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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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Optimal Timing for Structural Break Analysis in Price Transmission Modeling

I am investigating the price transmission between two inflation indices within the same sector but at different stages of processing. For example, I might be looking at the relationship between raw ...
André Goulart's user avatar
2 votes
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42 views

Integral of stochastic processes

Suppose that i have a random variable $I(t) = \int_0^{t} N(s) e^{\sigma W(s)} ds$ where $N(s)$ is the number of arrivals at the time s ( notice that is not the total arrivals until time s, just the ...
Bruno Llacer trotti's user avatar
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Gambler ruin's: Probability of k consecutive win before j consecutive loss

Assume that a stock has a probability of $p$ to win, a probability of $q$ to lose, and a probability of $(1-p-q)$ to remain every day. What is the probability of $k$ consecutive wins occur before $j$ ...
Zhihao Xu's user avatar
1 vote
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14 views

What do we call a Poisson point process with an instantaneous log-rate being a Wiener process?

I have implemented a stochastic process for simulating demand of service that wanders in its average rate. This is a useful scenario for evaluating a controller that tries to optimize availability and ...
Galen's user avatar
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2 votes
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How to tune the unadjusted Langevin algorithm?

I want to start investigating the (unadjusted) simulation of the Langevin process $${\rm d}X_t=b(X_t){\rm d}t+\sigma{\rm d}W_t,$$ where $$b:=\frac{\sigma^2}2\nabla\ln p.$$ I don't want to simulate ...
0xbadf00d's user avatar
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How to guess a random walk to achieve max sample correlation?

Define this 1-D discrete random walk start from 0: roll a die (the die may or may not be fair, the fixed probability of each face is unknown to the observer/guessor)...
cat's user avatar
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2 votes
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Mathematical Introduction to Theory of Time Series Analysis

Assume that the reader has strong background in stochastic calculus (including and beyond continuous time stochastic processes like martingales and Markov chains and others, the construction of Levy ...
2 votes
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23 views

Autocorrelation of the lognormal Black-Scholes process

The Black-Scholes model with constant volatility $\sigma$ and interest rate $r$ is defined as $$ dS_t/S_t=rdt+\sigma dW_t $$ I derived the autocorrelation of the spot process $S_t$ for future times $0&...
Andras Vanyolos's user avatar
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27 views

Transformed Ornstein Uhlenbeck process

Say I have 𝑋 that follows an Ornstein-Uhlenbeck process: $𝑑𝑋_𝑑=πœ™X_t𝑑𝑑+πœŽπ‘‘π‘Š_𝑑$ Let $π‘Œ_𝑑=exp(𝑋_𝑑)$. How can I calculate the autocorrelation function of $Y_t$?
Isi's user avatar
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How to show that a stable discrete stochastic process converges to a stationary process?

So I have a discrete stochastic process defined by $x_{k+1}=Ax_k+Bw_k$ where $w_k$ is zero mean Gaussian white noise with covariance $R_w$, and where $A$ has its eigenvalues in the unit disk. I can ...
Minecraft dirt block's user avatar
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What is the most common notation to describe cyclostationary stochastic processes?

Can a cyclostationary stochastic process be described as: $\{X_t[i]:i\in\mathbb{N}\}\quad\forall~t\in\{1,2, \dots,T\}$ where variability at the $t^\text{th}$ index across all repetitions/seasons ...
joaocandre's user avatar
1 vote
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Is it possible to describe repeating data patterns as a stochastic process?

Generally, can repetitive patterns in sensor readings (e.g. temperature measurements at different locations over time) be seen as some kind of stochastic process? That is, if similar patterns repeat ...
joaocandre's user avatar
3 votes
1 answer
143 views

Can MCMC sample any probability distributions?

I have three fundamental questions related to MCMC. I would appreciate the help on any one of those. The most fundamental question in MCMC field, which I can't find a reference, is: Can MCMC generate ...
George Lu's user avatar
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23 views

Gaussian Process with RBF kernel when $𝑋_0=𝑋_1=0$ deterministically

I originally have a Gaussian process over $[0,1]$ with mean function $m(x)$ and covariance function being the RBF kernel. However, I actually know that $𝑋_0$ and $𝑋_1$ are $0$ deterministically. Is ...
George Lu's user avatar
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1 answer
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Expected number of failures in the next period T

I'm trying to solve a specific problem and after much search I could not figure it out by myself. I have a series of observations of failure times of some machines t1, t2, t3... i.e. the total life ...
abinos's user avatar
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7 votes
1 answer
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Is there any physical process producing a logistic distribution?

