Questions tagged [stochastic-processes]

A stochastic process describes the 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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12 views

Projection operator: why squared norm of the sum of them is equal (or smaller) than the sum of the squared norms?

I am working through the proof of Lemma 2 in this paper (page 25, need it for my own research) and I am stuck at the very first step. Here, I will formulate a bit simplified version of this step. ...
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The Simple process [closed]

For a simple birth process{N(t);t>=0} if birth rate is 20 units per day. obtain the probability that the population is of size (i) 15 in 4 days (ii) 30 in 4 days
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For a Brownian motion, what is the probability that $B(t)$ 'hits' $a$ before it hits $b$, for given $a < 0 < b$?

My attempt: Let $X_1,X_2,\ldots $ be iid random variables with $P[X_i =-1] = P[X_i=1] = \frac12$. If we let $S _n =\sum_{i=1}^n X_i$, then for integer $a< 0< b$, the probability that $S_n$ hits $...
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Expected first time that $|B(t)|=1$ for a standard Brownian motion

I want to calculate $\mathbb{E}[T]$ where $T = \inf \{t \geq 0 \mid |B(t)| = 1\}$ and $B(t)$ is a Brownian motion with mean $0$. I saw some similar posts but for a one-sided hitting time, and in those ...
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Model that form relation between high frequency input and output variables, but observations are available at mixed frequency

I want to explore some forecasting models that form relation with daily output data and daily input data , but learns through monthly output data and daily input data. That means, output observations ...
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How to compute $P\left(\max_{{M_s + 1} \leq j\leq M_u} Y_j < \max_{1\leq j \leq M_s} Y_j\right)$?

Assume that $M_t$ follows a Poisson process and $T_i,\quad i=1,2,\cdots$ and $Y_i, \quad i=1,2,\cdots$ are i.i.d. random variables showing the arrival of events and interarrival times of the process $...
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Markov Process and a continuous PDF

I read an economics paper, and I got quite confused about the setting of the model: they assume that the labor productivity $\varepsilon$ (defined on a discrete state space) follows a Markov process. ...
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Autocovariance of a Stochastic Process

Let us define a stochastic process $X_t$ as follows: $$X_t = \sum_{i=0}^p \alpha_i \epsilon_{t-i} + \sum_{i=0}^q \beta_i \delta_{t-i}$$ where {$ε_t$} and {$δ_t$} are mutually independent normally ...
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Invertibility of a Moving Average Process (MA)

Let us consider a moving average process given by : $$X_t = \epsilon_t + \epsilon_{t-1}$$ Is it possible to show that this process is not invertible by expressing $\epsilon_t$ in terms of $X_t , X_{t−...
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Intended selection bias

Sampling or selection bias is often presented as something that has to be overcome, avoided, or at least appropriately considered because it's a problem otherwise. I wonder how often situations arise (...
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Convergence of Vector argmax

Suppose we have some vector v in $\mathbb{R}^d$. At each timestep t, one of its d elements, chosen randomly, is changed by an external process that moves v towards some unkown v* with high probability ...
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Finding expected value from expectation of squared distance

This problem is actually a part of a much larger biology problem that I am working on. However, I will leave out the unrelated parts. Consider a sequence of points $\{(x_j, y_j)\}$ where neighboring ...
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What is the underlying proccess of the probability of a stock reaching a determined strike price?

Suppose the price of a stock follows a random walk, the price of a derivative "Stock is over $x at time t" would also follow a random walk? I talk about probabilities in the question ...
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Permutation condition of Kolmogorov Extension Theorem for stochastic process

can someone explain the first condition of the kolmogorov extension theorem as exposed here https://en.wikipedia.org/wiki/Kolmogorov_extension_theorem? And perhaps provide a counter example, i.e. ...
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Distance between stochastic processes controlled by power spectral density

Let $f_1$ and $f_2$ be two stochastic processes over the same domain with finite power spectral densities (PSDs) $S_1$ and $S_2$ respectively. Can I bound a distance between $f_1$ and $f_2$ based on ...
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How was the law of total probability used here for this conditional probability to get this result?

If $X(t)$ is observed at a random time $U \sim \text{Uniform}(0, 1)$, then, by the law of total probability, we have that $$P(X(U) = k \mid X(0) = 1) = \int_0^\infty P(X(u) = k \mid X(0) = 1) g_U(u) \ ...
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Are the Beta and Bernoulli Processes Normalized Random Measures with Independent Increments?

