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Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Lovasz-Local-Lemma, question in the proof (independence of complements)

In the common proof of the Lovasz-Local-Lemma (e.g. from Wikipedia), when bounding the numerator, there is always this equality $$P(A \vert \bigcap_{B \in S_{2}} \bar{B}) = P(A)$$, basically saying that $A$ and $\bigcap_{B \in S_{2}} \bar{B}$ are…
Locust
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Realizations of stochastic processes

Wanted to ask something that might be a bit obvious. Is there any formal way to reject the hypothesis that two given timevseries are realizations of the same stochastic process? Thanks
Panagiotis Kyriakis
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Are all cadlag processes of finite variation?

I have the statement from the book: "As nondecreasing functions have left limits, a right continuous nondecreasing process is cadlag. Therefore, it is clear that $W^ +{\subset}W$, where $W^+$ is the set of nondecreasing processes and $W$ is the set…
c-walk
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Example of a reversible Markov chain which has a stationary but non-reversible distribution?

For a Markov chain, I define a reversible distribution to be a distribution wrt which the MC is reversible to. A stationary distribution is defined as a distribution that once reached will stay. A reversible distribution is a stationary…
Tim
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Does Strict Sense Stationarity imply Weak Sense Stationarity?

For a stochastic process, does being Strict Sense Stationary (SSS) imply being Weak Sense Stationary (WSS) since WSS process is easier to fulfill?
kubicwerke
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What condition can make every initial distribution have a limit distribution?

In a Markov chain (you can add additional conditions here, such as discrete-time, homogeneous, finite-state, .... But the less additional condition, the better ), what sufficient and/or necessary condition can make every initial distribution have a…
Tim
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Two stochastically continuous processes with the same finite dimensional distribution on a dense setvhave the same fdd everywhere?

Two stochastically continuous processes on $[0,T]$ with the same finite dimensional distribution on a dense subset of $[0,T]$ have the same finite dimensional everywhere? The processes live on different spaces. I suspect that this is true since…
user3503589
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Initial distribution under which the distribution of $X_t$ doesn't converge?

In a Markov chain $(X_t)$, is it possible to find an initial distribution for $X_0$, s.t. the distribution of $X_t$ doesn't converge in some sense (such as wrt total variation) as $t \to \infty$? The MC can be continuous or discrete-time, finite or…
Tim
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Derivative of Branching Process with nonzero extinction probability

I need to prove that for any Branching Process with $\mu > 1$ and extinction probability $a < 1$ with generating function $\phi$ , that $$\phi '(a) < 1 $$ With the assumption that $X_0 = 1$ I am having trouble with this problem as my knowledge of…
Nils F
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Intuition for definition of terminating renewal process

When the interarrival distribution F of an associated renewal process happens to have $F(\infty)<1$ (I'm assuming this to be the limit), we have a terminating renewal process. Why this name? what's the intuition for it?
An old man in the sea.
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proof of : $ E^Q \left[ Y|F(s) \right] = \frac{1}{Z(s)}E^P\left[ Y Z(t) | F(s) \right]$

To prove this Lemma : $$ E^Q \left[ Y|F(s) \right] = \frac{1}{Z(s)}E^P\left[ Y Z(t) | F(s) \right]$$ with $P$ and $Q$ two equivalent probability measures and $Z(t)$ is the expectation of the Radon Nikodym derivative $Z(t) = E^P…
JasBeck
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Probability of stopping at certain value for weighted random walk

Let $(X_i)_{n\in\mathbb{N}}$ be independent with $\mathbb{P}(X_i=1)=p=1-\mathbb{P}(X_i=-1)$, $i\in\mathbb{N}$.Let $a,b\in\mathbb{N}$. For $n\in\mathbb{N}$, let $S_n=\sum_{i=1}^nX_i$ and finally $\tau$ be the first hitting time of $\{ -a,b\}$ by…
amars
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Relation between reversible distribution and limiting distribution?

For a discrete time Markov chain, its limiting distribution is defined to be the same for all the initial distributions. A distribution over the state space is called a reversible distribution, if it satisfies the detailed balance equation. I think…
Tim
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Definition of $\rho$-mixing and its relation to strong mixing?

From Wikipedia: Suppose $\{X_t\}$ is a stationary Markov process, with stationary distribution $Q$. Denote $L²(Q)$ the space of Borel-measurable functions that are square-integrable with respect to measure $Q$. Also let $ℰ_tϕ(x) =…
Tim
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Is the Ornstein–Uhlenbeck process stationary in the wide-sense?

This post shows that OU process is AR(1) empirically. It looks like the eigenvalues of the coefficient are all less that one i.e. the process is wide-sense stationary. I find this a bit conterintuitive because OU process clearly has a strong trend…
user1559897
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