# 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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### How to derive transition matrix in this stochastic process?

I am new to stochastic processes and trying to solve a question related to finding a transition matrix of some experiment. The question is a A sequence of experiments is performed, in each of which ...
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### Show that $\{(X_{n},X_{n+1})\}_{n\geq 0}$ is a Markov Chain where $\{X_{n}\}_{n\geq 0}$ is a Markov Chain

Show that $\{(X_{n},X_{n+1})\}_{n\geq 0}$ is a Markov Chain where $\{X_{n}\}_{n\geq 0}$ is a Markov Chain. Remark: We know that $\mathbb{P}(A|\emptyset)$ is undefined, I am right? This fact is ...
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### Poisson process - time of arrival of a client, given that one client arrived at first interval

I have the following situation: I have a Poisson Process with $λ=7$ (seven customers / hour). This process describes the arrival of customers in a store. The store is open from 9:00 AM to 19:00 PM. My ...
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### Strict Sense Cyclostationary and Shifting the X with $\theta$

Fellow stackexchangers, I did my best to put a topic that describes the question that I am going to ask. I am reading Probability, Random Variables, and Stochastic Processes by Papoulis and I am ...
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### Can a stochastic process be decomposed as follows?

So for instance an AR(1) process \begin{equation} x_t=\rho x_{t-1}+u_t \end{equation} where, say, $u_t\sim IID(0,1)$ for $t\in \mathbb{N}$, can be expressed using backward iteration as follows: \begin{...
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### Renewal process vs inhomogeneous Poisson process?

I am analyzing a dataset with recurrent events, and considering two candidate models: A model based on a renewal process (time is measured from the previous event) A model based on an inhomogeneous ...
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### Strategic Multi Armed Bandit

As a part of my project, I have been tasked with formulating a multi-armed bandit problem with strategic arms. What I have found out is a Gittin's index approach to the problem provides a solution ...
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### Interpolation using Gaussian processes

This is about Gaussian process interpolations, where the given data are f(0) = 1, f(0.4) = 3 and f(1) = 2. Assume that the covariance function used is the exponential covariance, where the expectation ...
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### Estimate partially observed Poisson process

I try to estimate the intensity of a Poisson process $P_1$, but it is not fully observable. There are some "obervers" coming to the system which follow another Poisson process $P_2$. In ...
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### Time series - Stationarity and invertibility?

Sometimes when I take material from time series to study, it appears out of nowhere "for a process to be stationary it is necessary for the roots of the characteristic polynomial to fall outside ...
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### Stochastic Differential Equation with drift multiplied by everywhere discontinuous random process

(WARNING: this has been crossposted on physics.stackexchange (questions/588606). It has been suggested to post also here, let me know if it is against the rules). Let the stochastic process $\{X_t\}$ ...
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### Key elements of a Poisson Process? [duplicate]

I'm new to Stochastic Processes, in general. However, I see that they come in varying forms of complexity. A gaussian process (and gaussian process regression) are quite complicated and can easily fit ...
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### Converge of Scaled Bernoulli Random Process

Suppose a random sequence is defined by $X_n := n B_n$, where $B_n$ is a Bernoulli sequence such that $\mathbb{P}(B_n = 1) = 1/n$. I am interested in the convergence properties of this random process ...
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### What correlation structure is necessary to ensure a random walk is almost surely bounded?

Say I have a stochastic process $\{X_t\}_{t \in \mathbb{N}}$ such that their cumulative sum $\{S_t\}_{t \in \mathbb{N}}$ is a random walk process: $$S_t = \sum_{i = 1}^t X_i$$ If each $X_t$ is i.i.d ...
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### Expected number of running heads in coin toss

How to find the expected number of running heads of a specific length (say 'k' exactly) in 'n' tosses of a coin (fair/biased). For example, consider the output of a coin toss as follows "...
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### Are the following model assumptions on a data stream too restrictive?

Suppose that you were to model a "generic" continuous-time real-world data signal $X$ taking values in a bounded continuum $K\subset\mathbb{R}^d$ (e.g. the body temperature of a patient or ...
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### Is the MA($\infty$) process with i.i.d. noise strictly stationary?

I have a MA($\infty$) process defined by $$X_t = \sum_{k=0}^\infty \alpha_{k} \epsilon_{t-k}, \qquad t\in\mathbb{Z}$$ where the sums converge a.s. and the $\epsilon_t$ are i.i.d. centered noise ...
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### Is a weakly stationary AR(p) process also strictly stationary if the noise is i.i.d.?

An AR($p$) process is any causal and weakly stationary solution to the equations $$X_t = \beta_1 X_{t-1} + \dotsc + \beta_p X_{t-p} + \epsilon_t, \qquad t \in \mathbb{Z}$$ where the polynomial \$B(z)...
Is there a test that can distinguish the strictest form of the random walk, $$P_{t}=P_{t-1}+\varepsilon_{t}, \varepsilon_{t} \sim \mathrm{IID}\left(0, \sigma^{2}\right)$$ where each step is assumed to ...