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 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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9 views

Parameters estimation of non-stationary random process using different runs

Let $X(t)$ be a non-stationary continuous time-dependent random process with a known model but unknown parameters. I'd like to know if it's possible to estimate the parameters of $X(t)$ not by using ...
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1answer
45 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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32 views

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
32 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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9 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
92 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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185 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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1answer
29 views

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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1answer
40 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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20 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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29 views

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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3answers
213 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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0answers
64 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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26 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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26 views

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
41 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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45 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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94 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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1answer
59 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
39 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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48 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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12 views

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
35 views

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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32 views

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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8 views

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 ...
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1answer
29 views

Change the classifier decision by using the probability estimates

I have a stream of documents composed of $1$ to $n$ pages. The objective is to segment the stream of documents. Every first and last page of a document is classified as either the beginning $b$ or ...
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14 views

Random walk with stochastic volatility

I've done some analysis on a financial random walk, and even post-transformations am finding heteroskedascity across longer time periods. I want to investigate whether this is due to stochastic vol ...
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63 views

Which is the difference between a distribution and a process (Poisson)?

I'm doing my PhD in geomechanics. I thought we use a Poisson-Weibull distribution (for the variability of a parameter at the rock), but reading more about the subject I think maybe is a ...
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1answer
36 views

$\operatorname{var}[\frac{b}{a} B(a-b)-b B(b)]$ with $b\leq a$ and $b\geq 0$; $B=$brownian motion

I want to calculate: $\operatorname{var}[\frac{b}{a} B(a-b)-b B(b)]$ with $b\leq a$ and $b\geq 0$; $B=$brownian motion. I started like this: $(\frac{b}{a})^2 \operatorname{var}[ B(a-b)]+-b ^2 ...
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1answer
35 views

Why X-process is called a process?

I have recently learnt about kernels in machine learning. And I have been introduced to many different processes e.g. Gaussian process, Wiener process. Now my question is why a set of functions has ...
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19 views

Comparison of a numerical and a stochastic maximization of a Cauchy likelihood

I'm trying to provide a comparison of a numerical and a stochastic maximization (using uniform sampler) of a Cauchy likelihood in terms of the sample size via drawing a plot. My problem is associated ...
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45 views

Discrete Time Markov Chain - Inventory

Let $D_n$ be the demand for an item at a store on day $n$. Suppose that $D_n$ is a sequence of independent and identically distributed (i.i.d.) random variables with probability mass function: $p_k = ...
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If Maria performs more observations per unit of time than Maximilien, how can he estimates the Maria's results from his own?

General problem Having a sequence of values $v_0, v_\Delta, v_{2\Delta}, \ldots, v_{N\Delta}$, which are measured every $\Delta$ units of time, usually we are interested in the prediction of the ...
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27 views

Mean length of time spent queueing in $M/E_2/1$ system?

Context: Consider a $M/E_2/1$ queueing system, where the customer arrival rate is $\lambda$ and the service time distribution has a gamma distribution with parameters $2$ and $\mu$, i.e. with p.d.f. ...
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56 views

Continuous time Markov chain forward equation

This is a homework question and I need suggestion how to approach it. We have given the transitions $\ i\rightarrow i+1$ with rate $\lambda(i)$ where $\ i \ge 1$ $\ i\rightarrow i-1$ with rate ...
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1answer
62 views

Calculating discrete hazard rates problem

I am working on an assignment for a Stochastic Modeling class and am stuck on the following question: Let $X$ have probability mass function $p_j = P \lbrace X = j \rbrace $ for $j \geq 1$. Let ...
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1answer
85 views

Mean service time of a $M/E_2/1$ queueing system?

Consider a $M/E_2/1$ queueing system, where the customer arrival rate is $\lambda$ and the service time distribution has a gamma distribution with parameters $2$ and $\mu$, i.e. with p.d.f. ...
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0answers
27 views

Absorption probability in 1D RW with asymmetric step sizes, $ x<0 $

What is the probability of absorption at $ 0 $, as a function of position $ x $, for a 1D random walk (on $ \mathbb{Z} $) with asymmetric step sizes? For example, suppose that you can take two steps ...
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1answer
60 views

Probability of Jar containing Balls type question with replacement

A jar has 50 balls 1 to 50 each one having distinct number written on it. Bob, owner of the jar,each day he takes out one ball out of the jar randomly ( with equal probability) and put it back. Q1. ...
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43 views

Polya urn waiting time distributions

From an urn containing 1 ball each of n different colours, a ball is drawn, its colour noted and then it is returned to the urn along with k balls of the same colour. What is the distribution of the ...
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1answer
75 views

Relationship between average and characteristic function of a Gaussian process

I'm having trouble understanding an equality given in a book ("Speckle Phenomena in Optics" by Joseph Goodman p.145) for a zero mean, stationary Gaussian process: $\overline{\exp(i ...
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1answer
132 views

Integrating Gaussian white noise over a Gaussian density

I have encountered the following integral: $$ \int_{- \infty}^{+ \infty} X(\theta) \frac{\exp \big \{-\frac{\theta^2}{2\sigma^2} \big \}}{\sigma\sqrt{2 \pi}}d\theta $$ where, for fixed $\theta = ...
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186 views

I want to show $M(t)=e^{\frac{t}{2}}\sin(B(t))$ is a martingale by using Ito s formula

Show that $M(t)=e^{\frac{t}{2}}\sin(B(t))$ is a martingale by using Ito s formula. $B(t)$ is brownian-motion. i must show $s \le t $ then $E(M(t)|\mathcal F_{s})=M(s)$ but i dont know how use ito ...
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118 views

How to solve $\mathrm dX(t)=X(t)^2 \mathrm dt+X(t)\mathrm dB(t)$ with condition $X(0)=1$?

I want to solve the stochastic differential equation $$\mathrm dX(t)=X(t)^2 \mathrm dt+X(t)\mathrm dB(t)$$ with condition $X(0)=1$.
2
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1answer
99 views

Using Ito's lemma for Brownian motion

I am a little confused by Ito's lemma. I reviewed the basic application for geometric brownian motion. I'm now trying to apply it to a different functional form to make myself better. My mind leaps ...
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57 views

What is filtration?

I am studying Brownian motion and came across the concept of Filtration. However I can't understand how this concept relates to Brownian motion. My notes contained some gibberish about being ...
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1answer
45 views

Loss rate calculation in a Poisson process

I have a bit a naive question about Poisson process. Suppose I have a station that receives a number of packets $n(i)$ each time slot $i$. If the number of received messages is less than $M$ then the ...