most probable window

I have a series of observations, with an associated probability that an event is occuring at timestep t, something like: [0.8, 0.8, 0.3, 0.9, 0.2] Events can ...
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How do I write a random walk with drift ARIMA? [closed]

I modeled oil prices and got the following coeffecients for my arima model with drift. Is this the right way of writing the model?
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Do measurable maps preserve stationary ergodicity?

In a recent effort to establish stationary ergodicity for a certain stochastic process, I just happened to come across a statement, which I find to be little bit confounding. Given two measurable ...
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How to recognize an ARMA process?

By looking at the autocovariance, how could you recognise what discrete model (MA(q), AR(p), or ARMA(p,q)) is more appropriate to describe your data?
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Approximating AR(1) by finite order MA process - convergence results

I am currently struggling with a result pertaining to the finite order MA approximation of a simple AR$\,(\,1\,)$ process defined on a double sided time-index set $\,T=\mathbb{Z}$. I would be very ...
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How to determine the rate of input in a queue M/M/c?

I know the exit rate ($\mu$) and the average waiting time in the queue ($W_q$). I need solve to rate of input ($\lambda$) in a queue. $\rho = \frac{\lambda}{c\mu} < 1$ $\pi_0 = \left[\left(\sum_{...
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Brownian Motion proof: difference converging to 0 almost surely

I am reading a proof where it is assumed that $$ \lim_{n \to \infty} \sup_{0<s\leq s_0}\left| \frac{t_n(s)}{s}-1 \right|=0 , \hspace{30mm} (1)$$ where $t_n(.)$ is some sequence of functions. ...
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Irreducible (communicating) classes [closed]

The Markov chain $(Xn; n\geq)$ has state-space $S = (0, 1, 2, . . .)$, with $p_{i,0} = \frac{1}{4}$ and $p_{i,i+1} = \frac{3}{4}$ $\forall i \geq 0$, so that the transition matrix is P =$\...
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Biased coins and Markov processes

Good day, I am attempting an optional exercise and I am finding it hard to interpret the problem in terms of matrices and vectors. Coin 1 has probability 0.4 of coming up heads, and coin 2 has ...
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How to prove that $X_t=\int^t_0f(u)dW_u$ and $X_t-X_s$ are independent?

Let $X_t=\int^t_0f(u)dW_u$ for a deterministic function $f$ and $W_t$ is a brownian motion. How can I compute $E[\exp(\lambda_1 X_s + \lambda_2(X_t-X_s))]$ and prove that $X_s$ and $X_t-X_s$ are ...
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Stochastic Differential equation: CAPM

Let $R = (R_1, \dots , R_M)'$ denote a vector of excess returns of $M$ assets observed at $n$ time points, $0 < t_1 < t_2 < \cdots < t_n < T$, within a time span $T > 0$. We wish ...
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First passage time distribution via Monte Carlo simulation

The problem: I want to assess the first passage time distribution via Monte Carlo Simulation, where the first passage time is defined as: $$\tau=\inf\left\{t: X_t > l\right\}$$ where $l$ is the ...
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Homoskedasticity of the residuals

When, I fit an ARMA model to data, I look at the standardized residuals plot to assess if they behaves like uncorrelated random variables with zero mean and costant variance (if the model is good). ...
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What's the variance of an AR(1)/ARCH(1)

The main question is: an AR(1)/ARCH(1) process has the variance of the ARCH(1)? I've tried to compute the unconditional variance of an AR(1)/ARCH(1) model, so an AR(1) in which the noise is modelled ...
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Probability Density function of Poisson distribution

This is an assignment I got for my course on Stochastic Processes: Let us consider a random variable X distributed as a Poisson P (λ) where λ ∼ [0.5, 1]. (a) Which are the unconditional ...
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States of Markov chain and stationary distribution

