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 Survival Models model the time at which a random variable will first pass a certain point?

Using standard survival models (e.g. Joint Survival Models), I could calculate the hazard and survival functions for individual cohorts at different time points in the future. Thus, I could make the ...
firstpassage's user avatar
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Estimating the parameters when only characteristic function is known

Recently I was working with a process named Variance Gamma with Stochastic Arrival (VGSA) and trying to fit this process on a given data. To obtain VGSA, as explained in Carr et al. [2001], we take ...
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1 answer
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Toad Movement Model Example

This model of toad movement is presented in https://www.sciencedirect.com/science/article/pii/S030438001630850X. I'm confused at to how the probabilities are set out and was wondering why if P(return) ...
Christian Angelopoulos's user avatar
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Independence of 2D gaussian process derivatives

Suppose I have a gaussian process which takes 2D inputs x and y and gives a 1D output z. I understand based on Calculating the expression for the derivative of a Gaussian process that each of the ...
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A recurrent Markov Chain implies its k-step version is also recurrent?

I am curious about whether a Markov Chain $X_n$ is recurrent implies that for any $k > 0$, $X_{kn}$ is also recurrent. Here are my observations. If $X_n$ is transient, $X_{kn}$ must be transient by ...
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Independence of Shocks in ARCH(1): A Doubt from Hayashi’s Book

I am reading Hayashi's Econometrics book, and on pages 104 and 105 he defines the ARCH(1) model for a time series $g_i$ as: \begin{aligned} g_i &= \sqrt{h_i} \varepsilon_i, \\ h_i &= \zeta + \...
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When is a function of an ergodic stationary process itself ergodic stationary?

I am working with a function which has the form $f(X_1, \dots, X_n)$, where $\\{X_n\\}$ is an ergodic stationary process. Theorem 5.6 in "A first course in stochastic processes" by Karlin &...
Kristan's user avatar
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Is there any reference about Ergodic Theorems applicable to stochastic processes with strong dependence?

Consider the stochastic process $(X_{n})_{n\in\mathbb{N}} = (A^{+}_{n},A^{-}_{n})_{n\in\mathbb{N}}$ defined over the same probability space $(\Omega,\mathcal{B},\mathbb{P})$ such that the occurrence ...
user1234's user avatar
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Two random variables X1 and X2 may be partially dependent i.e. X1 is independent of X2 but X2 is dependent on X1?

$X(t)$ is a stochastic process defined on the time interval $(0,T)$. Discretizing the time interval one can specify a random variable $X(t_i)$ as: $$t_0= 0 < t_1,t_2,...,t_{n−1},t_n=T$$ And may be ...
Adrian Daniliuc's user avatar
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Integral Over functions Differential Entropy

Suppose there is some function: \begin{equation} f(t) = p(x) \end{equation} Where $p(x)$ is a PDF over $x$ at $t$. Some examples would be linear regression with error bounds or a Gaussian Process (...
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Stochastic Task Processing Times in Queueing Theory

I'm struggling with an operations research problem which has 3 stations containing 3 different task processing times and different coefficients of variation (for example, station 1 has 3 tasks with ...
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Is the following conditional density function equivalent to its unconditional counterpart? [duplicate]

Suppose we have a stochastic series $\{X_t\in\mathbb{R}, t=1,\cdots, T\}$. Further suppose that $G(X_t)=\mathbf{1}_{X_t\geq 0}$ where $\mathbf{1}$ is an indicator function. Can it be concluded that ...
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Estimating jump diffusion parameters from first passage time data?