In his Logit Models, J.S. Cramer writes the following (p. 23) Are there no physical processes producing a logistic distribution?
Durden's user avatar
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4 votes
1 answer
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In what ways is Gaussian Process Regression both parametric and non-parametric?

Gaussian Process Regression is considered a "non-parametric" model. However, the term "non-parametric" is often used imprecisely to mean different things, leading to questions ...
socialscientist's user avatar
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11 views

What is the spectral density matrix, in the context of vector stochastic processes

I cannot seem to find any good/easy to read resources on the spectral theory, and in particular for multivariate stochastic processes. I want to know: Any resources explaining spectral theory ...
Dylan Dijk's user avatar
1 vote
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30 views

Extended Hidden Markov Models (HMM) parameter estimation

For simpler HMMs, we can use algorithms like Viterbi training (not decoding) or Baum Welch to estimate the parameters that best describe the observed data. How do we do the same when using a more ...
AlexS123's user avatar
5 votes
0 answers
119 views

Limiting distribution of the Wishart process

Consider the Wishart process: $$ dS_t = \sqrt{S_t} \, dB_t Q + Q^\top \, dB^\top_t \sqrt{S_t} + (S_t K + K^\top S_t + \Omega \Omega^\top) \, dt $$ Or the restricted version where $\Omega = \sqrt{\...
Arthur B.'s user avatar
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1 vote
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25 views

Ergodicity-definition for general statistic

I'm struggling with the definition of ergodicity within time series. Consider a time series denoted as $X = (X_i)_{i\in\mathbb{Z}}$, where each $X_i$ represents a random vector defined on the same ...
Albert Paradek's user avatar
-1 votes
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33 views

Help with Gambler's ruin problem, can't solve abstraction [duplicate]

I'm having difficulty solving this exercise. When I assume that p=0.4 and player A's fortune is 99 dollars and B's fortune is 1 dollar, I can find that the probability of player A losing to player B ...
Insomnia's user avatar
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0 answers
34 views

Likelihood of polynomial time series

I want to model a time series process as follows: There are a total of T periods. I want to model audience dynamics. There is an initial audience of $N_0$ that evolves over minutes indexed by $t$. ...
Luis's user avatar
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1 vote
0 answers
28 views

Learning from urn draws when the urn can be replaced

Suppose there is an urn consisting of black and white balls. This urn is filled with a proportion $a$ of black balls, where $a\sim U[0,1]$. Balls are consecutively drawn from that urn (with ...
snowtape's user avatar
1 vote
0 answers
22 views

Connection between Cox-PH Regression and Martingales

I am having trouble linking these concepts together: Here is a Cox PH (Proportional Hazards) Regression Model: $$h(t|X) = h_0(t) \exp(\beta^T X)$$ where: $h(t|X)$ is the hazard function for an ...
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Invariant event defined in terms of stationary stochastic sequence

In "Almost Sure Convergence" by Stout, there is indicated that the concept of invariant event (and further, the concept of ergodicity) can be defined in terms of given stationary stochastic ...
Mentossinho's user avatar
3 votes
1 answer
62 views

Drawing dots problem

Suppose a person has a $0.2$ chance of drawing a red dot, a $0.4$ chance of drawing a blue dot, and a $0.4$ chance of drawing a green dot. If they draw a blue dot, they get a second chance to draw ...
Si Chen's user avatar
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8 votes
4 answers
587 views

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 ...
Uk rain troll'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 ...
Starlord22's user avatar
2 votes
1 answer
102 views

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
1 vote
0 answers
27 views

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 ...
MymanPJ's user avatar
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0 answers
35 views

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 ...
qqhgsjah8221's user avatar
0 votes
1 answer
23 views

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 + \...
user346624's user avatar
1 vote
0 answers
52 views

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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0 answers
46 views

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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1 vote
1 answer
65 views

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: $$t_0= 0 < t_1,t_2,...,t_{nβˆ’1},t_n=T$$ And may be ...
Adrian Daniliuc's user avatar
1 vote
0 answers
26 views

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 (...
Snowy Baboon's user avatar
1 vote
0 answers
11 views

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 ...
user405518's user avatar
0 votes
0 answers
24 views

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 ...
Carl's user avatar
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0 answers
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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
3 votes
1 answer
62 views

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
2 votes
1 answer
122 views

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
0 answers
181 views

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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0 answers
24 views

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
161 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,$...
Fredrik P's user avatar
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1 vote
0 answers
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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:\, ...
user553010's user avatar
2 votes
0 answers
52 views

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
0 answers
39 views

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
1 vote
0 answers
28 views

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
256 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

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