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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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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Moving Average with non-increasing coefficients

Suppose I have a Moving Average $MA(q)$. $$X_t = \sum_{j=0}^q \psi_j \epsilon_{t-j}, \quad (\epsilon_t)_{t\in \mathbb{Z}}\,\,\ i.i.d.$$ I know that there is no imposition regarding the monoticity of ...
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How to simulate non-gaussian stochastic paths

(Edited to be clearer) I am trying to replicate simulating Geometric Brownian Motion (GBM) but instead of the stochastic increment following a normal distribution, I would like it to follow a ...
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Variance of random walks in time series analysis [duplicate]

“For a random walk stochastic process, the variance is infinite.” Do you agree? Why?
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About ergodic stochastic process

$x_t=2.1+0.73x_{t-1}+ε_t$ $ε_t \sim iid(0,σ_ε^2)$ Given the stochastic process with deep Gaussian process above, is x_t an ergodic stochastic process? If possible, I would like to know the reason.
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The gamble problem to find limiting stationary distribution

[This question was asked here before, but I did not get satisfactory response] I was trying to solve the following problem: I find out the transition matrix is: $$P = \begin{bmatrix} 1 & 0 &0 ...
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Time series that are correlated in levels but not first differences

Is it correct to say that the two time series are correlated in levels, but not first differences? $$x_{t}=\phi t+u_{t}\\ y_{t}=\rho t+\epsilon_{t}$$ where $\epsilon_{t}$ and $u_{t}$ are mutually ...
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How does covariance stationarity even exist?

I've been wondering recently about covariance stationarity. Say we have a stationary series with statsmodels' ADF and KPSS results: ...
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Predicting $x_t$ knowing something about $\Delta^2 x_t$

I have this exercise question, what is your prediction of $x_{10}$ knowing that $\Delta^2 x_t = \epsilon_t $, knowing $x_9 = 1.56$ and $x_8 = 1.64$. I take this to mean that $$ x_t = x_{t-2} +\...
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Alternative to SimPy for continuous event simulation?

The python library, SimPy, is pretty explicit that it only handles discrete event simulation. Though it is theoretically possible to do continuous simulations with SimPy, it has no features that help ...
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Numerically solving the Forward equation to estimate SDEs

In books [1] dealing with inference for SDEs, why is the approach of numerically solving the forward PDE to obtain numerical estimate of the PTD not studied? One could then use this PTD to perform a ...
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Stationarity and ergodicity of a process conditional on a finite trajectory

Let us say we are interested in a single time series, e.g. the daily closing share price of Tesla. We can model it as a realization of a stochastic process $\{Y_t(\omega)\}$. It corresponds to a ...
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First order PDF of a stochastic process

I've started studying about stochastic processes and I need some help in this question. A random number generator is making numbers by this process: First number (X0) is a sample from Normal Standard ...
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Stochastic modelling, distribution and ergodicity of a particular time series with a given finite history

Let $\Omega$ be a sample space. A stochastic process $\{Y_t\}$ is a function of both time $t \in \{1, 2, 3, \ldots\}$ and outcome $\omega \in \Omega$. For any time $t$, $Y_t$ is a random variable (i....
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Question about autocorrelation of time series

I have a time series $S(t)$ for $t\in I $ where $I=[0,t_\text{max}]$ is an interval. The time series is very regular, $S(t) = \sin(t)$, although my following question is independent of the explicit ...
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What is wrong with using t-SNE for predictions?

If I have a dataset with hundreds of samples and thousands of features, and t-SNE does a good job of separating classes compared to others classifiers, I don't understand why I can't rerun the ...
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How to change some parameters of a process to make it WSS?

