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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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Why does the correlation function of this stochastic differential equation starts at different points?

I am working with the following differential equation: The equation is $$x=\beta +\sqrt{2D} \xi(t)$$ where $\xi(t)$ is a white noise term, with a reflecting wall boundary conditions. After solving ...
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Popular stochastic model for behavior of staying at the same position?

I am looking for a popular stochastic model employed for a trajectory of a fish which tries to keep staying at the initial position against water pressure from time-varying directions. The trivial ...
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Behavior of Markov chain formed from n dice rolls [on hold]

[![I am really interested in knowing the behavior of the Markov Chain {Yn} ][1]][1] [1]: https://i.stack.imgur.com/o5ew9.png Also what will be the limiting probability of P(Yn=0).
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24 views

limit and stationary distribution of a Markov chain

Consider a Markov chain on the non-negative integers with transition probabilities 􏰀$1/2$ if $y=x+1$ and $1/2$ if $y=0$. Find $\lim_{n \to \infty} P(X_{n}=0)$. Is this limit the same as the ...
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25 views

probabilities related to a transient single-server queue

Consider an $N = 1$ server queue with arrival rate $\lambda > 0$ and service rate $\mu = 1$. If the process is transient, find $\rho{_{x0}}$ for $x ≥ 1$. My attempt: The process is transient if $\...
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12 views

Autocorrelation of stochastic process with python

So I am trying to simulate a SDE and find the corresponding correlation function. The equation is $$x=\beta +\sqrt{2D} \xi(t)$$ where $\xi(t)$ is a white noise term. After solving it using Euler-...
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15 views

Correspondence between time series models in continuous vs. discrete time

I am interested in an overview over the connection and correspondence between time series models in continuous vs. discrete time in finance. E.g. take ARMA(p,q) or GARCH(s,r) or ARMA(p,q)-GARCH(s,r) ...
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24 views

Is this problem worked out correctly? [closed]

Calls are received at a company call center according to a Poisson process at the rate of five calls per minute. (a) Find the probability that no call occurs over a 30-second period. (b) Find the ...
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19 views

Compute the Limiting Distribution

Consider the transition matrix $ P = \begin{bmatrix} 1-p&p\\ q&1-q \end{bmatrix} $ for general $2$-state Markov Chain $(0 \le p, q\le 1)$. Find the limiting distribution ...
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15 views

Expectation of arrival times

Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth ...
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Stochastic process $X_t=\varepsilon_t\varepsilon_{t-1}$ is not white noise, where $\varepsilon_t \stackrel{iid}{\sim} N(0,\sigma^2)$

Let $\{\varepsilon_t\}$ satisfy $\varepsilon\stackrel{iid}{\sim} N(0,\sigma^2)$ Let $X_t$ be defined as $$X_t=\varepsilon_t\varepsilon_{t-1}$$ Is $\{X_t\}$ stationary? Is it white noise? ...
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22 views

How can I compute expected return time of a state in a Markov Chain?

I was watching a YouTube video regarding the calculation of expected return time of a Markov Chain. https://www.youtube.com/watch?v=X_Ll0-Ytu7U&vl=en I haven't understood the calculation of $m_{...
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44 views

What is the difference between time, arrival-time, and inter-arrival-time is poisson process?

Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth ...
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1answer
73 views

Finding the mean and variance of an infinite server queue

I am presented with the following homework problem: Let $X(t)$, $t > 0$, be the infinite server queue and suppose that initially there are $x$ customers present. Compute the mean and variance of $...
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1answer
15 views

Deriving expression for expected offspring in branching process

I am looking at branching processes in Dobrow 2016 (p. 160), where the author states that the "mean of the offspring distribution" is $\mu =\sum_{k=0}^{\infty} k a_k$. I want to know why the ...
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30 views

Distance between point process realizations

Is there a valid distance metric for measuring how similar are the realisations of two point processes? E.g. let's say we simulate two histories $ h_1, h_2 $ in the time interval $ [0, T] $ for two ...
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139 views

Continuous time Fourier representation

I have learned that the Fourier transform of a continuous-time unit-periodic stochastic process is: $$x(t) = \sum\limits_{k=-\infty}^{\infty} a_k e^{i2\pi kt} \quad \quad \text{ where } \quad \quad ...
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47 views

Why is the best prediction for a gaussian process a linear prediction?

