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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Transformation on state-space that preserves Markov property

I am solving a problem in Mathematical Statistics by Jun Shao Let $\{X_n \}$ be a Markov chain. Show that if $g$ is a one-to-one Borel function, then $\{g(X_n )\}$ is also a Markov chain. ...
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Expected time to $k$-th occurence

I was trying to solve the following exercise: Suppose that the sequence of independent events $\{A_i\}$ satisfies $$\phi(n) = \sum_{i=1}^{n} P(A_i) \rightarrow \infty \text{ as } n \rightarrow ...
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Is this a martingale?

Let $W_t$ be a standard Brownian motion with $W_0 = 0$ and let $Z_t$ solve the stochastic differential equation $dZ_t = 2 Z_t W_t \mathrm{d}W_t$. This has solution $$ ...
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Given particle undergoing Geometric Brownian Motion, want to find formula for probability that max-min > z after n days

Consider a particle undergoing geometric brownian motion with drift $\mu$ and volatility $\sigma$ e.g. as in here. Let $W_t$ denote this geometric brownian motion with drift at time $t$. I am looking ...
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How do you check if a sequence of numbers is truly random? [duplicate]

Suppose a source produces an indefinite sequence of positive integers. How can you check whether the numbers are generated truly randomly?
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Proving that an unfair game is martingale

The answer is probably gonna be very straightforward but I'm missing it. I want to prove the following claim: Suppose we play a game where we win with probability $p$ and lose with ...
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712 views

Questions about geometric distribution

I have some trouble understanding the record value for a sequence of i.i.d. random variables of geometric distribution. Following quotation is from Univariate discrete distributions By Norman Lloyd ...
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The Laplace transform of the first hitting time of Brownian motion

Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is ...
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Questions about the concept of strong Markov property

I am trying to understand the concept of strong Markov property quoted from Wikipedia: Suppose that $X=(X_t:t\geq 0)$ is a stochastic process on a probability space ...
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Expected value for a Poisson Process

A machine works for an exponentially distributed time with rate μ and then fails. A repair crew checks the machine at times distributed according to a Poisson process with rate λ; if the machine is ...
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Correlated Brownian motion and Poisson process

Is there an easy way to construct, on the same filtered probability space, a Brownian motion $W$ and a Poisson process $N$, such that $W$ and $N$ are not independent ?
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The Poisson Distribution

Customers arrive at a casino as a Poisson process of rate 100 customers per hour. Upon arriving, each customer must flip a coin, and only those customers who flips heads actually enter the casino. ...
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Markov Chains - 2 clarification questions

I'm just getting started with Markov chains and have a few simple questions: Is it possible to define a period for a reducible Markov chain? If so, how? Can we define balance equations and a ...
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Poisson process

Let $\{X(t), t\ge 0\}$ and $\{Y(t),t\ge 0\}$ be independent Poisson processes with parameters $\lambda_1$ and $\lambda_2$, respectively. Define $Z_1(t)=X(t)+Y(t)$, $Z_2(t)=X(t)-Y(t)$, $Z_3(t)=X(t)+k$, ...
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Equilibrium distributions of Markov Chains

I often get confused about when a Markov chain has an equilibrium distribution; when this equilibrium distribution is unique; which starting states converge to the equilibrium distribution; and ...
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251 views

Exchanging order of stochastic integral and $L_2$ norm

Suppose there is a second-order real-state stochastic process $X: \Omega \times T \rightarrow \mathbb{R}$ with $T= \mathbb{R}$ and probability space $(\Omega, \mathcal{F}, P)$. I was wondering if the ...
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What is the difference between various kalman filters?

What is the difference between additive and multiplicative kalman filters, as well as some other kinds? I'm also looking for reference texts and articles that describe the algorithms, so ...
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290 views

Stochastic Urn Process using a Pareto distribution

N urns are assigned m balls in a stochastic process based on a Pareto distribution. The process is as follows: X is a Pareto random variable (xminimum = 1, alpha is a parameter) if X > N, throw the ...
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Interpretation of sigma algebra

My question is how to interpret sigma algebra, especially in the context of probability theory (stochastic processes included). I would like to know if there is some clear and general way to interpret ...
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180 views

example on variance of stochastic processes

I saw this expression in a book and I cannot understand how did he get this expression. Suppose $Z_t$ and $D_t$ are some stochastic processes and we have these expressions, $Z_{t_k} - Z_{t_{k-1}} = ...
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162 views

Limit probability for some Hitting time of a Feller Process

I wanted to know if it was true that if we are given a one-dimensional Feller process taking values in $\mathbb{R}$ and a hitting time $\tau_A=\inf\{t>0 s.t. X_t\in A\}$ with $A$ a open set (this ...
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A question regarding the hitting time formula in brownian motion

Let $\tau_a=\inf\{t: B_t=a\}$, the hitting time of the standard Brownian motion to reach the boundary $a$. This is easily derived $$E(e^{-\lambda \tau_a})=e^{-|a|\sqrt{2\lambda}}$$ But I am having ...
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Independent exponentially distributed random variables

I got this problem within others as homework, and I don't know how to do it. Does anyone know how to start solving it? Thank you! Any help would be appreciated. Sorry for my writing in LaTex because ...
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What is the difference between all types of Markov Chains?

