Questions tagged [stochastic-processes]

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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Bernoulli process with nonstationary probability

Say we have a process $X_t\vert P_t\sim \mathrm{Bin}(n,P_t)$ where $X_t$ is observable but $P_t$ is not. Also, the success probability $P_t$ might vary over time and I don't assume some ...
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If $X(t)=A\sin(\omega t + \theta)$ with $\theta\sim U[0,2\pi]$ then is $f(X(t))$ jointly WSS with $X(t)$ for a function $f(z)$?

Suppose a continuous time process $X(t)=A\sin(\omega t + \theta)$ with $A$, $\omega$ fixed and $\theta\sim uniform[0,2\pi]$. It is easy to see that this is a strict sense stationary process. Now, let ...
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Car Driving Behavior: using non-instant transition with a Markov Transition Matrix

I have a dataset on driving behavior/trips (workable example below), with information on the departure location, arrival location, duration of the trip and length of the trip. The states of the ...
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artificial neural network

ann.compile(optimizer='adam', loss='binary_crossentropy', metrics=['accuracy']) ann.fit(x_train,y_train, batch_size=32, epochs=100) this is few lines of my code my questions is adam optimizer works ...
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Hurst estimation in small samples

I'm trying to estimate the Hurst exponent of a time series which I believe behaves as a fractional Brownian motion. My problem is that all the estimation methods I have found so far (r/s, Whittle, etc....
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Struggling to understand the difference between a DTMC and a CTMC

My textbook, Modeling and Analysis of Stochastic Systems, third edition, by Kulkarni, introduces continuous-time Markov chains (CTMCs) as follows: In Chapters 2, 3, and 4 we studied DTMCs. They ...
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How to simulate the supremum of a Gaussian Process

I have a problem where I need to estimate the quantiles of the supremum of a Gaussian Process certain point $t_0$ in time. This should be achieved by simulations. I have a centered Gaussian Process $...
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Poisson counting process subinterval distribution

Suppose $N(t)$ is a homogeneous Poisson counting process with a constant parameter $\lambda$. Given positive real numbers $T$ and $\tau$, and non-negative integer $n$, what is the probability that $N(...
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probabilty , events , stochastic process [closed]

Problem 1 Suppose that the universal set S is defined as S = {1, 2,⋯, 10} and A = {1, 2, 3}, B = {X ∈ S : 2 ≤ X ≤ 7}, and C = {7, 8, 9, 10}. a. Find A ∪ B. b. Find (A ∪ C) − B. c. Find A¯ ∪ (B − C). d....
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Minimum variance hedge ratio - returns or differences for Monte Carlo

So If I want to simulate for two stocks a monte carlo simulation since I want to show that in the end a portfolio with the minimum variance hedge ratio has the lowest standard deviation do I use as ...
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Minimum Variance Hedge Ratio for Prices and Returns

So from my understanding Hull (2012) f.e. shows that the optimal hedge ratio minimizes the variance of the returns. But what happens to the variance of the prices? Is the Minimum variance hedge ...
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Number of Susceptible individuals in stochastic SIR ICM

I am using SIR-ICM to model an outbreak in large population (around 10 million). I want to know the size of susceptible individuals to use in this model. I am in a doubt to use the actual population ...
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Variance of the autocovariance function at a particular lag

For a uniform random number(u) having mean $\mu$, the autocovariance at a lag $\tau$ is given by $$C(\tau)=\frac{1}{N-\tau} \sum_{i=0}^{n-\tau} (u_i-\mu)(u_{i+\tau}-\mu)$$ For the uniformly ...
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How do I find the common factor from a stochastic process ARMA model

How would you factorise this ? There should be a common factor between the Lag polynomial of the auto regressing moving average A(L) and the for the lag polynomial moving average B(L). It should be ...
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How does polynomial trend get 'detrended' by the d-differencing?

