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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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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2answers
44 views
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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1answer
19 views
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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0answers
22 views
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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1answer
23 views
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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10 views
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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1answer
73 views
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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49 views
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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54 views
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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0answers
19 views
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
33 views
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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10 views
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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0answers
35 views
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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1answer
27 views
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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0answers
40 views
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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29 views
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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15 views
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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2answers
31 views
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
$$
where $k$ is constant and $0<...
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0answers
12 views
Diffusion with Stochastic Resetting: How large must t be in order for the reset waiting time to follow an exp(r) distribution?
I've read Diffusion with Stochastic Resetting by Evans and Majumdar (2011): Link Here
Right before equation (3) on the second page, the article states
Consider the
particle at some fixed time $...
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0answers
10 views
Stochastic model with random number of spatial locations and translation with time?
I'm looking for a topic which I struggle to put into words. Its' a reasonable consideration which I expect has been carefully studied. I hope someone can tell me the name of it and offer some guidance ...
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1answer
76 views
Memoryless Property of a Markov Chain of Order 1. Is AR(1) memoryless or of infinite memory?
A stochastic process constitutes a discrete Markov Chain of order 1 if it has the memoryless property, in the sense that the probability that the chain will be in a particular state i, of a finite set ...
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0answers
106 views
Proving whether a series is stationary
I want to prove whether the following equation is stationary or not:
$$
x_t = (x_{t-1} + \epsilon_t) (1+k(x_{t-1}+\epsilon_{t})^2)^{-1/2}
$$
Also written like:
$$
x_t = (x_{t-1} + \epsilon_t) \frac{1}...
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0answers
12 views
Derivative of Time-Transformed Stochastic Process
Given a continuous time stochastic process X(t), we can define the functional transformation,
$$f(X)(t)=(X(t))^2−2X(t)$$
and evaluate the Hadamard derivative. Given a transformation on the real ...
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1answer
33 views
Are Gaussian Mixture Models stochastic or deterministic?
Each time we generate a gmm model, we obtain slightly different clusters. Can we hence say gmm is stochastic? We obtain the same clusters if a random seed is set; does this mean given a random seed, ...
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0answers
60 views
Multiple interval ratio of E[X / (X + Y)]
I have a sequence of interchanging on- and off-intervals, each pair identified by index $i$. The duration of the on-interval $i$ is represented by random variable $X_i$, and the duration of the off-...
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0answers
3 views
stochastic ordering of counting processes
Let $N_1(t)$ be a delayed renewal process and $N_2(t)$ be an ordinary renewal process such that $N_1(t)\geq_{st}N_2(t)$. Consider a renewal process $Z(t)$ with the same inter-arrivall distribution as $...
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2answers
36 views
How would you go about modelling a sport with a fixed maximum score line (e.g. volleyball or tennis?)
In short, I don't have a great maths background but have always mucked about with modelling sports and making predictions (as I enjoy it).
I have always used the Hal Stern "The Probability of Winning ...
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0answers
19 views
Bound for the bias of ergodic averages for non-stationary Markov chains
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_n)_{n\in\mathbb N_0}$ be a filtration on $(\Omega,\mathcal A)$
$(E,\mathcal E)$ be a measurable space
$X$ be a $(E,\...
4
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0answers
46 views
When can a Gaussian Process solve an SDE?
Considering an SDE of the form:
$$dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t$$
... (where $W_t$ is a Weiner process) is there a set of necessary and sufficient conditions on the structure of the ...
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0answers
10 views
Continuous-time Observation process in Filtering (why differential equation?)
Suppose we start with a stochastic dynamical system equation
$$ dX_t = a_t(X_t)dt + dW_t $$
and we want to augment it with an observation process (noisy measurements) to get a continuous-time ...
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0answers
12 views
Polynomial chaos expansion and ODEs
I am trying to figure out how to use PCE. My background is dynamics and analysis.
So if I understand correctly the main idea is that we have a random variable $X$, whose distribution we do not know, ...
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0answers
49 views
Is the autocovariance of a random walk with drift same as that of without drift?
A random walk without drift is not stationary. Because its autocovariance function depends on time. A random walk with drift is not stationary as its mean is not constant.
But what is the ...
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1answer
56 views
Probability of first time to an event
We have a stream of events over time. Suppose that $f_t$ is the probability density that an event happens at time $t$. For example, $f_t$ can be the probability density that any bus arrives at time $t$...
