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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Checking whether a given formula is correct for a homogeneous Markov chain

I am new to cross validated so I hope my question belongs here. I saw in a paper where I study someone claiming the following: Given a $ \{ X_n \}_{n=0}^{\infty} $ be a homogeneous Markov chain ...
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Interpreting graphs from an Audjusted Dickey Fuller in R: library(plm)

I performed an Augmented Dickey Fuller test in R. However I do not understand how to interpret the results of the graphs. Can someone help me with that? ...
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How the process parameters changes with the length of data aggregation?

Is there any general relationship for a process(e.g. ARMA, O-U process) applied to financial data over different time intervals. e.g.In this question there is an answer telling the O.P. to aggregate ...
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How to fit a discrete distribution that can only be sampled from to count data?

My question is similar to this one. Assume we have a distribution from which we can only sample, but have no information on its pmf and consider further some count data: ...
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37 views

Predicting the maximum of a function given a set of samples

The main aspects of the question are highlighted in bold Let $f: \mathbb{R}^n \mapsto \mathbb{R}$ be a function. Supposing that we have access to a set of samples $(X,Y)$ obtained by sampling the ...
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Estimate transition matrix from many short Markov chains

I have a situation where data from the following process is observed: For $i = 1, \dots, n$ let $(X_{i,1}, \dots, X_{i,m_i})$ be a sequence of $m_i$ random variables coming from a discrete-space ...
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$c(n)$ is trend, $r(n)$ is fluctuation. Should $\text{cov}[c(n),r(n)]/\text{var}[r(n)]$ be close to zero?

Suppose $y(n)$ is a random time series given as function of the discrete-time variable $n$. Suppose we can decompose it into $y(n) = c(n) + r(n)$, where $r(n)$ is a strict stationary residual ...
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Expected time between two events

I'm having trouble with the following problem: Consider a game between two players A and B. Player A must complete three tasks each of which take an exponentially distributed amount of time with ...
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What does fixed regressor say about our linearity condition?

The linearity condition states that $\mathbb{E}[y_i]=(\vec{x}_i)^{T}\vec{\beta}$ for all $i$. Now, if we have fixed regressors, $\{\vec{x}_1,\vec{x}_2,\cdots\}$, our linearity condition only says for ...
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Generating Brownian motion on a manifold using charts

Suppose I have an $n$-dimensional manifold $M$ with a chart $\left(x,U\right)$. Are there any known methods for simulating Brownian motion on $M$ by first simulating a process in ...
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What methods can be used for tractable computation of probabilities for evolutionary model of non-independent entities?

I'm trying to extend a simple model which works as follows. We have n 'original' entities which each have a colour. This population evolves by the following events, which occur at exponential rates: ...
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20 views

How do I choose the initial features vectors for a Stochastic Gradient Descent trained SVD++ algorithm?

I'm reading the SVD++ Netflix Recommender Systems paper because I want to be able to properly assess this approach to building a recommender system. How should I choose the initial values of $q_i$ ...
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Kolmogorov Forward and Backward Equation Intepretation

Let $\lambda_i$ be the sojourn rate of state i, $q_{ij}$ be the transition rate form i to j, and $p_{ij}$ be the transition probability from i to j. The Kolmogorov Forward and backwards equation are ...
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Hypothesis testing to discriminate between two renewal processes

We have time [0,T] to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of ...
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Simple question about Ornstein-Uhlenbeck process

My question comes from this paper. The picture bellow provides a summary of the equations. Suppose prices of two stocks satisfy (2.1) SDE. Then X(t) is expressed as (2.2) and can be modeled with as ...
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26 views

Bayesian Optimization for a Stochastic Target that changes over time

Let's say there is a single slot machine that: costs zero to play can only be played once per day has a payout that is conditionally normal and is a function of the date and time. I want to use ...
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What is the difference between a truncated normal distribution and a half normal distribtion in a Stochastic Frontier Analysis?

I am trying to replicate a SFA where the error term u is assumed to have a cumulative normal distribution function truncated from below at zero. In my opinion, that refers to a truncated normal ...
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34 views

How to properly show the efficiency of a process?