Consider the R.P., $X(t)=Au(t-T)$ where, $$A \sim N(\mu,\sigma^2)$$ $$T \sim Exp(\lambda)$$ and $u(t)$ is the unit step function. If $A$ and $T$ are independent we'll have, $$E(X(t))=\mu(1-\exp\{-\...
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Application of Time reversibility property of stochastic processes

Are there any theoretical or applied consequences of a stochastic process being time reversible? I know a Markov chain being time-reversible implies the existence of a stationary distribution. But in ...
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Proving the expectation of a variable in a stochastic process

Problem Information packets arrive at a server with a poisson process having rate $\lambda = 2$ per hour. The server processing time for a packet follows the distribution : $f(x) = 1, 0\leq x\leq1$ ...
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Plim of time average of a stochastic process, across independent realisations

I have a stochastic process $\{X_t\}_{t\in\mathbb{z}}$. Further, I have $N$ independent realisations of this process. What guarantees that $plim_{T\rightarrow\infty} \frac{1}{T}\sum_{t=1}^{\infty}X_{t}...
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Some help on this question about stochastic epidemic models without removals would be greatly appreciated

I get this question up until the underlined part, would someone be able to explain the rest of it to me as its had me stumped for a little while now.
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Is this statement of a stationary density function correct?

I'm planning to use a discrete-time stochastic process defined in the following paper: Nicolau, J. (2002). Stationary Processes That Look Like Random Walks—The Bounded Random Walk Process in Discrete ...
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If $(X_t)$ is a Gaussian process, what is $Cov(X_s^2,X_t^2)$?

Let $(X_t)_{t \in \mathbb R^+}$ be a Gaussian process $\mathcal N(\mu(t), \sigma^2(t))$ with covariance function $K$. Let $s \leq t$. Can we express $Cov(X_s^2,X_t^2)$ in terms of $K, \mu(t), \sigma^2(...
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How do you consider the negative states i.e. ...-3,-2,-1, when solving for the stationary distribution of this MC?

I have shown that the MC is aperiodic and irreducible, if that helps.
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How can one find a system of SDE's from a probability density function?

Suppose I have a joint distribution function say $p(x,y,z)=f_{X, Y, Z}(x,y,z)$. Is it possible to find a system of stochastic differential equations or a single stochastic differential equation from ...
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Sample path boundedness of Gaussian Processes

Consider a Gaussian Process $GP(0,k(x,x'))$ with zero mean and bounded, continues covariance function $k(x,x')<c,\quad \forall x,x\in\mathbb{R}^n$. Are the sample paths of this process (almost ...
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Show that two random vectors have the same distribution

I'm unsure of a result that looks simple, but I want to make sure I'm not getting it wrong. Let $(\epsilon_t, t \in \mathbb{Z})$ be a process such that $\epsilon_t$ i.i.d.. I want to show that for any ...
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Renewal process: Stationary and independent increments?

For a renewal process, we know that the inter-arrivals are independent but not exponentially distributed, as opposed to the Poisson process for which the inter-arrivals are exponential. We also know ...
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How can I generate probabilistic forecasts to do probabilistic classification?

I have a collection of univariate, irregularly spaced, financial time series. Each series is labeled by its class. The image below shows some example data. A note on the data: The time series could ...
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Meaning of "$\stackrel{p}\longrightarrow$" in math notation (arrow with a p over it)

I have a problem with the concept of a symbol (an arrow with a P over it). Can anyone clarify this for me? For example what the symbol means in this relationship: $$\large Y_t^{(n)} \stackrel{p}\...
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A nondeterministic covariance-stationary process approximated by an ARMA process

We know that the Wold Decomposition Theorem says that any purely nondeterministic covariance-stationary process, $x = [x_t : t \in \mathbb{Z}]$, can be written as a linear combination of lagged values ...
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Winning at tennis

What is your probability of winning a game of tennis, starting from the even score Deuce(40-40), if your probability of winning each point is 0.3 and your opponent's is 0.7? My answer: I think the ...
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Modeling Urns and Balls System as a Markov Chain

Suppose I have $q$ urns each of which hold up to $n$ distinguishable balls, but only $1$ of each type of ball (there being $n$ types of balls too). I would like to make any particular configuration of ...
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Show that two white noise definitions are equivalent

Given $(\varepsilon_t)\sim WN(0,\sigma^2)$ a white noise. By definition $$E(\varepsilon_t)=0,\,\, E(\varepsilon_t^2)=\sigma^2 \quad \forall t$$ and $$E(\varepsilon_t \varepsilon_s) = 0, \quad s\neq t$$...
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A conceptual question about the limitation of the MA processes