James et al. 2009 introduced the notion of "Normalized Random Measures with Independent Increments." Do the Beta process and Bernoulli process belong to this family?
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Two-stage stochastic linear optimization

I am familiar with the notion of two-stage stochastic optimization but I have not found any constructive examples so far, so I am stuck now on how to actually implement this on a given problem. The ...
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Find the correlation function of stochastic process given differential equations

Assume two systems for which the following differential equations hold between their input and output signals. $$a \dfrac{dv(t)}{dt}+b v(t)=x(t)$$ $$\dfrac{dy(t)}{dt}=v(t)u(t)$$ Also, assume that the ...
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Linear Combination of stochastic processes is also a stochastic process?

I have a general question about stochastic processes. Is the linear combination of any number of stochastic processes, also a valid stochastic process? What about non-linear combinations? Any book/...
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How can I verify that my solution is correct when finding the PDF and CDF of a random process?

If I have $X(t) = \sin(\omega t + \theta)$ where $\omega$ is constant and $\theta$ ~ $U[0,2\pi]$, and I need to find the PDF and CDF of $X(t)$, then how can I confirm my solutions for the PDF and CDF ...
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Where can I learn the math behind how airlines dynamically price tickets?

It's easy to think of factors that airlines take into account, and how the price will vary over time. There are no shortage of articles or medium posts about that. But how can I frame this as a ...
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AR(1) - Stationarity condition

Consider the well-known AR(1) model: $$x_t = \phi X_{t-1} + \epsilon_t$$ where, as usual, $\epsilon$ is an independent white noise process. I have read many sources. All of them get away saying that ...
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How do I find a frequency of a sequence in a markov chain?

I have a 4x4 markov chain transition matrix T. Lets call the states A,B,C,D. I am looking for the large time expected frequency of a given sequence, say A-B-B-D. How exactly am I supposed to find this?...
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How do I calculate the probability that some system will survive for some years before some event will occur?

I have a failure rate problem involving structural beams. The scenario is that a set of $n$ structural beams are supporting a weight of $W$ kilograms. For the purpose of simplification, we can assume ...
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Probability $X_n=k$ for a Markov Chain over the integers

Let $\{X_n\}$ be a Markov chain with state space $\mathcal{S}=\mathbb{Z}$, and $X_0=0,$ and $p_{0,1}=p_{0,-1}=1/2,$ and $p_{i,i+1}=1$ for all $i \ge 1,$ and $p_{i,i-1}=1$ for all $i\le -1.$ The ...
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Modified Renewal Process

I'm having trouble understanding how to deal with the modification to the queue. My thoughts were to use Poisson thinning to generate the infinitesimal matrix but that seems like a long shot. Any ...
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Fisher information of an Ornstein-Uhlenbeck process

I would like to compute the Fisher information of an Ornstein-Uhlenbeck process $X_t = Y_t - \beta Z_t$ where $Y_t$ and $Z_t$ are two time-series. My log-likelihood function in this case is: $$\...
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How are differential equations and stochastic differential equations different?

In the simplest terms, how are differential equations and stochastic differential equations different? As far as I can tell, SDEs are PDEs or ODEs, where the derivative of some function wrt itself is ...
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Cental limit theorem, Chebyshev's inequality, and convergence of distributions through rescaling

I've been thinking about this issue for a few days and although read some of relevant questions on this site, still couldn't get it off my mind. Suppose we have $n$ i.i.d random variables $X_i$ with ...
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Area under Gaussian Process Regression for an interval?

For no practical reason whatsoever, simply curious: Say you need to use Gaussian Process Regression to model some $x,y$ relationship. Now, suppose that you're interested in integrating over some ...
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Additive forecast errors across uneven time steps?

Assume that I have some stochastic dynamical model of the form: $$x_{t+1}=M(x_t)+\epsilon$$ where $M:x\rightarrow x$ is a deterministic function, the subscript $t$ denotes a certain time step, and $\...
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Why is sampling 50% of observations in stochastic gradient boosting equivalent to bootstrap sampling?