I want to understand why, for Gaussian processes, the best prediction is linear. I do not understand its proof. For Gaussian process $X(t), t\in I $ the best prediction is linear. Proof: We only ...
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16 views

Does this decomposition theorem of stationary process exist?

I have this vague impression that there was a theorem on Wikipedia about stationary process, saying that if $(x_k)$ is a strictly stationary process, then there exists a decomposition in the form of $...
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1answer
27 views

Custom Space State model using DLM in R

DLM package in R can model linear space state models of the form: I have a different category of equation which is also a linear polynomial equation of order 1 with constant coefficients. I would ...
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15 views

Second-Order stationarity condition for complex-valued autoregressive process

Let $\{c_n\}$ be a complex-valued discrete autoregressive process of order $p$, $\mathsf{AR}(p)$, such that: \begin{equation} \label{cn} c_n = \sum\nolimits_{i=1}^{p}\rho_i c_{n-i} + w_n, \quad n \in (...
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1answer
35 views

Mean of $X_t = \epsilon_t\epsilon_{t-1}$

Consider the following stochastic process: $$X_t = \epsilon_t\epsilon_{t-1},~~~~~~~~~\epsilon_t \sim N(0, \sigma^2)$$ Determine whether the process is covariance-stationary, strictly ...
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53 views

Learning a Gaussian Process from function observations (not GP regression)

Suppose we have a set of observations, where each observation represents a function. For example, our set is $\{f_1, f_2, ..., f_n\}$ where each $f_i = \{(x_1, y_1), (x_2, y_2), ..., (x_{p_i}, y_{p_i})...
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How to model count data with decay

I'm trying to understand how I might model count data where there's diminishing marginal utility and a stochastic process. So, let's say we're modeling the number of "useful intelligence tips" given ...
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33 views

Prove limiting distribution goes to stationary distribution: $\lim_{t\to\infty} \pi_{j}(t) = \overline{\pi}_{j}$

This is a problem I'm struggling with on continuous-time Markov chains. Here, we are considering a continuous Markov process with phase space $\{1, 2\}$ (there are only two states). Moreover, $\...
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14 views

Simulating a (discretized) Cox process via binomial sampling

Let X be a Cox process (doubly-stochastic Poisson process) with fixed intensity(rate) $\lambda=50$ , and choose some small time interval $dt=0.01$ . Is the proper way to simulate this, by letting Y ...
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1answer
46 views

Select optimal points for Gaussian process with a well-known target function

I'm currently trying to select the optimal points for a Gaussian Process Regression, and the important thing is that i already know the whole target function. Therefore, it's not Online Learning ...
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6 views

Combined uncertainity given probability measurement belongs to one of two classes

I have two classes of measurements $A$ and $B$ characterized by different uncertainty distributions. I also have a probability $P_A$ that a given measurement belongs to class A and not B ($P_A = 1 - ...
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67 views

Sum of N random variables from the same distributions [duplicate]

Given $n$ independent random variables from the same distribution, how to obtain the distribution of their sum? For example, the distribution of $n$ normal distribution is $N(n\mu, n\delta^2)$. ...
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22 views

Average and Variance of the Rate of Change in a Continuous Variance

I have a continuous variable, $P_t$ whose evolution is unknown. However, I obtain a history of it i.e. $P_0, P_{dt}, P_{2dt}, ...... , P_T$. For a continuous process variable, I know that the rate of ...
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Recommendations for Data Fitting Mean Reversion Processes

I have a time-series with clear mean-reverting properties over some time-scale. I have a very long measurement of this series, so can see that, whilst it always reverts to a fixed mean, the ...
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16 views

Difference between a stationary Stochastic process and a stochastic process with stationary increments

Could someone please explain to me the difference between a stochastic process which has stationary increments and a stationary stochastic process. As far as I have understood, if a stochastic ...
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48 views

Distribution of a one realization of a stochastic process [closed]

Suppose $X$ is a stochastic process such that $X(t) \sim N\left(\mu(t), \sigma^2(t)\right)$ for all $t$ and $\mu$ and $\sigma$ are some smooth functions and we are given one realization of this ...
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Efficient simulation for distribution of time until coin toss pattern

Suppose we flip a biased coin (with probability $p$ being Heads) repeatedly until a certain pattern (e.g., HHHTT) appears. We are interested in the number of flips $N$ required. It is well-known that $...
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Downsample a stochastic process without losing correlation statistics?

