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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Lindeberg-Feller CLT follows from Martingale CLT?

I've been studying about central limit theorems, in particular the Lindeberg-Feller CLT and other extensions. In most textbooks and sources online the martingale central limit theorem comes as an ...
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1answer
28 views

Kullback-Leibler divergence when the $Q$ distribution has zero values

For discrete probability distributions $P,Q$, the Kullback-Leibler divergence of $Q$ from $P$ is defined to be $$D_{\mathrm{KL}} ( P \mathop{\|} Q ) = \sum_i P(i) \ln \left( \frac{P(i)}{Q(i)} ...
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1answer
19 views

Period of a Markov Chain: Why is this one aperiodic?

Here is the problem from a stochastic processes book: Consider a Markov Chain on {0,1,2} having transition matrix 0 1 2 0| 0 0 1| 1| 1 0 0| 2|.5 .5 0| ...
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26 views

Optimization of stochastic differential equations

Is there a way to optimize or maximize a set of differential equations. such that each equation is represented by a time series S_((t+1),μ) = μ*(S_(t+1)-S_t) + S_t and μ = 2/(i+1), i=1,...,n. Then I ...
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1answer
34 views

Joint Density Function of uniform and gamma density [closed]

Let $U$ be uniformly distributed over the interval $[0, L]$ where $L$ follows the gamma density $f_L(x) = xe^{-x}$ for $x\ge 0$. What is the joint desnity function of $U$ and $V = L - U$?
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49 views

Independent Exponentially Distributed RV's [closed]

Consider a post office with two clerks. John, Paul, and Naomi enter simultaneously. John and Paul go directly to the clerks, while Naomi must wait until either John or Paul is finished before she ...
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17 views

Smoothness of marginal distribution of a diffusion in the initial condition

Let $\{p_t\}_{t \ge 0}$ be a one-dimensional diffusion process (on [0,1] ) with drift $\mu(p) = C_1(1-p)-C_2p+p(1-p)s(p),$ where $s$ is a Lipschitz function and $C_1,C_2 \in (0,1)$, and diffusion ...
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1answer
14 views

Power spectral density of the system output

$w(t)$ and $z(t)$: two stationary random processes $z(t) = Pw(t)$. $P$: a stable, LTI system. How to show: $$ S_z(jw) = P(jw)S_w(jw)P(jw)^*$$ $S_z(jw)$ is the power spectral density of ...
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1answer
25 views

Convergence of random variables in $L^1$

So $g$ is a continuous real-valued function and are given that the sequence of random variables $Y_n$ converges to $Y$ in $L^1$, $E[|g(Y_n)|]<\infty$ and $E[|g(Y)|]<\infty$. Show that $g(Y_n)$ ...
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6 views

Formula for $X_t - X_{t+h}$ where $X_t$ is $MA(q)$ process

Lets say that we have $MA(q)$ process $X_t = Z_t + \psi_1 Z_{t-1} + \dots+\psi_q Z_{t-q}$. $Z_t$ are IID (normal with mean $0$ and standard deviation $\sigma$). Now I need to find form of $X_t - ...
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1answer
42 views

Stochastic integral where the integrator is zero in probability

We are given a continuous semimartingale $Y$ and a continuous process $B$ of finite variation. Hence, we know that $\langle B \rangle$, the quadratic covariation of $B$, is zero in probability. I now ...
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1answer
47 views

Expectation and Convergence of Sum of Random Variables [closed]

Let $X_1, X_2, ...$ be a sequence of independent random variables with $$\mathbb{P}[X_i=1]=\mathbb{P}[X_i=-1]=\frac{1}{2}$$ Let's now consider the sum $S_n=\sum_{k=1}^{n} X_k$. I need to show three ...
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1answer
53 views

limit distribution of weak convergent sequence

Let $\{X_n(t) \mid t \in T\}$ be a bounded stochastic proces for some non-empty set $T$, we assume that the finite dimensional distributions converge in distribution and we let for $k\in \mathbb{N}$, ...
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25 views

Conditions for a Markov process to have independent increments [duplicate]

I consistently see "Let $\{X(t)\}$ be a stochastic process with independent increments..." in various texts, though I have yet to find any conditions under which we can guarantee a process to have ...
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57 views

Conceptual Understanding of a Simple Random Process

I have a simple discrete time random process that with probability $0.5$ chooses a deterministic sequence so that $X(t) = -1$, for $t<1$ and $X(t) = +1$ for $t \geq 1$, similarly with probability ...
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1answer
34 views

The Pólya urn model describes a martingale

Suppose an urn contains one blue and one red ball and that we perform the following random experiment: In each round $n\in\mathbb{N}$ we randomly draw a ball If the drawn ball is blue, we replace it ...
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1answer
89 views

$P^n$ transition matrix of a Markov chain

The setup: We have an unlimited supply of balls and $k$ boxes. In every step, we randomly (all of them have the same probability) choose a box and put a ball in it. Let $X_n$ be the number of ...
3
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1answer
46 views

Squared Bessel Process and Ito Lemma

$dX_t = \delta dt+ 2\sqrt{X_t} dW_t$, where $W_t$ is a standard Wiener process, Define $\tau =\frac{\sigma ^2}{2\nu(2 − \delta)}\left(1 − \exp \left(−\frac{2\nu t}{2−\delta}\right)\right)$ If ...
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2answers
16 views

Do you think this transition Matrix is correct?

