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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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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19 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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25 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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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 ...
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15 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 ...
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1answer
36 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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9 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
$$
\epsilon_t \overset{iid}{\sim}\mathcal{N}(0, \sigma^2)
$$
$$
\epsilon_t' ...
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42 views
Definition of Stochastic Process as Probability measure in a Prob. Space
I've found this question, with a very good answer, but they don't broach my question.
In Oksendal's Stochastic Differential Equations book, it's written «the stochastic Process is a probability ...
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1answer
15 views
Mean and Variance: first-order stochastic dominance
Suppose $X$ and $Y$ follow the same distribution, with same mean.
And $Var(X)<Var(Y)$. Then, does $X$ first-order stochastically dominate $Y$? Intuitively, I think this will hold. But I do not know ...
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10 views
First moments of GBM-like process with non-normal shocks
First consider a standard GBM process of the form,
$$\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 ...
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139 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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21 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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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 ...
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25 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 ...
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0answers
18 views
Excursion areas and bridges for F and chi (square)
My question relates to excursion areas (and excursion bridges, in general) for
$\chi$-, $\chi^2$- and/or F-distributed stochastic processes (or more general random fields). I'm finding it difficult ...
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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
14 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 ...
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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 $...
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votes
2answers
131 views
Dealing with different definitions of the Ornstein-Uhlenbeck process
I've run up against a wall in reconciling two different definitions of the Ornstein-Uhlenbeck process, and would appreciate some help.
On the one hand, as discussed here, we can define an Ornstein-...
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0answers
31 views
Predicting probability of next event happening
My Data is:
TimeStamp <- time stamp of the event occurring
Length <- length is the duration of the event
ID <- identifier where the event is occurring ( 25 IDs)
TimeStamp | Length | ID
The ...
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55 views
Why does the variance of a Brownian motion increase linearly with time?
Brownian motion is said to follow a path where each value is normally distributed with mean $\mu t$ and variance $\sigma^2 t$.
What is the basis for the relation that variance varies directly ...
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1answer
40 views
Layman's explanation on stochastic and statistical models
What's the differences between stochastic models (process) and statistical model (analysis). As I understand, a stochastic model (process) simply means it involves random variables, which is basically ...
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0answers
27 views
What is the relation between stationary distribution, limiting distribution, ergodicity and detailed balance equation in a markov chain?
I have studied them from various sources and I am not able to make a strong conclusion with respect to their relation with each other.
This is what I understand by these terms
Ergodic : If all ...
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8 views
Parameter estimation of time-dependent R.Vs(random processes),
I have taken a number of probability and statistics courses which cover estimation and basic random processes but something which is not clear is how you can do parameter estimation for time-dependent ...
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1answer
58 views
Convert a state-space model with exogenous input to one without
I have a state space model of the form
\begin{align}
x_{t+1} &= Ax_t + Bu_t + w_t\\
y_t &= Cx_t + Du_t + v_t
\end{align}
where $u$ is the exogenous input. Also, $ w_t \sim N(0, Q)$ and $v_t \...
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8 views
Layman's explanation for Beta Stacey Dirichlet Process
I would really appreciate it if someone would explain the Beta Stacey Dirichlet process and its application scenarios in layman's terms.
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0answers
15 views
Difference between Beta process and Beta regression?
In a layman language, how the difference can be defined between statistical process and the same named GLM such as poisson processes and poisson regression or beta processes and beta regression? How ...
1
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1answer
36 views
how to solve this markov chain problem?
This is a problem in the book of "introduction to stochastic process ". Any help to solve this problem ??
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0answers
35 views
Distributions over continuous distributions [closed]
I know a few stochastic processes that are described as 'distributions over distributions'. For example, the Dirichlet process draws discrete distributions from a default model (see e.g., p3 of these ...
2
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0answers
37 views
Is this inhomogeneous Poisson process?
I asked this question in math stack exchange but since it is more related to probability and statistics, I thought I can ask here.
Consider the population growth model where $P′(t)=rP(t)$, where $P(...
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1answer
41 views
Does the MLE-Kalman prediction maximize the likelyhood of the prediction?
The question is the following. Say I have observations of a Gaussian stochastic process ($\{x_i\}_{i=1}^n$) for which is convenient to use the state space formalism (and Kalman recursions) to describe ...
2
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1answer
79 views
Initial vector $h$ in Bayesian stochastic volatility models (Jacquier, Polson and Rossi, 1994)
I was going through the paper Jacquier, Polson and Rossi (1994): Bayesian Analysis of Stochastic Volatility Models. While the model seems straight forward to implement. I'm not able to find how the ...
