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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Reporting sensitivity/specificity using a random process?

I'm using a method that involves cross-validation to make predictions on my dataset. As it splits the data randomly, I will end up with different results (I believe this is an example of a stochastic ...
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mean queue delay ( nonpreemtive priority)

I'm trying to solve a problem where all arriving items (arrival exponential $\lambda = 1/5$) are divided into into groups, those who are served within 5 units of time and those who have their service ...
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Poisson Process

I would appreciate a hint on this problem: A pedestrian wishes to cross a single lane of fast-moving traffic. Suppose the number of vehicles that have passed by time $t$ is a Poisson process of rate ...
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stochastic network optimization

I'd like to optimize the flow of materials through a network. There are vertices (i.e. physical locations) and edges (i.e. links between the physical locations). Inputs: locations transactional ...
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13 views

How to prove a relation holds almost surely?

Suppose we have two random process $X(t)$ and $Y(t)$. If $\forall t \quad X(t)\ge Y(t)$, Can I conclude that $X(t) \ge Y(t)$ almost surely(a.s)? Thank you very much in advance.
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Find the distribution of the supremum of the Brownian motion

Let $A(t)$, $t \in [0,1]$ be a Gaussian process with zero mean and co-variance kernel $\mathrm{Cov}(A(t_1),A(t_2))= \min (t_1,t_2),\, \forall t_1,t_2 \in [0,1]$. Find $$P\left[\sup_{t \in ...
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Application of Lévy–Khinchine formula

How can we express the characteristic functions of Wiener and Poisson processes by using the Lévy–Khinchine formula? I don't know how to find the characteristic functions of particular Levy ...
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Discrete time generator of stochastic process

While looking at one paper about Metropolic Hasting optimal convergence rates, I came accross a discrete time generator of Markov chain. It is defined as follows: $$G V(x)=nE\left [ \left( ...
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A way to check the accuracy of a Markov chain?

I posted the same question on MSE since I was not really sure whether to post it here first or not. Anyway since I still did not get any answers I will be posting this here hoping for some help. Say ...
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1answer
15 views

Ensemble in stochastic process

I am learning a time series and forecasting course.In the book "The Analysis of Time Series by Chris Chatfield" it says that We only have single outcome of the process and a single observation on ...
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What is a stationary function?

Snoek et al, have a recent paper "Input Warping for Bayesian Optimization of Non-Stationary Functions" (http://arxiv.org/abs/1402.0929) which mentions "stationary functions". I understand what a ...
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Discrete optimization with a very large solution neighborhood to explore

I have a problem whose feasible (discrete) solutions can measured by a cost function. I am thinking of using some optimization technique to get better solutions from a rough initial approximation. I ...
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Covariance of $cov(5W_7+6W_9,W_7)$ where $W_t$ is a standard Brownian motion

I'm having trouble deducing the value for the problem in the title. Here is what I have done so far. (Given a standard Brownian motion (BM) $W_t, t\geq0 $ with $W_0 = 0$ and $\sigma^2=1$) The ...
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1answer
28 views

Markov Chains : Can anything be said about what happens in between two transition?

In time homogeneous discrete Markov chains we take a set period for a single transition. In examples we see sometimes depending on the examples the transition period being a a month a week etc. I'm ...
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23 views

Stochastic linearization by irregular waves of ship roll motion equation

I'm interested in finding some way for doing "stochastic linearisation by irregular waves of ship roll motion equation". I found some publications about it but its hard for me to understand, how to ...
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Scheduling Algorithm for a multi-server queue problem

I have 4 servers, n customers and m reports. At any time, a customer may request one of m reports. There are only 4 servers which are capable of generating reports. Each server can only process one ...
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38 views

Interpretation of the partial autocorrelation function for a pure MA process

I have been working with some time-series theory and I noticed something that I can understand "mathematically", but not based on the intuitive explanations of what the partial auto-correlation ...
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22 views