I'm no statistician but my background is in computer science. At work, we are trying to improve the efficiency of a system where 5 people (A-E) each produce one part of a report and send it to 2 key ...
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Calculate covariance of slow and fast variables

Say you have two time series $X_t$ and $Y_t$ where $X_t$ is given by an $AR(1)$-process and $Y_t$ is a deterministic function of $X_t$: $$Y_t = f(X_t).$$ Also assume that the fluctuations of (the ...
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12 views

Estimating correlation of two HW1F processes

I have been thinking of an efficient way to estimate 2 HW1F processes efficiently. I assume two processes to be separate short rates (for Libor & Euribor). I was just planning to use the ...
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53 views

Optimization of stochastic computer models

This is a tough topic for myself to google since having the words optimization and stochastic in a search almost automatically defaults to searches for stochastic optimization. But what I really want ...
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69 views

monte carlo simulation using exponential distributions

I'm trying to simulate a stochastic model of deterministic exponential population growth, where $dN/dt = rN$ where $N$ is population size and $r$ is rate ($t$ time). I'm assuming there's no carrying ...
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78 views

The distinction between stochastic independent variable and measurement error in independent OLS variable

Assume that OLS regression of the form: $$Y_t = X_t'\beta + u_t$$ Suppose $X_t$ are stochastic, thus standard Gauss-Markov assumptions need to be accommodated. Given that: $$\text{E} {(\hat\beta)} ...
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Expressing a non-unique stationary distribution for a markov chain?

I am working ahead of my stochastic processes class, so some of what is written below may be inaccurate. I am working a problem that asks me to compute a stationary distribution for a Markov Chain, ...
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Ergodicity and percentiles

The mean ergodicity of a homogeneous/stationary random field/process is such a property under which given a sufficiently long realization, the mean of the realization converges to the mean of the ...
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Will there ever be an unhappy Tribble in Oz?

Here's an amusing problem brought to me by a student. Although it was originally phrased in terms of mutually annihilating bullets fired at regular intervals by a gun, I thought you might enjoy a ...
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How can I model a process which follows a pdf that changes w.r.t time?

I'm interested in modeling the probability of a gym having had k number of arrivals at time = t. Clearly this should be modeled by some type of time cont. stochastic process but a poisson process will ...
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26 views

Gradient descent for a noisy system

I have a system with tuning parameter $w$. To evaluate this system I use cost function $f(w)$. I try finding the optimum value for $w$ using Gradient Descent starting from $w_0$. The problem with ...
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21 views

Why is the periodogram of differenced white noise not flat?

I'm a final year undergrad who was doing a project that involved the implementation of a frequency-domain volatility estimator. I haven't a lot of stats background so not understanding a point that ...
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22 views

Kolmogorov Equations for a 3 state model. CTMC with a 3x3 generator matrix. Solving for $p_{11}$

I have a matrix $Q= \left[ \begin{array}{ccc} -3&3&0\\ 2&-5&3\\ 0&4&-4 \end{array} \right] $ where the state space is $S=[0,1,2]$ I need to solve the ...
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51 views

Joint Density for Renewal Processes

I'm trying to derive the joint density for the time-average age Z and time-average residual life Y for a renewal process, and use that result to determine if Z and Y are independent. If we call ...
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Deriving the age process CDF from a renewal process

My question is whether I correctly derived the CDF of the steady-state age process. For context on why I' am asking this question is problem 6.1(a) from Introduction to Stochastic Processes by Lawler ...
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Understanding stationarity in stochastic processes and time series

I am having trouble fully grasping the concept of stationarity in time series. Here is what I have gathered so far. A stochastic process is a collection of random variables with mean $\mu$ and ...
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Checking mean ergodicity of random field given the covariance function

Given a homogeneous 2D random field with a known covariance function, what is the easiest way to check if it is mean-ergodic? In my case the covariance function is rather complex and given by: ...
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Inference of discrete-valued multivariate time series from asynchronous ticks

I am looking for a relevant model to do inference of a large multivariate time series whose values arrive asynchronously. To be more precise, let $X = (X_1,\ldots,X_N)$ be the multivariate time ...
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What this slowly-varying nonstationary behavior of $x(n)$ implies for $y(n) = f[x(n)]$?