Suppose $X_t$ is an ARIMA(p, d, q) process, then so is $X_t + m(t)$ where $$ m(t) =a_0 + a_1t +...+ a_{d-1}t^{d-1} $$ is some polynomial of degree $d-1$. How does such a polynomial trend get '...
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Impact of correlation bounds for Monte Carlo simulations

As the lognormal distribution imposes bounds of attainable correlations as discussed in Attainable correlations for lognormal random variables my question would be what happens if say we want to do a ...
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Monte Carlo Error for Minimum Variance Hedge Simulation

So I was running a monte carlo simulation for two assets and a portfolio consisting of 1 quantity of the first asset and short a fraction x of the second asset to hedge, where the fraction is ...
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Equal Limiting Distributions in a Markov Chain Implications

I am studying Stochastic Models, and I came across the concept of limiting distributions, and I just wanted to make sure that I have understood it correctly. If the limiting probabilities are equal ...
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Comparing Two Poisson Processes

I'm trying to solve this problem: Regional and international planes arrive at an airport following independent Poisson processes with rates $\lambda$ and $\mu$, respectively. Each regional plane has ...
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Does the unconditional mean of a non stationary ARMA process exists?

Assume that we are dealing with an $ARMA(1,1)$ model: $$ y_{t} = \theta y_{t-1} + \epsilon_{t} + \alpha \epsilon_{t-1} $$ where $$ \epsilon_{t} \sim WN(0, \sigma^{2}) $$ Then, we can rewrite the model ...
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Show that this expression is a Poisson Process

How can I show that a Poisson process can be represented by $N(t) = \sum_{i=1}^{\infty} \mathbb{1}(T_{i} \leq t)$ . Note that $T_{i} = \sum_{j=1}^{i}X_{j}$ where $X_{j}$ is the inter-arrival time of ...
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how to model random changes in line segments

I have a discrete time stochastic process where an interval from 0 to L consists of smaller sub-segments, where the boundaries are always at integers, and L is some large integer. At each iteration, a ...
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why independence assumption not applicable on time series analysis?

I just started my course in time series analysis. I saw a statement there: "Statistical methods that depend on independence assumption are no longer applicable in time series analysis". Why it so?
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Autocovariance lag dependence imply variance finite?

Suppose {$X_t,t \in T$} Gaussian Process, such that: $E(X_t)=\mu$, where $\mu$ constant $\forall t \in T$ (constant mean) $\gamma_X(X_s,X_t)=\gamma_X(X_{s+h},X_{t+h})$ $\forall t,s \in T$ and $h \in \...
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Is *conditional* stationarity (of a stochastic process) a useful concept?

Typical stationarity is defined in terms of some notion of invariance of a stochastic process w.r.t. a time shift operator. I think you could talk about conditional invariance: $ P(X_{t_1}, ... X_{...
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Where is the “Markov property” in Markov Random Fields? Are Markov Random Fields stochastic processes at all?

I am learning about Markom Random Fields. As I understand them they are constituted of some Random variables $X_{i}$ and some potential functions $\phi(X_s)$ where $X_s$ is a set of some random ...
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Is it possible to combine SGD/ Minibatch GD and Batch Gradient Descent to find the global optimum quickly?

Say the model is bouncing around the optimum with SGD. Is there a way to know that it's near the minimum, pause the model, and continue with an extremely small learning rate
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Average of the product of two autocorrelated variables

I am not expert in statistics, so I hope this question doesn't sound too trivial. I have a discrete random variable that evolves in discrete time, . The stochastic process behind this variable has ...
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Don't understand one particular normal distribution notation from opaper Stochastic Variational Deep Kernel Learning, help needed

I'm in progress working with paper "Stochastic Variational Deep Kernel Learning" NIPS 2016 and I have the problem with understanding the meaning of this normal distribution notation from part 2 ...
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How to estimate model coefficients for option-like response: ie. response is related to variables in the form y = max( formula, 0)

Rewrite Hopefully someone can point me to a resource on how to estimate the parameters I'm trying to model. I've had trouble giving a title to my question and googling for resources. Suppose $y$ ...
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Intuitive explanation of Gaussian Process Regression

How would you intuitively explain the idea behind Gaussian Process Regression to someone unfamiliar with stochastic processes? Especially the point where you discuss modeling covariance, choice of ...
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Method to generate 2D arrays with specified autocorrelation?