I know that there is a literature on approximating the first passage time distribution of jump diffusion processes. I know there is also a literature on estimating parameters of jump diffusion ...
ThinkConnect's user avatar
2 votes
1 answer
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Wold's decomposition theorem for stationary processes

The posts How come the deterministic part of Wold decomposition does not violate stationarity? More about the deterministic part of Wold decomposition express some concerns as regards the "...
Alecos Papadopoulos's user avatar
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1 answer
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More about the deterministic part of Wold decomposition

This is a follow-up on this question of mine. Wold's representation theorem states that every covariance-stationary time series $\{Y_t\}$ can be written as the sum of two time series, one ...
Richard Hardy's user avatar
1 vote
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How do Incorrect Statistical Assumptions affect Estimation? [closed]

As a learning example, I am trying to see how adversely statistical analysis is impacted when the distribution of errors is incorrectly specified. Here are some specific situations I thought about: ...
Uk rain troll's user avatar
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Why is mean of innovations restricted to zero in definition of VAR process?

The VAR process is defined as: $$\begin{align} \mathbf{y}_t = A_1\mathbf{y}_{t-1} + \dots + A_d\mathbf{y}_{t-d} + \boldsymbol{\epsilon}_t, \quad t \in \mathbb{Z} \end{align}$$ where $\boldsymbol{\...
Dylan Dijk's user avatar
2 votes
1 answer
159 views

Compute share moving between deciles of a stationary AR(1) process

I want to compute the probability $P_{ij}$ to move from decile $i$ in one period to decile $j$ in the next period in the distribution of a stationary AR(1) process $$Y_t = \rho Y_{t - 1} + \upsilon_t,$...
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Question about the mean first passage time

A homogeneous Markov chain $\{X_n\}_{n\in\mathbb N}$ with discrete state space $\mathcal{S}$. Consider the minimum number of steps to visit $k\in \mathcal{S},$ $$\tau_{k}:=\text{min} \left\{n\ge 1:\, ...
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Determine if Poisson process rate changes

Imagine a person, P, calls into a call center multiple times an hour. With 50% probability, P will call with some low rate (like an average of ~3 calls an hour) the whole time. With 50% probability, P ...
KHAAAAAAAAN's user avatar
1 vote
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An example of a random variable $y\in L^\dagger_2$ having more than one linear combination, $y = \Sigma_{i}\alpha_i x_i = \Sigma_{i}\beta_i x_i$

In the answer for the following exercise: Let $\{x_1,...,x_n\}$ be a finite collection of random variables with $E(x_i^2) \lt \infty$ ($i = 1,..., n$). Show that the set of all linear combinations $\...
Tran Khanh's user avatar
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Difference in Gamma Distributions have Poisson? [duplicate]

Today I learned about a Double Stochastic Process for the first time. Apparently a Cox Process is a Double Stochastic Process. Here is my attempt to summarise this: Cox Process: A point process (I ...
Uk rain troll's user avatar
3 votes
1 answer
203 views

Poisson Distribution with Evolving Lambda Parameters?

Consecutive differences in Poisson arrivals have an Exponential Distribution. In modelling this kind of data, I have usually seen the arrival rate (lambda) held as constant. Sometimes I have seen Non-...
Uk rain troll's user avatar
1 vote
0 answers
42 views

Expected arrival time for the first person in the decomposition of poisson process

Question: Customers arrive at a shop according to a Poisson process with rate $\lambda > 0$ per hour, with probability $p$ being male and $1-p$ being female. During the first hour $n$ people ...
Frank Lee's user avatar
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Convergence of Truncated and Subsequenced MA($\infty$) Processes with Square Summable Coefficients

Let $X_t=\sum_{j=0}^{\infty} \phi_j \varepsilon_{t-j}$ be an MA($\infty$) process with square summable coefficients. We know that if we truncate the process at $n$, creating the sequence: $$X_{t,n} = \...
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Waiting/jumping times of non homogeneous process on a state chain (almost Markovian)

When considering a chain of states from 0 to N in continuous time with constant up and down going transition rates, the jumping or waiting times are exponentially distributed. Now consider the ...
Nicouh's user avatar
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What is the correct definition of the Cesàro summation of autocovariances?