I have a stochastic variable $X(t)$ which changes at a discrete set of random times $t_1, t_2, \dots$. I can simulate this stochastic process to obtain a series $X(t_1), X(t_2),\dots$ However, the ...
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38 views

How to find the distance confidence between the sum of random variables and a given target value?

Given a distribution $X$($E[X]$, $Var[X]$) and a target value T, I am wondering whether there is a bound on $P(|\sum_{i=1}^{i=N}X_i - T| < \epsilon)$. For example, $P(|\sum_{i=1}^{i=N}X_i - T| &...
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Nested CV with Online Learning

I have a time series binary classification dataset. I am implementing an online learning Logistic Regression algorithm in Sklearn and am cross validating with Sklearn's TimeSeriesSplit method. I am ...
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9 views

What are some easy to understand books for discrete stochastic process simulation using R?

What are some easy to understand books for discrete stochastic process simulation using R programming language? I mean for the starters?
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16 views

Random walks-heavy tailed case

Let $\beta > 0$ and $S_{0}=0$, and let $S_{n}=\xi_{1}+\dots+\xi_{n}$,$n \geq 1$, be a random walk with i.i.d. increments $\{\xi_{n}\}$ having a common distribution $P(\xi_{1}=-1)=1-C_{\beta}$ and $...
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1answer
39 views

Combining probabilities to find most probable window

I have a series of observations, with an associated probability that an event is occuring at timestep t, something like: [0.8, 0.8, 0.3, 0.9, 0.2] Events can ...
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55 views

How do I write a random walk with drift ARIMA? [closed]

I modeled oil prices and got the following coeffecients for my arima model with drift. Is this the right way of writing the model?
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22 views

Do measurable maps preserve stationary ergodicity?

In a recent effort to establish stationary ergodicity for a certain stochastic process, I just happened to come across a statement, which I find to be little bit confounding. Given two measurable ...
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1answer
25 views

How to recognize an ARMA process?

By looking at the autocovariance, how could you recognise what discrete model (MA(q), AR(p), or ARMA(p,q)) is more appropriate to describe your data?
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Approximating AR(1) by finite order MA process - convergence results

I am currently struggling with a result pertaining to the finite order MA approximation of a simple AR$\,(\,1\,)$ process defined on a double sided time-index set $\,T=\mathbb{Z}$. I would be very ...
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13 views

How to determine the rate of input in a queue M/M/c?

I know the exit rate ($\mu$) and the average waiting time in the queue ($W_q$). I need solve to rate of input ($\lambda$) in a queue. $\rho = \frac{\lambda}{c\mu} < 1$ $\pi_0 = \left[\left(\sum_{...
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14 views

Brownian Motion proof: difference converging to 0 almost surely

I am reading a proof where it is assumed that $$ \lim_{n \to \infty} \sup_{0<s\leq s_0}\left| \frac{t_n(s)}{s}-1 \right|=0 , \hspace{30mm} (1)$$ where $t_n(.)$ is some sequence of functions. ...
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1answer
53 views

Irreducible (communicating) classes [closed]

The Markov chain $(Xn; n\geq)$ has state-space $S = (0, 1, 2, . . .)$, with $p_{i,0} = \frac{1}{4}$ and $p_{i,i+1} = \frac{3}{4}$ $\forall i \geq 0$, so that the transition matrix is P =$\...
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1answer
58 views

Biased coins and Markov processes

Good day, I am attempting an optional exercise and I am finding it hard to interpret the problem in terms of matrices and vectors. Coin 1 has probability 0.4 of coming up heads, and coin 2 has ...
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17 views

How to prove that $X_t=\int^t_0f(u)dW_u$ and $X_t-X_s$ are independent?

Let $X_t=\int^t_0f(u)dW_u$ for a deterministic function $f$ and $W_t$ is a brownian motion. How can I compute $E[\exp(\lambda_1 X_s + \lambda_2(X_t-X_s))]$ and prove that $X_s$ and $X_t-X_s$ are ...
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39 views

Stochastic Differential equation: CAPM

Let $R = (R_1, \dots , R_M)'$ denote a vector of excess returns of $M$ assets observed at $n$ time points, $0 < t_1 < t_2 < \cdots < t_n < T$, within a time span $T > 0$. We wish ...