Let $X$ be a Markov chain with a state space $S={\{0,1,2,... \}}$ and a transition matrix $P$ with given $p_{i,0}=\frac{i}{i+1}$ and $p_{i,i+1}=\frac{1}{i+1}$, for $i=0,1,2,...$. Find out which states ...
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Probability that exactly 12 buses will arrive within 3 hours

Let's suppose there are two buses $A$ and $B$. They draw up at the bus stop under the Poisson distribution with intensities $3$ and $5$ times per hour. (a) What's the expected length of time after the ...
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Proving that given Markov chain is homogeneous. Find state space and transition matrix

Let $X_i$ be the results of a consecutive throws of a die. Let $Z_n=3(X_1^2+\cdots+X_n^2) \bmod 5$. Show that the sequence ${\{Z_n \mid n\geq1\}}$ is a homogeneous Markov Chain. Find a state space and ...
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Continuous Stochastic Processes examples

I am trying to understand various types of stochastic processes. In order for that to happen, I needed some simple examples to be built so that I can build an intuition about them. According to the ...
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Discrete Stochastic Processes examples

I am trying to understand various types of stochastic processes. In order for that to happen, I needed some simple examples to be built so that I can build an intuition about them. According to the ...
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Some simple examples to understand various types of Stochastic Processes

I am trying to understand various types of stochastic processes through some analogical examples using simple experiments like a coin toss or die roll. The book of Hwei Hsu (Chapter-5, Page-162-165, "...
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Can a state have zero periodicity? [closed]

I am getting my concepts cleared in Stochastic process. I understand the concept of periodicity. Just to make it clear, suppose there is a finite Markov chain with states $1,2,3$. Let their ...
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Are all transient states positive recurrent?

In my Stochastic process lecture notes, I noticed that in problems related to a finite Markov chain, the mean recurrence time are calculated only for recurrent states. I calculated the mean recurrence ...
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Does convergence in distribution imply asymptotic stationarity?

Let ${\bf \tilde{x}}_1, {\bf \tilde{x}}_2, \ldots$ be a (possibly non-stationary) stochastic sequence of $d$-dimensional random vectors that converges in distribution. Does it immediately follow that ...
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Reconstructing a Random Process from Moments

If I had the moments of a single random variable $ X $, then I could compute the distribution by Laplace transforming the generating function, $$\begin{align} \rho(X) &= \mathcal{L}^{-1} \big\{ \...
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Covariance non-zero mean AR(1)

Why when I compute the autocovariance function of a non-zero mean AR(1), X(t)-u=Φ(X(t-1)-u)+ε the presence of the mean does not change my result and so the formula should be the same of a zero-mean ...
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Stochastic vs Adversarial Multi-Armed Bandit Problems

I know that the multi-armed bandit can be formalised in multiple ways - two of them being the stochastic and adversarial ways. I am familiar with the fact that adversarial way is a game theoretic ...
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Optimal scaling of the Random Walk Metroplis-Hastings algorithm and the speed measure of the limiting diffusion

Let $d\in\mathbb N$ with $d>1$ $\ell>0$ $\sigma_d^2:=\frac{\ell^2}{d-1}$ $f\in C^2(\mathbb R)$ be positive with $$\int f(x)\:{\rm d}x=1$$ and $g:=\ln f$ $Q_d$ be a Markov kernel on $(\mathbb R^...
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Residence times of the telegraph process ?

The telegraph process is a two state stochastic process defined by the master equation $$ \dot{\pi}_0(t) = \tau^{-1} \pi_1(t) - \sigma^{-1} \pi_0(t) $$ $$ \dot{\pi}_1(t) = \sigma^{-1} \pi_0(t) - \...
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Formulating a Transition matrix for Markov Process

I am dealing with a medical process which is as follows. There are 10000 Veterans who are enrolled in this study. All 10000 have medical condition called onychocryptosis which is a fancy term for ...
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Generate Gaussian process with squared exponential covariance function

In a (stationary) Gaussian Process, values which are closeby are more similar than values far away from each other. The correlation function tends to zero as distance increases. Often, one models the ...
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Distribution of Conditional Brownian Motion