I have been looking for some good material covering Markov Chains but everything seems so difficult to me... After reading about the subject, I figured out that there is basically three kinds of ...
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Sum and product of Martingale processes

Given two Martingale processes $(X_t)$ and $(Y_t)$, are their sum $(X_t+Y_t)$ and their product $(X_t \times Y_t)$ also Martingale? If not, will the two $(X_t)$ and $(Y_t)$ being independent grant ...
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Independent stochastic processes and independent random vectors

The definition for the two processes to be independent is given by PlanetMath: Two stochastic processes $\lbrace X(t)\mid t\in T \rbrace$ and $\lbrace Y(t)\mid t\in T \rbrace$ are said to be ...
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Diffusions - global and local

Suppose $dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$ is a diffusion. Is there a sense in which the dynamics are "dominated" locally by the diffusion term, and dominated globally by the drift term? If $\mu$ ...
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Almost surely in definition of Martingale

$(X_n), n \in \mathbb{N}$ is a stochastic process. I saw in one definition of Martingale that $$E [X_{n+1} |X_0 , X_1 , . . . , X_n ] = X_n \quad a.s., \forall n \geq 0.$$ I understand what "almost ...
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Using Central Limit Theorem

Can anyone help me with it: Using the central limit theorem for suitable Poisson random variables, prove that $$ \lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}=1/2$$ Thanks!
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Finding Characteristic Function

Can anyone help me with this problem? The random variable $X_n$ takes the values $\frac{k}{n}$, $k=1,2,\ldots,n$, each with probability $\frac{1}{n}$. Find its characteristic function and the limit as ...
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Binomial distribution with parameters p and N, p is distributed to a beta distribution with parameters r and s

For each given $p$, let $X$ have a binomial distribution with parameters $p$ and $N$. Suppose $p$ is distributed according to a beta distribution with parameters r and s. Find the resulting ...
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What's the procedure to condense several states into one in Markov chain?

I am wondering the validity and the procedure to condense/group several states into one in markov chain, namely, if it is possible, how to transform the state vector and the transition matrix? Many ...
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stochastic matrices?

Let $A$ be a symmetric stochastic matrix, such that the sum over the columns, for each row, is 1, and all elements are positive. $A$ dimensions are $n \times n$ Let's say that $B$ is a matrix which ...
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Recommendation on stochastic process books

I was wondering if someone could recommend good books on stochastic processes with measure theory treatment with not much or no measure theory treatment for each, it would be nice to have some ...
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Infinite line of people

Let us assume that we have an infinite line of people, and each person can either move forwards or remain at the same place. They move only one step at a time. (They are jumping from one position to ...
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Difference between time series and stochastic process?

What is the difference between a "time series" and a (discrete-time) stochastic process?
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yet another urn problem

There is an urn of N balls, each of a unique color. In each step one takes out 2 balls without replacement, changes the color on the 2nd ball to that of the first ball and returns them back to the ...
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proof of equivalent definitions of recurrent states for Markov chains

I am wondering if anyone could prove the following equivalent definition of recurrent/persistent state for Markov chains: 1) $P(X_n=i,n\ge1|X_0=i)=1$ 2) Let $T_{ij}=min\{n: X_n=j|X_0=i\}$, a state ...
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Time until x successes, given p(failure)?

I hope this is the right place for help on this question! I expect this should be easy for this audience (and, no, this isn't homework). I have a task that takes $X$ seconds to complete (say, moving ...
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Weak stationarity of a process modulus

Is it true that if a continuous-time stochastic process $X_t$ is weakly stationary then $|X_t|$ is also weakly stationary?
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Stochastic interpretation of Einstein Equations

Einsteins theory of gravitation, general relativity, is a purely geometric theory. In a recent question I wanted to know what the relation of Brownian Motion to the Helmholtz equation is and got a ...
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Relation of Brownian Motion to Helmholtz Equation

one can obtain solutions to the Laplace equation $$\Delta\psi(x) = 0$$ or even for the Poisson equation $\Delta\psi(x)=\varphi(x)$ in a Dirichlet boundary value problem using a random-walk approach, ...
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581 views

Gaussian Distributed Random Variables

I know the sum of independent Gaussian distributed random variables also has a Gaussian distribution. say we have a sequence of Gaussian random variables not necessarily independent $ X_0, X_1, X_2, ...
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Hitting times of reversible markov chain with known steady state probabilites

Consider a reversible markov chain Xn whose steady state distribution is known, can we find the expected hitting time to a subset A of the states starting from some state i ? Additionally you can ...
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Variation of a function

What exactly is the variation of a function ? Is it a distace or an element of some space The total of a real valued function $f\colon [0,t] \mapsto \Re $ is as below say $\pi = \{0=t_0,t_1,\cdots , ...
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Stochastic integral and Stieltjes integral

My question is on the convergence of the Riemann sum, when the value spaces are square-integrable random variables. The convergence does depend on the evaluation point we choose, why is the case. Here ...
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Good introductory book for Markov processes

Which is a good introductory book for Markov chains and Markov processes? Thank you.
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A type of stochastic jump process

Let $X \geq 1$ be an integer r.v. with $E[X]=\mu$. Let $X_i$ be a sequence of iid rvs with the distribution of $X$. On the integer line, we start at $0$, and want to know the expected position after ...
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Going to the Movies!

I was looking at movie times today and was struck by the oddly-spaced showing times. For example, at the local Loew's Theater "Tron: Legacy 3D" (127 min.) is playing on two screens at the following ...
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An application of the Optional Sampling Theorem

let $S(k), k\geq 0$ a discrete random process. Suppose $S(N)$ is with probability one either 100 or 0 and that $S(0)=50$. Suppose further there is at least a sixty percent probability that the price ...