Folks, I am trying to figure something out here without success. Suppose $x(n)$ is a random discrete-time signal (or random time series) containing an arbitrary number of samples (say, $N$ samples). ...
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What is the mathematics underpinning geo-spatial analysis?

I've spend some time in this filed and yet on thing still plagues me. My understanding is that a random field is represented by some n multi-dimensional joint probability density function that is ...
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Problem on Poisson Process

I am doing some problems related with the Poisson Process and i have a doubt on one of them. The problem is stated as follows: A doctor works in an emergency room. The emergencies arrive according a ...
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76 views

Find the distribution of $ N = \min \left\{k: \prod_{i = 1}^{k}U_i \lt .6\right\}. $

I'm cross-posting this from math.SE because it's not getting any love over there. However, if that's considered heresy, I can delete the posting over there. The Statement of the Problem: Let $ \{ ...
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Simulating a stochastic integral

I am trying to solve exercise 3.9.10 on p. 66 of Ubbo F. Wiersema's "Brownian Motion Calculus" (John Wiley & Sons, 2008), which asks to simulate the stochastic integral $$ \int_0^1 B(t)\ dB(t) $$ ...
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Solving for a difference equation for $s_{t}$

Given $f_{t}=u_{t} - \bar{P}$ and the law of motion for $u_{t} = \rho u_{t-1} + \epsilon_{t}$, where $0<\rho<1$, $\epsilon_{t}$ is mean-zero iid and can be interpreted as a domestic price level ...
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Confused by definition of stationary stochastic process

Borrowing heavily from definition of stationary stochastic process here, I am having a hard time understanding why $$F(X_{n_1}, X_{n_2},...,X_{n_k}) = F(X_{n_1+n},...,X_{n_k+n}),$$ for every $k \ge ...
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recommended text for self study on stochastic processes

I am interested in learning stochastic processes. I have two goals in mind. The first, as a statistician working in industry, is to get an understanding of applied stochastic processes under my ...
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Affine processes/Car (Compound autoregressive) processes - why bother?

I recently stumbled over the definition of affine processes, which (if I understand correctly) are the same as Car processes. Definition: A q-dimensional process $w_{t+1}$ for $t\in\{-\infty, ..., ...
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martingales, stochastic processes

Suppose Xn, $n\geqslant0$ is a Markov chain on $\varphi =\left \{ 0,1,2,...,d \right \}$ and $P(x,y)=\frac{\binom{2x}{y}\binom{2d-2x}{d-y}}{\binom{2d}{d}} $. States 0 and d are absorbing states for ...
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Question in deriving Ito's Isometry

I am trying to wrap my head around stochastic integrals, and I am having trouble understanding the proof of Ito's Isometry, $$ I^2 = \sum_{j=0}^k \Delta^2(t_j)D^2_j + 2\sum_{0 \leq i < j \leq k} ...
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Deriving transition matrix from infinitesimal generator, continuous time Markov chain

I' am reading Introduction to Stochastic Processes by Lawler and I' am a bit confused how demonstrates you get the transition matrix $\textbf{P}_t$ from the infinitesimal generator $\textbf{A}$. I'll ...
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Trying to understanding how finite-state space, continuous time Markov Chains are defined

I' am reading Introduction to Stochastic Processes by Lawler and am struggling to understand how continuous time, discrete state space processes are defined. Quote from the book, A (time-homogeneous) ...
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Unsure whether my continuous time Markov Chain distribution is correct

I' am reading Introduction to Stochastic Processes by Lawler and have hit problem 3.3(c) that I' am not sure I have correct. $\textbf{3.3}$ Suppose $X_t$ and $Y_t$ are independent Poisson processes ...
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Two-alternative forced choice [closed]

Suppose that $p[r|+]$ and $p[r|-]$ are both Gaussian functions with means $\langle r \rangle_+$ and $\langle r \rangle_-$ and common variance $\sigma_r^2$. How can I show that $$P[correct] = ...