I'm a little confused regarding a mathematical definition (Cesàro summation) and its application to stationary time series. First, consider the definition given by Wikipedia, adapting to autovariances....
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Efficiency terms in Stochastic Frontier Analysis can be greater than 1?

Let us consider a Cobb-Douglas production function with $Y$ being the output and $X$ being the input (assume for simplicity only one input) and a composite error term: $$ Y = e^{\beta_0}X^{\beta_1}e^{...
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Coupling from the past for continuous state space

Coupling from the past (CFTP) is a Markov chain Monte Carlo method for sampling exactly from a distribution. Briefly, in its vanilla form, CFTP simultaneously simulates a Markov chain starting from ...
fool's user avatar
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A Pólya urn random walk with step probability $p$ and an absorbing state. Is there for this random walk a $p>0.5$ where absorption is almost certain?

This question relates to: How to pick the winner in the "Play the Winner" treatment assignment scheme (Urn model) which is like a Pólya urn model where the additions of balls into the urn ...
Sextus Empiricus's user avatar
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1 answer
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Sufficient conditions for second-moment ergodicity

Let $(Y_t)_{t \in Z}$ be a covariance-stationary stochastic process. According to Hamilton (page 46-47), we say that the process is Ergodic for the mean if $$\overline{Y}\equiv \frac{1}{T}\sum_{t=1}^...
user346624's user avatar
3 votes
1 answer
152 views

Calculating the cumulative distribution function and the probability density function of an interval with ratio of a shorter and longer segment

The interval $[0, 2]$ is divided into two parts by randomly marking a point in $[0, 1]$ according to the rectangular distribution. Let $X$ be the length ratio $L_1/L_2$ of the shorter segment $L_1$ to ...
Ste0l's user avatar
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3 votes
1 answer
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Definition of recurrent stochastic process, in general

This interesting question: Recurrence definition for a Markov chain gives the definition of a recurrent state for some discrete process. I was wondering, in the case of a continuous (time) process, ...
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Remove non repeatable stochastic noise from an image

Assume that you have these coordinates inside an image. The algorithm for creating these crosses comes from FAST-algorithm for corner detection. But the problem with FAST-algorithm is that some of ...
euraad's user avatar
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2 votes
2 answers
176 views

How to understand the definition of Markov Chain $P(X_{n+1}\in B\mid \mathcal{F}_n)=p(X_n,B)$?

The definition of Markov Chain in Durrett (Probability: Theory and Examples, 2019, Section 5.2) is: $$P(X_{n+1}\in B\mid \mathcal{F}_n)=p(X_n,B), $$ where $p$ is the Markov transition kernel ...
dodo's user avatar
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1 answer
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Confused while studying stationarity and autocovariance

I started to study Time Series Analysis and have stumbled on a roadblock. When introducing the autocovariance function, the instructor mentions that we assume stationarity in the data that we are ...
insipidintegrator's user avatar
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For a stochastic system, should we optimize its time average cost or total cost? What is the difference between the two goals?

For a stochastic system, the time average cost like: $ \lim_{T \to \infty}\frac{1}{T}\sum_{t=0}^{T-1}\mathbb{E}\{cost(t)\}$ and total cost like: $ \lim_{T \to \infty}\sum_{t=0}^{T-1}\mathbb{E}\{cost(t)...
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Question about mean hitting times

A homogeneous Markov chain $\{X_n\}_{n\in\mathbb N}$ with discrete state space $\mathcal{S}$. Set $$\tau_{k}:=\text{inf} \left\{n\ge 0:\, X_n=k \right\}.$$ where $\tau_{k}$ is defined to be $+\infty$,...
Kevin's user avatar
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2 votes
1 answer
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A question related to the convergence of mathematical expectations restricted to an Interval centered on zero

Let $(X_j)_{j= \mathbb 0}^\infty$ a fixed realization of strictly stationary AR(1) process: $$X_j = 0.9 \,X_{j-1}+ \eta_{j}, \quad (\eta_j) \overset{iid}{\sim} N(0,1)$$ For each $n$, consider $B_n\sim ...
André Goulart's user avatar
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Expected value of a variable that depends on two other random variables, and one of these random variables depends on another random variable

I am trying to solve the problem using conditional expectations. The expected value H is depends on the waiting time T and a set threshold X (a real number that is a constant random variable during ...
ton_K's user avatar
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0 answers
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What do I have to show still to prove that $x_t$ is stationary and ergodic?

