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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I want prove $E[\int_{\Lambda}h(y)\mathcal M(t,dy)]=t\int_{\Lambda}h(y)\lambda_L(dy)$ and $\dots$

if $\Lambda$ is a Borel set such that $0 \notin \bar \Lambda$ Then. $$E[\int_{\Lambda}h(y)\mathcal M(t,dy)]=t\int_{\Lambda}h(y)\lambda_L(dy)$$ and $$E[(\int_{\Lambda}h(y)\mathcal M(t,dy)-t\int ...
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If $X^T(t)=X(t\land T)$ is said to be the process $X$ stopped at $T$. I want prove following statment

Let $X$ be a stochastic process defined on a probability space $(\omega ,\mathcal F,P)$ endowed with a filtration $(\mathcal F)_{t \ge0}$ and let $T$ , $T^\prime$ be $\mathcal F_{t}-$stopping times. ...
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34 views

Calculating expectation function and covariance function

Let $E_n(t)$ denote the empirical cdf based on iid uniform $u[0,1]$ random variables $U_1,...,U_n.$ The corresponding uniform empirical process $(e_n(t),0\leq t\leq 1)$ is given by ...
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42 views

Modeling a process with decay and refilling

My question is about the approach that needs to be taken for modeling a particular process with . I have looked around for similar questions or answers but didn't find any. I got some links to Markov ...
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1answer
16 views

Steady state Markov Chain

is it able to count the steady state of problem with recurrent subchain.. for example if there are A B C D things and they are all recurrent. do they have steady state?? and also.. how to count ...
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1answer
29 views

multi stage binomial “process”

I wish to model the retransmission time of a file that divided into K blocks. I know the successful blocks of first transmission obey the binomial distribution $$ X_1 \sim \text B(K,p) $$ , p is the ...
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1answer
26 views

Multiplication of two random distribution

I am trying to find the resulting PDF , when two random functions are multiplied. First function obeys normal distribution and second function obeys cauchy distribution. Can anybody tell me how to ...
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21 views

I want Find finite dimensional densities for an $\mathbb R^d-$ valued Gaussian process $X$with specified mean and covariance functions

Write the finite dimensional densities for an $\mathbb R^d-$ valued Gaussian process $X$with specified mean and covariance functions.
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33 views

Stochastic Processes

I have a couple questions about stochastic processes. My professor didn't really give in depth explanations and non of his lecture slides do much explaining either. Say S(x) is a stochastic variable ...
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17 views

Simulation of a process consist of Brownian motion and Poisson process

I am trying to simulate the following process: h(t)=B(t)+e[P1(t)-P2(t)] in which B(t) is a Brownian motion and P1, P2 are Poisson process with ...
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94 views

Is a time series the same as a stochastic process?

A stochastic process is a process that evolves over time, so is it really a fancier way of saying "time series"?
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165 views

Expected value of a product of two compound Poisson processes

I'm working on my master thesis now and I've been struggling with a problem for some while now and no one seems to be able to help me or point me in any direction. So now I reach out to see if someone ...
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43 views

How do I solve this stochastic differential equation?

So I have a second order stationary process $Y(t), \infty < t < \infty$ which has a continuous sample function, mean $\mu_Y = 1$ and covariance function $r_Y(t) = e^{-|t|}, -\infty < t < ...
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1answer
40 views

How do you call a Markov chain with idempotent transition probability matrix?

I don't recall ever seeing a term to refer to Markov chains for which all the transition probabilities matrices are equal ($P^{(n)}=P\quad\forall\,n\in\mathbb{N}$) but I'm sure there should be one.... ...
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119 views

Convergence of likelihood implying convergence of marginal likelihood?

I will ask my question through a toy motivating example. It is well known that a Poisson process is the continuous time analog to a Bernoulli process (for example: ...
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38 views

Split Poisson Process AND severity

I have a Poisson process whose statistics are interarrival times ($\bf X$), number of arrivals ($\bf N$), and arrival times ($\bf T$). Later, the process is split by a Bernoulli process that ...
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10 views

Variance of the second order stationary process's mean

So i have a second order stationary process with the following covariance function $r_X(t) = \alpha e^{- \beta |t|}, -\infty < t < \infty$ Now, $\bar{X} = \frac{1}{T} \int_{0}^{T} X(t) dt$ ...
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10 views

Stochastic differential equations with colored noise

I'm interested in how to model colored noise. I'm aware of the Generalized Langevin Equation but not terribly familiar with the details of it. (I've worked extensively with the Langevin where the ...
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23 views

Does the noise term in a SDE need to be Gaussian?

Most of the examples I've seen for stochastic differential equations are of the form: $$ dX_t = \mu(X_t, t)dt + \sigma(X_t, t) dW_t $$ where $dW_t$ is a Wiener process, i.e., the independent ...
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59 views

What is the term for E[x*y']

I know this is probably a very simple question, but I recall learning that for 2 random vectors, x & y, with mean mx & my, E[(x-mx)(y-my)'] is the covariance & E[xy'] is the correlation ...
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1answer
35 views

How to predict the time series data

I have no background of advanced stats. I am an engineer and I have the following data. I am representing it as a decent graph for better understanding. I want to forecast the collision for the next ...
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1answer
20 views

Fit stochastic differential equation to data

Could I have some review of the method I used to fit following SDE: dX = f(t) dt + s X dW Fitting method: Calculated sample for sdW from our data as: sdWt = ...
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2answers
24 views

How to understand the “arriving rate” in a homogeneous Poisson process?

We know that when the arriving rate of a Poisson process $X(t)$ becomes constant, then the process becomes a homogeneous Poisson process. I have trouble understanding what "a constant arriving rate" ...
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13 views

Approximating a random field

I have got a bunch of $n$ ($\approx 100$) pixelized maps. Each pixel is a single figure. Each so-called map can be represented by a matrix whom each element is a pixel. Let's say that there $p\times ...
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52 views

Recommend textbook for probability theory and stochastic process

Would you mind recommend a textbook for the following topics? It's a graduate level course for students in finance/economics. Probability theory (no measure theory please) Conditional expectation ...
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18 views

Probability generating function of poisson point process

Assume you have a 1D non homogenous PPP $\Xi$ with intensity $$\lambda(x)=\lambda x^{\frac{2}{\alpha}-1} \ x \in \mathbb{R}^+$$ where $\alpha$ is positive integer. Now define $$\gamma_k = ||x_k||$$ ...
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48 views

Cointegration - Why can't I estimate a VAR on the differences?

When talking about variables that are I(1) (the first difference is stationary), Lutkepohl book says: "...in general, a VAR process with cointegrated variables does not admit a pure VAR representation ...
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38 views

Mean and variance of Cox process

Consider the (doubly-stochastic) Poisson point process with rate $ \lambda(t) = \rho e^{-t/\tau} $ where $\rho\sim\Gamma(\alpha,\beta)$ is a Gamma-distributed random variable. I require the mean ...
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9 views

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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10 views

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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54 views

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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42 views

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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31 views

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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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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23 views

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
25 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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1answer
37 views

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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18 views

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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1answer
39 views

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
35 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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39 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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46 views

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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72 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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11 views

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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31 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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18 views

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
49 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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2answers
70 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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1answer
49 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 ...