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.
0
votes
0answers
3 views
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 ...
0
votes
0answers
5 views
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?
0
votes
0answers
15 views
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 $...
2
votes
1answer
32 views
Combining probabilities to find most probable window
I have a series of observations, with an associated probability that an event is occuring at timestep t, something like:
[0.8, 0.8, 0.3, 0.9, 0.2]
Events can ...
1
vote
0answers
18 views
How do I write a random walk with drift ARIMA? [closed]
I modeled oil prices and got the following coeffecients for my arima model with drift.
Is this the right way of writing the model?
3
votes
0answers
16 views
Do measurable maps preserve stationary ergodicity?
In a recent effort to establish stationary ergodicity for a certain stochastic process, I just happened to come across a statement, which I find to be little bit confounding.
Given two measurable ...
0
votes
1answer
21 views
How to recognize an ARMA process?
By looking at the autocovariance, how could you recognise what discrete model (MA(q), AR(p), or ARMA(p,q)) is more appropriate to describe your data?
1
vote
0answers
19 views
Approximating AR(1) by finite order MA process - convergence results
I am currently struggling with a result pertaining to the finite order MA approximation of a simple AR$\,(\,1\,)$ process defined on a double sided time-index set $\,T=\mathbb{Z}$. I would be very ...
0
votes
0answers
12 views
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_{...
1
vote
0answers
14 views
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. ...
1
vote
1answer
38 views
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 =$\...
0
votes
1answer
46 views
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 ...
0
votes
0answers
15 views
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 ...
2
votes
0answers
28 views
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 ...
2
votes
1answer
38 views
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 ...
0
votes
0answers
12 views
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).
...
1
vote
0answers
49 views
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 ...
1
vote
1answer
81 views
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 ...
1
vote
0answers
40 views
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 ...
0
votes
0answers
51 views
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 ...
3
votes
1answer
90 views
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 ...
0
votes
0answers
21 views
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 ...
0
votes
0answers
49 views
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 ...
0
votes
0answers
30 views
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, "...
0
votes
1answer
22 views
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 ...
1
vote
2answers
135 views
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 ...
4
votes
1answer
63 views
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 ...
1
vote
0answers
14 views
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,
$$\begin{align}
\rho(X) &= \mathcal{L}^{-1} \big\{ \...
0
votes
0answers
33 views
Covariance non-zero mean AR(1)
Why when I compute the autocovariance function of a non-zero mean AR(1),
X(t)-u=Φ(X(t-1)-u)+ε the presence of the mean does not change my result and so the formula should be the same of a zero-mean ...
0
votes
1answer
42 views
Stochastic vs Adversarial Multi-Armed Bandit Problems
I know that the multi-armed bandit can be formalised in multiple ways - two of them being the stochastic and adversarial ways. I am familiar with the fact that adversarial way is a game theoretic ...
6
votes
1answer
103 views
Optimal scaling of the Random Walk Metroplis-Hastings algorithm and the speed measure of the limiting diffusion
Let
$d\in\mathbb N$ with $d>1$
$\ell>0$
$\sigma_d^2:=\frac{\ell^2}{d-1}$
$f\in C^2(\mathbb R)$ be positive with $$\int f(x)\:{\rm d}x=1$$ and $g:=\ln f$
$Q_d$ be a Markov kernel on $(\mathbb R^...
0
votes
0answers
5 views
Residence times of the telegraph process ?
The telegraph process is a two state stochastic process defined by the master equation
$$ \dot{\pi}_0(t) = \tau^{-1} \pi_1(t) - \sigma^{-1} \pi_0(t) $$
$$ \dot{\pi}_1(t) = \sigma^{-1} \pi_0(t) - \...
0
votes
0answers
33 views
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 ...
1
vote
2answers
60 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 ...
2
votes
1answer
25 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 $\ X(0)= 0$.
...
2
votes
0answers
34 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 \...
1
vote
1answer
40 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 ...
0
votes
0answers
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 ...
2
votes
1answer
84 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 ...
1
vote
0answers
67 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 ...
1
vote
0answers
59 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 ...
1
vote
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 ...
1
vote
1answer
39 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 ...
0
votes
0answers
26 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 ...
0
votes
0answers
48 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 ...
0
votes
1answer
34 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 ...
1
vote
0answers
41 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 ...
0
votes
0answers
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 ...
0
votes
0answers
88 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 ...
0
votes
2answers
35 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<...