In the stochastic gradient boosting paper, Friedman (2002) writes that sampling half of the observations before each iteration is "roughly equivalent to drawing bootstrap samples at each ...
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Predicting stationarity of a time series

I have a time series, for example equity prices. This series is weakly stationary (for instance an AR process that does not violate the stability condition). How can I make predictions on how likely ...
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Plain English explanation of Ito's integral?

I'm looking for a plain English explanation of Ito's integral. I don't need an exhaustive proof, derivation, etc. Just a simple ~this is effectively what it does and why it's better than a Riemann sum ...
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When does a stationary distribution for a CTMC not exist?

Recalling that a Continuous-time Markov Chain (CTMC) is defined by its generator matrix $Q$, when does a stationary distribution for a CTMC not exist? Or when does there not exist a probability vector ...
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Nonparameteric Empirical Estimator For Stochastic Process

Motivation: If $X$ is a random-variable defined on some probability space $(\Omega,\Sigma,\mathbb{P})$ then Glivenko-Cantelli lemma guarantees that the empirical distribution $\frac1{N}\sum_{n=1}^N \...
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Are there good examples of martingale processes that are not simple random walks?

Are there non-trivial examples of martingale processes that aren't simple random walks? I'm trying to better understand the difference between martingales and simple random walks. They look pretty ...
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Does the convergence rate never increase of a Stationary Ergodic Random Processes under sub-sampling?

Summarize the problem Given A Stationary Random Processes (strict sense) $X_i$ I define two Stationary Ergodic Random Processes by $$ \bar{X}_n = \frac{1}{n} \sum_{i=0}^{n-1} X_i \ \ \text{and} \ \ \...
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Dirichlet Process vs Gaussian Process?

I'm studying the Dirichlet Process (DP) and looking to the Gaussian Process (GP), which I've had more experience with, to help make connections. The GP can receive $X_{train}$, $X_{test}$, and $Y_{...
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Stopping Time for a sequence of Random Variables

If $M$ and $N$ are stopping times of sequence $\{X_n\}$ where $n \geq 1$. Then are $min (M, N)$ and $max(M,N)$ also stopping times of the sequence $\{X_n\}$ ? Is there any rigorous way to prove ...
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How to model a stochastic process from data?

For a detailed background of the problem I am solving, please click here. In summary, the aggregate losses of 34 companies across a country were gathered. These losses were sampled spatially across a ...
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Determine the time series process behind many different realizations

We have an unknown generator process. This could span from the simplest case of a Gaussian random walk to the most intricated and convoluted models you can think of. However, we don't know which is ...
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What does it mean to perform backprop “through the operations of an SDE solver”?

I am reading this cool paper about Neural SDEs as GANs. I've gotten through all of it and I understand fairly well. I've taken a couple classes on SDEs so I'm comfortable with the math. What I don't ...
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Is this continuous-state example correct?

When defining stochastic process, I am given the following example: Customers arrive one at a time to a service facility and queue and wait for attention by the one server who attends to each ...
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R Determine similarity of multiple time series of stochastic events

I am using R to perform my analyses. I have five datasets: 365 rainfall measurements in a year; 4 x 365 height values in runoff drains located close to each other in the same year. The graph plots of ...
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Estimating stochastic volatility shock for TFP

I am trying to estimate a stochastic volatility shock for Total Factor Productivity (TFP) in a similar way to Fernandez-Villaverde and Rubio-Ramírez (2010) and Fernandez-Villaverde et al. (2011). $$...
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Expect hitting time of a discrete time random walk with complex step size distribution

Suppose a random walk starts from $S_0=0$. The iterative equation is $$S_{t+1}=\max\{S_t+y_{t+1}-k,0\},$$ where $k$ is a fixed value that is larger than 1, and $y_t$, $t=1,2,\cdots$, are i.i.d. and $$...
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Limit distribution for a linear discrete-time stochastic process: limit of the sum of linearly transformed uniform distributions

I have posted this in math stack exchange, but I figured maybe this is a better forum for this kind of question. Suppose we have a stochastic linear process: $$x_{k+1} = Ax_{k} + Bw_{k} \qquad \text{...
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What are examples of statistical experiments that allow the calculation of the golden ratio?

There are some very simple experiences that can be done by a kid at home, whose result allows one to statistically approach famous numbers such as $\pi$ or $e$. An example where $\pi$ shows up is ...

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