6
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2answers
174 views
The Fishing Problem
Suppose you want to go fishing at the nearby lake from 8AM-8PM. Due to overfishing, a law has been instated that says you may only catch one fish per day. When you catch a fish, you can choose to ...
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19 views
What is the correct terminology for simulating markov chains with state transitions observed from data
I'm struggling to find literature for a process I'm working on because I'm lacking the correct technical language to describe it.
I have a markov chain with finite states which members of my ...
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85 views
How to estimate passengers destinations from flightradar data?
We have a graph with vertices corresponding to airports and edges corresponding to flights between those airports. On edge between airports A and B we have and number of passengers transferred from A ...
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1answer
24 views
How to estimate the kernel of a Markov process with continuous state-space, from a finite sample?
With a discrete state-space discrete time markov chain, given a sequence of sample data $X_{1} \dots X_{n}$, I might estimate the transition probabilities $P_{ij}$ using relative frequencies. From our ...
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1answer
25 views
Estimating the Mean of a Function of a Stochastic Process
My question is about estimating the mean of a function of a stochastic process:
A function $f(X_t, X_{t-1}, ... , X_{t-n})$ takes in a fixed number of events $X_t, X_{t-1}, ... , X_{t-n}$ from a ...
2
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0answers
38 views
What is the name of this stochastic process?
Suppose that $\{Z_t : t \in [0,1]\}$ is a standard Brownian Motion process. It's well known that $X_t = Z_t - tZ_1$ is a Brownian Bridge, because it's a continuous Gaussian process, with mean function ...
5
votes
0answers
26 views
Is average stopping time a continuous function of Bernoulli parameter?
Consider an infinite sequence $X = (X_i)_{i \in \mathbb N}$ of i.i.d Bernoulli random variables with (unknown) parameter $p \in (0,1)$, and let $N$ be a stopping time on $X$. Is it always the case ...
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0answers
30 views
Why does a spline or a GP estimate the mean function?
I've got a periodic function $m(t)$ where $t \in [0, 1]$.
Now, let's say that I take n independent samples from a really weird distribution (say it's a zero inflated distribution where the ...
4
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1answer
71 views
Correlation between Ornstein-Uhlenbeck processes
Consider the Ornstein-Uhlenbeck process, $U(t)$, whose evolution follows:
$$
\mathrm{d}U(t) = -\theta U(t) \mathrm{d}t + \sigma \mathrm{d}W(t),
$$
where $\theta \in (0,2)$ is the mean-reversion rate, $...
1
vote
0answers
31 views
Why we say that In GP, choice of kernel reflects the belief of smoothness of the process?
In Gaussian Processes in Machine Learning (chapter 4 pdf), the book shows that the smoothness of kernel is corresponding to the mean square smoothness. (I mean continuity or differentiability) I know ...
0
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1answer
77 views
N-Urns N-Color ball modelling as Markov Chain
I am trying to model a system which can, mostly, be simplified to elements of different groups changing groups among themselves. I want to understand how frequently the elements change group and how ...
1
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1answer
26 views
Definition of nonarithmetic law
I came across the term nonarithmetic, but I don't now what that means. It is a condition for a Proposition of a Paper I am reading. There it is said:
Assume that the law of $\ln A_0$ is nonarithmetic....
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30 views
What is the best test for stationary volatility?
I'm familiar with the tests for stationary mean and unit root &c. How does one best determine that the data possess weak or second-order stationary? This an important requirement of ARCH-type ...
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0answers
28 views
Maximum Likelihood Estimation (MLE) for Markov Chain Rates $R_{ij}$ a.k.a. $Q_{ij}$
The following past question deals with using MLE to estimate the transition probabilities matrix $P_{ij}$ directly. I, however, am looking to estimate the rates matrix $R_{ij}$ a.k.a. $Q_{ij}$, which ...
2
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0answers
26 views
How to actually draw a distribution from a given Dirichlet process?
I am wondering if there's an algorithm to actually draw a distribution from a given Dirichlet process. The closest thing I've came across is the stick-breaking construction of a Dirichlet process, but ...
2
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0answers
52 views
Covariance of Gaussian process?
Problem:
Consider the random process defined by the Ito integral
$$ X_t = \int_0^t f(\tau)\, dB_\tau $$
where $f(\tau)$ is a deterministic real-valued function and $B_\tau$ denotes the canonical real-...
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36 views
Karp luby sampling
Can anybody explain Karp Luby based sampling in simple words.
Thanks a lot in advance.
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0answers
30 views
Modeling by means of a negative binomial process
The negative binomial distribution with parameters $p\in(0,1)$ and $t>0$ is sometimes defined as the distribution of the number of failures before the $t$th success. This is supported on the set $\{...