We know that linear time-series techniques are frequently used in macroeconometrics. The Wold Representation Theorem states that any covariance-stationary process may be expressed as an MA process ...
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Coefficients for higher dimensional NIPC (non intrusive polynomial chaos expansion) heavily biased towards 0 degree function

Currently, I am trying to apply the PCE to the thermal fin problem as solved using finite element. Although I cannot attach the code I use, suffice to say I solve a linear system $AU=F$ where $F$ is ...
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Simulating paths of stochastic process from density

I need yout help! I have a stochastic process $X_t$ and I know its density function $f(x,t)$, which is defined for $x>t$. I'm looking for a code in R that simulates the paths of the process, so I ...
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What would be a continuous-time version of a VAR process?

It is often said that a AR(1) process can be viewed as a discretized version of the continuous-time Ornstein-Uhlenbeck process. Can we really claim this to be valid considering that the Ornstein-...
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Purpose of negative indices in time series

In the book Time-series analysis by Hamilton we find the following passage: A time series is a collection of observations indexed by the date of each observation. Usually we have collected data ...
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Why is the difference between 2 time series drawn from the same process not White Noise?

I take the difference between 2 time series (each with 200,000 observations) drawn from the same ARMA(2,1) process and find that (at least the first 1000 observations of) this difference looks like ...
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Derivation of autocovariances of squared shocks

I have two shocks: $\varepsilon_{1t}$ has constant volatility $E[\varepsilon_{1t}^2]$ = $\sigma^2_1$ while $\varepsilon_{2t}$ has time varying volatility $E[\varepsilon_{2t}^2]$ = $\sigma^2_{2,t}$. I ...
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The sum of $n$ zero-mean values is of $O(\sqrt{n})$ at max

Why is the sum of $n$ zero-mean values is of $O(\sqrt{n})$ at max? If $X$ is a random variable with a standard deviation $\sigma$, the standard deviation of $n$ such i.i.d. random variables is $\sqrt{...
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White Random Process(White Noise) and Mean Value

Can someone give a definition of a white random process? I was trying to understand the penultimate paragraph of this answer Is it necessary for white noise to have zero mean which states that there ...
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Wait so probabilities of 0 or 1 CAN change? $P(A|B)=1$ does not imply $P(A)=1$ because $0 < P(A=B) < 1$?

Nassim Nicholas Taleb says here no probability that is 0 or 1 should ever change. Despite these 6 questions Does an unconditional probability of 1 or 0 imply a conditional probability of 1 or 0 if ...
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Can we conclude that Stochastic Gradient Descent is a "superior" algorithm that Gradient Descent?

On a very informal level, if we were to compare the (classical) Gradient Descent Algorithm to the Stochastic Gradient Descent Algorithm, the first thing that comes to mind is: Gradient Descent can be ...
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Does there exist a bounded measure for a certain stochastic process which follows a non-semigroup algebraic structure?

It is known that if we have a Markov process $(X_t)_{t \geq 0}$, then for all bounded measurable $f$ and for all $s,t \in \mathbb{R}$, there exists a bounded measure $P_sf$ such that $$ E[f(X_{t + s}) ...
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What is the expectation of a variable that has two cases (being either 0 or a constant) depending on a normal distributed variable

I need to determine the expectation of a variable: $\mathbb{E}(b(v))$ The value $b$ has two cases: it is either 0 when $v<0$ or a constant c when $v\geq 0$. The value $v$ has a normal distribution. ...
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Proof of Stochastic Gradient Descent

When it comes to the classical Gradient Descent Algorithm, for optimizing Convex Functions - there is a standard proof that demonstrates the convergence of this algorithm provided certain conditions ...
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Understanding the notation of a MA process in a paper

I am trying to understand a equation in a paper that I am reading. First, considere de $MA(\infty)$ process definition: \begin{equation} (I)\quad \quad \quad X_t = \sum_{j=0}^\infty \psi_j \...
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