Let $\ X(t),t \ge 0$ be a Brownian motion process. That is, $\ X(t)$ is a process with independent increments such that: $$\ X(t) - X(s) \sim N(0,t-s), 0\le s \lt t $$ and $\ X(0)= 0$. ...
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Justification of acceptance probability in simulated annealing

In simulated annealing the acceptance probability for a new state in step $k$ is traditionally defined as $$ P(\text{accept new})= \begin{cases} \exp(-\frac{\Delta}{T_k}), & \text{ if } \Delta \...
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Is standard Brownian motion (AKA a Wiener process) weakly or strictly stationary?

Question Let $B(t)$ be a standard Brownian motion (AKA a Wiener process). Is $B(t)$ weakly or strictly stationary, particularly as defined here? My Thoughts We know, by definition, that its ...
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What are some higher level statistical methods used to measure the impacts of an advertising campaign (handling seasonality, time trends, etc.)?

I am trying to help my firm utilize better metrics for future growth. What variables are needed for the stochastic frontier production functions to measure if a firm is minimizing costs or maximizing ...
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What is the difference betwen a time non-homogenous Markov Chain and a non-linear Markov Chain? Example

A time non-homogenous Markov Chain is one in which the transition probabilities are not constant over time. A non-linear Markov Chain is a model that is not linear in parameters and satisfies the ...
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Law of Large Numbers for Covariance Stationary Processes… Difference and Relationship between LLN and Ergodicity

We have a covariance stationary time series. We must assume that the time series was produced by an ergodic process if we are to make the bridge between the realization of the time series that we ...
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Spatial regression: random and fixed effects [closed]

I'm working with spatial data (two rasters or matrix in the attached Figure), that is distributed in a 2D-space and each grid has a value. The two grids have the same number of cells. Variable "Y" is ...
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Is it better to have 1 washer room in an apartment with 20 washers or 10 floors with 2 washers each in a dorm?

The housing department of the university I study at gave a presentation on a new dorm being constructed slated for 2022. Being in the preliminary phases, they are currently designing how this building ...
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1answer
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Regression of Stationary Time Series in Non-Stationary Time-Series

Let's suppose that I have a time series $Y_t$ with dimensions $T \times 1$ with monthly frequency, and a matrix of external variables $\boldsymbol{X_t}$ of dimensions $T \times p$ where $p$ also ...
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Finite irreducible Markov Chain are recurrent

Question: Show that all state in a finite irreducible Markov chain are recurrent. Attempt: First I considered that a finite irreducible Markov chain is transient. since there are only a finite ...
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expected time to enter a state in birth-death process

I have a question regrading the expected time of entering a state in a birth-death process. Specifically I don't quite understand Page 378 Equation 6.3 of the book here. It is about birth-death ...
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Autocorrelation of a stochastic process

What does it mean to compute the autocorrelation between two different time istances in a time series? I don't well understand the concept, which is the sense of measuring between two time istances of ...
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Nature of a stochastic process [closed]

I am a econometrics student, and I've to understand what type of process the one in the image below is. I am not able to go back to the main classic stochastic processes (AR, MA, ARMA, ARCH/GARCH). In ...
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question regarding a stochastic process

Say let $Y_1, Y_2,...$ be a family of i.i.d random variables with a common $\mu$ and common variance $\sigma^2$. Let $N_t$ be a Poisson process with rate $\lambda >0 $. Assume that $N_t$ is ...
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How to solve / fit a geometric brownian motion process in Python?

For example, the below code simulates Geometric Brownian Motion (GBM) process, which satisfies the following stochastic differential equation: The code is a condensed version of the code in this ...
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Determining whether a model with random walk errors is stationary

If we have a model like an AR(1) except the errors are a random walk (i.e. not iid), then is the model itself stationary? So the model is: $$ x_t=kx_{t-1}+\epsilon_t $$ where $k$ is constant and $0<...