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0answers
51 views
Survival probability of a random walk with renewal timings
A random walker starting at time $t=0$ and location $x=0$ moves to the right ($x+1$) or the left ($x-1$). The $k^{\mathrm{th}}$ moves to the right and left occure at the times $\sum_{i=1}^{k} R_i$ and ...
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1answer
26 views
In a probability generating function, what exactly is the parameter of G(z)?
For instance, given $\DeclareMathOperator{\P}{\mathbb{P}}
\DeclareMathOperator{\E}{\mathbb{E}}
G(z) = \E z^X$, what exactly is $z$? and also what does the generating function actually give you? ...
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0answers
36 views
Definition of the integrated autocorrelation time
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\pi$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$
$(X_n)_{n\in\mathbb N}$ be a real-valued stationary stochastic ...
1
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0answers
16 views
Query about stochvol package in R
This is regarding the package stochvol in R.
See vignette
I am referring to section 3.1 of the above document. It says:-
The core sampling function svsample
expects its input data y to be ...
1
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1answer
40 views
What is the difference between a Random Vector (Joint r.v.) and a Random Process?
What is the difference between a Random Vector (Joint r.v.) and a Random Process?
Kindly, explain with a simple example (like toss of a coin, roll of a die, picking a card, etc.).
.
Note. As far ...
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0answers
14 views
Showing the 2nd Order Properties of 2 ARMA Processes are Identical
Given 2 processes
$$
Z_t = \epsilon_t + \theta\epsilon_{t-1}
$$
$$
Z_t' = \epsilon_t' + \theta^{-1}\epsilon_{t-1}'
$$
where
$$
\epsilon_t \overset{iid}{\sim}\mathcal{N}(0, \sigma^2)
$$
$$
\epsilon_t' ...
0
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0answers
46 views
Definition of Stochastic Process as Probability measure in a Prob. Space
I've found this question, with a very good answer, but they don't broach my question.
In Oksendal's Stochastic Differential Equations book, it's written «the stochastic Process is a probability ...
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1answer
20 views
Mean and Variance: first-order stochastic dominance
Suppose $X$ and $Y$ follow the same distribution, with same mean.
And $Var(X)<Var(Y)$. Then, does $X$ first-order stochastically dominate $Y$? Intuitively, I think this will hold. But I do not know ...
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0answers
12 views
First moments of GBM-like process with non-normal shocks
First consider a standard GBM process of the form,
$$\frac{dS_t}{S_t} = \mu dt+ \sigma dW_t$$
but instead of the normal $W_t \sim N(0,1)$ , instead we have that $W_t \sim EMG^-(0,1,\lambda)$. Where ...
3
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2answers
160 views
Conditional probability for consecutive Bernoulli trials
Independent trials, each of which is a success with probability $p$, are performed until there are $k$ consecutive successes. Let $N_k$ denote the number of necessary trials to obtain $k$ consecutive
...
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0answers
30 views
FDD of a stochastic process
For calculation of the FDD of an ou process, do I have to calculate all the pushforward measures? If its true can you please tell me the steps in those calculations?
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0answers
21 views
Can additional iterations of backward induction as described affect optimal policy?
Consider a game with the following properties:
Single player
Finite number of game states (after the player arrives at a terminal state, he or she can begin again from the start state; the player can ...
0
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0answers
36 views
Expectation of random sum of dependant variables
The expectation of random sum of independent identically distributed variables is given either by the law of total expectation or by Wald's identity. Are these generalised to tackle the random sum of ...
2
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0answers
37 views
Excursion areas and bridges for F- and chi(-square)- processes
My question relates to excursion areas (and excursion bridges, in particular) for $\chi$-, $\chi^2$-, gamma- and/or F-distributed stochastic processes (or more general random fields with positive ...
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0answers
4 views
What's the velocity and displacement of a free particle?
I'm reading Ornstein and Uhlenbeck (1930). They calculate the velocity of a free particle at time t given an initial velocity at time zero to be normally distributed.
They also calculate the ...
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0answers
17 views
More AR lags means more persistence?
Suppose that $x_t$ is an ARMA(p,q) stochastic variable and that $y_t$ is another stochastic process that satisfies
$$ y_t = \frac{1}{(1-\rho_1 L)\cdots(1-\rho_n L)} x_t, $$
where $L$ is the lag ...
0
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0answers
11 views
Name for a special kind of uniform law of large numbers for an empirical process
Let $x$ be a random vector over $\mathbb{R}^n$, for some $T \subseteq \mathbb R$ we have a function $g : \mathbb{R} ^n \times T \to \mathbb R$. Let $a_n$ be a positive sequence with zero limit and $...