Continuous AR model from a Discrete AR model

I have a model of an $\mathrm{AR}\left(2\right)$ process, thus: $X_n + a_1X_{n-1}+a_2X_{n-2}=Y_n \qquad \mbox{for} \qquad n=0,\pm1,\pm2,\ldots $ and I have the equivalent stochastic differential ...
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sample generative model from a chain/tree

I have a tree of states and I would like to sample from this tree based on pure birth process; however, I don't know how exactly I can do this; so far I have done this; I simplified my problem; the ...
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25 views

Probabilities in a Markov Model

I am reading a paper on Markov Models and I am trying to figure out how to compute the probabilities for the $\alpha$-pass. I am given an $N\times N$ matrix $A$, that has the probabilities of ...
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Confidence intervals and bootstrapping stochastic processes

I am currently using a stochastic method for prediction that only reports my parameter of interest $\widehat{T}$ and does not report confidence intervals, though I would like them. I understand that ...
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1answer
47 views

Estimation of AR(1) process

Suppose the stochastic process ${X_t}$ satisfies the equation $$X_t=\phi X_{t-1} + Z_t \tag{A}$$ where $\phi>1$ and $Z_t$ is a white noise. Then iterating forward we get that the only stationary ...
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Why representation of AR process comes up in estimation

Let ${X_t}$, $t=...-2,-1,0,1,2...$ be a stochastic process that satisfies: $X_t=\rho X_{t-1}+\varepsilon_t$ with $|\rho|<1$ and $\varepsilon_t$ is a white noise. In that case, we also know that ...
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56 views

Probability generating function for negative values of random variables?

What if we have negative integral values for a random variable?Then is it possible to write a probability generating function for it? All definitions I have seen so far is for non negative integer ...
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48 views

Showing $ λ_V(x) \leq min\{λ_1(x),\cdots,λ_n(x)\}$ Hazard function

Suppose $X_1, \cdots, X_n$ are independent, nonnegative continuous functions, each $X_i$ has hazard function $\lambda_i(x)$. If $V=\max\{X_1, \cdots, X_n\}$, I need to show that ...
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Crosscorrelation of stochastic process

Let $Z_1,Z_2 $ i.i.d. standard normal $$ X(t) = \begin{cases} 0, & \text{if } t<Z_1, t<Z_2\\ 1, & \text{if } Z_1\le t <Z_2 \text{ or } Z_2\le t <Z_1 \\ 2, & \text{if } t\ge ...
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Best Multiple Imputation Method for Multiple Surveys Mixed Together, Presented Randomly?

I am working with a dataset containing data from 15 different surveys. The surveys were presented all as one battery to participants, with questions from all surveys essentially placed into a pool and ...
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1answer
54 views

Transition rates in continuous time markov chain

A house has 2 rooms of similar sizes with identical air conditioners equipped with thermostats which turn on and off as needed to maintain the temperature in each room to a desired level of 22 ...
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21 views

Transition matrix in left-right hidden semi-Markov model

i'm developing a hidden semi-Markov model left-right . In a left-right model a sequence of $M$ states starts in state 1 and ends in state M, with no repetition of states. Since the model is ...
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1answer
106 views

relationship between ARMA and AR

I once heard some statements regarding the relationship between ARMA and AR process, such as An average of severl lags of an autoregression forms an ARMA process ...
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383 views

stochastic vs deterministic trend/seasonality in time series forecasting

I have moderate background in time series forecasting. I have looked at several forecasting books, and I don't see the following questions addressed in any of them. I have two questions: How would ...
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Sampling brownian motions

I wish to sample standard linear Brownian motions on the interval $[0,1]$. I do this by dividing the interval into $n$ equal sub-intervals, deciding $B(0)=0$, and letting ...
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51 views

Stability of a GI/G/1 queue with $\rho=1$?

