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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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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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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 ...
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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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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 ...
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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 ...
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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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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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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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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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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 ...
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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 ...
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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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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 ...
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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, $...
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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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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Karp luby sampling

Can anybody explain Karp Luby based sampling in simple words. Thanks a lot in advance.
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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 $\{...