Let $x_t=\varphi x_{t−1}+\varepsilon_t$ be the model with errors being white noise. If model is correctly specified and $|\varphi|<1$, why is $x_t$ not stationary and ergodic? Why cannot I use ...
MSKO's user avatar
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ARIMAX with Exogenous Variable modeled using Ornstein-Uhlenbeck Process

I have a question with regard to building ARIMA with Exogenous Variables (ARIMAX) models where one of the Exogenous Variables is to be modeled using the Ornstein-Uhlenbeck process. I recently read an ...
PeterOrnstein's user avatar
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Qustion about transient states (Continuation)

Let $$f_{kk}^{(n)}:=\Pr(X_n=k,X_v\ne k,1\le v\le n-1\mid X_0=k),~n\in \mathbb Z^+ .$$ Attributing to the comments of @Zhanxiong , I have added the other two cases to the Case 1. Case 1. Is there a ...
Kevin's user avatar
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1 answer
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Question about transient states

Is there a discrete time-homogeneous Markov chain $(X_n)_{n \geq 0}$ in which one transient state $i$ satisfies $\sum_{n=1}^{\infty}nf^{(n)}_{ii}<\infty ?$ where $$f_{ii}^{(n)}:=\Pr(X_n=i,X_v\ne i,...
Kevin's user avatar
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0 votes
2 answers
170 views

How are Markov chains memoryless when they have memory of size 1?

Markov chains are typically described as memoryless as the next state depends only on the current state but not any of the past states. But wouldn't true memorylessness mean that the next state does ...
csstudent1418's user avatar
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What is the distribution of completed parts by a given machine after t minutes in SimPy's Machine Shop example?

Question You can read the Machine Shop example in the SimPy documentation, however I have tried to put it into its mathematical description below so that reading Python is not necessary. Suppose a ...
Galen's user avatar
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1 vote
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$\mathbf{P}(S_{n(i)}=i \mid S_1=i),\mathbf{P}(S_{n(i)+1}=i\mid S_1=i),......$ are zeros?

$\left\{\xi_{n}\right\}_{n\in\ \mathbb{N}_{+}}$ is a sequence of independently and identically distributed random variables, each taking a finite number of integer values.$\mathbf{E}(\xi_1)\ne 0,$ For ...
Kevin's user avatar
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1 vote
0 answers
137 views

Monte Carlo simulation vs. Discrete event simulation

I'm trying to understand the difference between a Monte Carlo simulation vs. a Discrete event simulation. I learned from googling( for eg.: https://bookdown.org/manuele_leonelli/SimBook/types-of-...
user2450223's user avatar
1 vote
1 answer
99 views

How come the deterministic part of Wold decomposition does not violate stationarity?

Wold's representation theorem states that every covariance-stationary time series $\{Y_t\}$ can be written as the sum of two time series, one deterministic and one stochastic: $$ Y_t=\sum_{j=0}^\infty ...
Richard Hardy's user avatar
4 votes
1 answer
63 views

Notation for an ordered pair of stochastic processes

There are two stochastic processes, $\{ Y_{1,t} \}$ and $\{ Y_{2,t}\}$. If I take them as an ordered pair, what notation do I use: $\{(Y_{1,t},Y_{2,t})\}$, $\{(Y_1,Y_2)_t\}$, $(\{Y_{1,t}\},\{Y_{2,t}\}...
Richard Hardy's user avatar

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