Suppose I have a specified value of autocorrelation (say 0.9) at lag 1. Is there a way to generate a random 2D (NxN) array which will have this particular autocorrelation value at the same lag?
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Renewal processes, interarrival time distributions

I am dealing with renewal processes recently and I have some questions and I hope you can help me :). Why interarrival time distributions need to be independent in a renewal process?
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Renewal processes simulation using Monte Carlo

I am dealing with renewal processes recently and I have some questions and I hope you can help me :). I'll post each question separately. Is this a valid way to simulate a renewal process using Monte ...
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Poisson process, combined waiting time

Passengers arrive at a railway station according to a homogeneous Poisson process with rate λ. At the beginning (time 0), there are no passengers at the station. The train departs at time T. Denote by ...
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Renewal processes, relationship between distribution of the events amplitudes and the interarrival times

One of the basic assumptions of a renewal process seems to be that the distributions of the interarrival times are iid. However, this makes me think if there must exist a relationship between the ...
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k-nearest neighbor kernel density estimation for a an unknown stochastic process?

Suppose we have a sequence of random variables $\{X_t:t=1,\cdots,T\}$ following an unknown stochastic process (possibly stationary or non-stationary). Now I have two questions: 1- Would it be ...
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Can a random variable be expressed as a sum of deterministic and random variable?

Say we have a sequence of random variables $\{X_t:t\geq 0\}$ following an unknown stochastic process with distribution $X_t\sim N(\mu_X,\sigma_X^2)$. This idea came to me from the additive noise model....
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Linking Correlated Dependent Variable with Independent Variable

I have a Monte Carlo model that generates a distribution of possibilities $X_i$ for the non-normal stochastic process $Z$ it describes. The distribution of $X$ and $Z$ is fat tailed but for the most ...
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Parameter estimation for time-varying autoregressive processes in R

I want to estimate the parameters of an autoregressive process with time-dependent coefficients. For example TVAR(1) model with 1 lag: $$ X_t = \phi_t X_{t-1} + \sigma_tW_t $$ where $\phi_t$ and $\...
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Extending normalized probability generating function (pgf) in branching process

My question is in the context of branching processes and simply how to extend a normalized negative binomial probability generating function (pgf) from $\frac{1}{y}[G(s)]^y$ to $\frac{n}{y}[G(s)]^y$ ...
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Can we express the following unconditional probability as follows?

Some of you may be aware that I have been asking a nagging question for quite a while on this forum, in different shapes and forms. Although I may have been a nuisance, may I thank you as this has ...
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How to estimate a single AR model for multiple time series?

I have ACF of a time series data, I want to interpolate the ACF using AR model for which I need to calculate the coefficient of the AR model The problem is I have (6000x6000) matrix of ACF and I cant ...
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AR(1) with known initial and terminal condition: how to draw the innovations?

Suppose I have the following stationary $AR(1)$ process: $$ y_{t}=\alpha_{0}+\alpha_{1}y_{t-1} + u_{t} $$ where $u_{t} \sim \mathbb{N}(0,\sigma^{2})$, with $\sigma^{2}$ known. Suppose I have an ...
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The covariance function of a stochastic process is positive semidefinite

Let $\{X_t, t \in \mathbb{Z}\}$ a real-valued stochastic process and $\gamma : \mathbb{Z} \times \mathbb{Z} \to \mathbb{R}$ the autocovariance function. I would like to show it is a positive ...
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The question is related to stochastic process and the Markov chains

This is the complete question. Jim is currently living in Scranton. Each year that he lives in Scranton, he has a probability of 1/2 of staying in Scranton the next year. Otherwise, he has an equally ...
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In predictive regressions, can we assume that the predictor follows the Ornstein-Uhlenbeck process?

A predictive regression is a regression of the form \begin{equation} y_t=\beta x_{t-1}+\varepsilon_t \end{equation} where $x_{t-1}$ is generally assumed to be a highly persistent stochastic variable, ...
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Is a grid-like network more efficient than a random network?

Imagine two different types of street networks with the same number of nodes and where edges are weighted according to their length. Both network types cover the same geographic area (say 1km²) and ...
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Can you find the dynamic unconditional mean and variance of a process without assuming a model for the DGP?

Let us say $\{X_t:t \geq 0\}$ is a weakly stationary stochastic process with initial condition $X_0$. Suppose we know that the process is normally distributed, but we wish to estimate the dynamic mean ...
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Probability distributions associated to the logarithm numeration system

The most elementary logarithmic numeration system is defined as follow. Any random number $X \in [0, 1]$ can be represented uniquely as $$X=\log_3(A_1 + \log_3(A_2+\log_3 (A_3 + \cdots)))$$ with $A_k \...

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