The final theorem in Chapter 19 of Meyn and Tweedie's Markov Chains and Stochastic Stability tells us that if the mean inter-arrival time $\lambda$ of a GI/G/1 queue is greater than its mean service ...
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28 views

How can I specify cross-correlations between differential equations?

for my Master thesis I want to model co-movements between two commodity forward curve. First, I specify the model for a single forward curve $F(t,T)= \bar{F}_0(t)e^{s(T_i)-y(t,T-t)(T-t)}$ where ...
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Convergence time of a Markov chain

We know that a regular Markov chains converges to a unique matrix. The convergence time maybe finite or infinite. My interest is in the case where the convergence time is finite. How can we accurately ...
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285 views

How can I show that a random walk is not covariance stationary?

How can I show that a random walk ($y$ follows a random walk) is not covariance stationary? I tried to work on the formula below (with no results) Could you give me just a hint on how to proceed ...
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79 views

Beta-Mixing Time Series

There are plenty of resources on how to compute $\beta$-mixing coefficients for a time series and to check if a time series is $\beta$-mixing or not. However, I am struggling to actually find concrete ...
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38 views

Continuous time markov chain backward/forward equations

Using Kolmogorov's forward and backward equations, show that $p_{11}(t) + p_{21}(t) + p_{31}(t) = 1$ and $p_{21}(t) = p_{31}(t)$ where $p_{ij}(t) = P(X(t) = j | X(0) = i)$. My attempt: I can show ...
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Continuous Time Markov Chain Transition Rates

A hospital has two physicians on call, Dr Dawson and Dr Baick. Dr Dawson is available to answer patients' calls for time periods that are exponentially distributed with mean 2 hours. Between ...
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1answer
51 views

When is a ARMA(p,q) process ergodic?

We know that a ARMA(p,q) process is weakly stationary, iff there is no root of the characteristic polynomial of its AR part lying on the unit circle. But what is the necessary and sufficient ...
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1answer
64 views

How to obtain solution of differential equation in this simple linear birth-death process?

(Apologies for the poor title, I didn't know what what to type in) I am having a problem with the second part of this question (an example question from a past stochastic course I took): ...
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104 views

Poisson process and uniform distribution

Question: A single-pump petrol station is running low on petrol. The total volume of petrol remaining for sale is $100$ litres. Suppose cars arrive to the station according to a Poisson ...
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76 views

Discrete time Markov Chain - Long-term frequency

Let's say I have the following scenario: A mouse is put into a maze that's constructed as below: There are 9 rooms with connections between the rooms as indicated with a "gap" in the ...
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1answer
44 views

Big O notation preserved under convex functions?

Suppose that the random variable $X_T$ is $O_p(1)$ as $T \rightarrow \infty$. Does this imply that the random variable $\max\{0,X_T \}$ is $O_p(1)$?
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60 views

Explaining Gaussian Processes

I am finding it hard to understand Gaussian Processes. Can someone please explain it here in an accessible way? I do understand what Gaussian distribution is but couldn't understand Gaussian ...
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Implication of uniform stochastic boundedness?

Let $\theta \in \Theta \subseteq \mathbb{R}^d$ be a parameter vector. Let $Q: \Theta \rightarrow \mathbb{R}$ be a function mapping from the parameter space to the real numbers. Let $Z_T$ be a a ...
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1answer
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Maximum likelihood estimation (MLE) for Markov Chain initial distribution?

I am working on using MLE to estimate a Markov Chain, I have successfully estimated the transition matrix $A$, using the method provided in ...
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Generalised (nonhomogeneous) Poisson process

Define a generalised Poisson process as an arrival process that begins at time 0 and that satisfies: The independence property: the number of arrivals during two non-overlapping intervals ...
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How to create a multivariate Brownian Bridge

It is known, that a standard multivariate Brownian bridge $ y(\mathbf u) $ is a centered Gaussian process with covariance function $$ \mathbb E(y(\mathbf u) y(\mathbf v)) = \prod_{j=1}^d (u_j \wedge ...
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Distribution of volatility

Is there a test/way to determine the distribution of stochastic volatility? For example, I have a random walk where the increments are non-normally distributed and heteroskedastic. I would like to try ...