Here is the situation we are trying to model: given a car that has 3 states, labeled 1, 2 and 3. state 1: is when the vehicle is in good operating condition. state 2: repairs may be required to ...
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28 views

Random sum a.s convergence and convergence in probability

Let $X_n$ be a sequence of independent random variables such that $$\mathbb{P}[X_i=1]=\mathbb{P}[X_i=-1]=\frac{1}{2}$$ Consider the sum $S_n=\Sigma_{k=1}^nX_k.$ How can we show that for any ...
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9 views

How to decompose a Markov Modulated Poisson Process (MMPP)

I have two questions to ask here. The superposition of two independent MMPPs is also a MMPP. How to calculate the rate of a new burst and the rate of requests within one burst if these two MMPPs are ...
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29 views

Riesz decomposition of a nonnegative supermartingale

$X=(X_n,\mathcal{F_n})$ is a nonnegative supermartingale, and moreover $EX_n\to 0$, i.e., it is a potential. If $X_n=M_n-A_n$ is the Doob decomposition, then $$EX_n=EM_n-EA_n=EX_0-EA_n,$$ so by the ...
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1answer
42 views

Law of Iterated Logarithms

I know the Law of Iterated Logarithms states the following almost surely: $$\limsup_{t\to\infty} \frac{B(t)}{\sqrt{2t\log\log t}} = 1 $$ I was wondering if there are similar ones. For example, if I ...
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15 views

Predictive Density Independent in Gaussian Process Regression?

I am a little confused in Gaussian process regression. In a GP regression, let $Y=[Y_a, Y_b]\sim \mathcal{N}(0, K+\sigma^2I)$, where $Y_b$ is the target of training samples. The task is to predict ...
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27 views

Quadratic Variation of Increasing Process?

I am looking through my notes and I came across the following statement: Let $X_s$ be a positive local martingale and let $M_t = max_{0 \le s \le t} X_s$. Then since $M_t$ is an increasing process, ...
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24 views

Solving an expectation related to CIR process

I encounter the following question Let $X$ satisfy the SDE $$dX_s=k(\alpha-X_s)ds+\sigma\sqrt{X_s}dW_s$$ for $s\geq t$ with $X_t=x$, where $k,\alpha,\sigma$ are positive constants. Find the ...
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36 views

A characterization of quadratic variation for $L^2$ martingales

I am trying to prove the following statement but I am totally at a loss. Let $(A_t)$, $t \in \mathbb{R}^+$ be an adapted (with respect to the filtration $(\mathcal{F}_t)$) continuous integrable ...
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2answers
34 views

Stochastic Differential Equations - A Few General Questions

I just have a few questions about stochastic differential equations. I generally did a lot of pure math but signed up for a course on probability models and stochastic differential equations because I ...
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1answer
25 views

A distribution of a stopped Wiener process

Let $(W_s)_{s \geq 0}$ be a Wiener process and $\tau$ be a random variable with an exponential distribution with parameter $1$. Suppose that $W$ and $\tau$ are independent. Determine the distribution ...
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31 views

A Doob-Meyer decomposition related question

First I will state the question and then I will show my answer, which I obtained by imposing an additional condition on the processes involved. I would like to get some help on how to solve the ...
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85 views
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What is the difference between Calculus and Analysis? In Stochastic processes?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
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28 views

capacity of biased random walk in $\mathbb{Z}^2$

Let $P_{x,y}$ the probability that a random walk starting from $x$ will ever visit $y$. Consider a biased random walk in $\mathbb{Z}^2$. Let $A_k$ be the set of vertices having a distance less than ...
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1answer
20 views

How do we sample from a Gaussian process

I have one particular question on Gaussian processes. A Gaussian process is fully characterized by $\mu$ and $\Sigma$. However, I do not understand how can we sample a (random) function from the so ...
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18 views

Brownian motion reflected on a sphere

Consider a standard Wiener process (in 3 dimensions) $(W_t)_{t>0}$, such that $W_0 = x_0 \neq 0$. I am trying to determine the transition density of $(W_t)$ reflected on a sphere of radius $a < ...
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26 views

Convergence in finite-dimensional distributions of some integral

Let $(X^n_t)_{t \geq 0}$ be a sequence of random real-valued processes that converges in finite-dimensional distributions, i.e. for all $k \in \mathbb{N}$ and for all $0 \leq t_1 < \dots < t_k$ ...
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20 views

Mean and variance of Gamma distribution

How do I calculate the mean and the variance of a Gamma distribution? I was told to prove the variance was sigma/lambda(^2), I don't know how to find the variance much less the variance.
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41 views

Brownian motion with drift (stopping time and supremum)

Suppose $(B(t))_{t \geq 0}$ is a Brownian motion and $(B_{\mu}(t))_{t \geq 0}$ is a Brownian motion with drift, which is defined by $$B_{\mu}(t) := B(t) + \mu t, \ \ \ \mu <0. $$ With $T_{a} := ...
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1answer
13 views

Connection between transition probability and SDE?

Can someone highlight what is the connection between the transition probability of a continuous time stochastic process $X_t$, i.e. $p(x,t\vert x_0,0)$ and the stochastic differential equation of the ...
3
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1answer
55 views

Why is $\mathbb{P}(F\geqslant G) = \int_{\mathbb{R}} \mathbb{P}(F \geqslant g | G=g) \, D_G(g) \text{d}g$?

For random variables $F,G$ I have problems with understanding the equation $$\mathbb{P}(F \geqslant G) = \int_{\mathbb{R}} \mathbb{P}(F \geqslant g | G=g) \, D_G(g) \text{d}g, $$ where $D_G$ is the ...
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1answer
49 views

Help with Semimartingale decomposition.

I'm having trouble with the following question: Let $\{W_t\}_{t\geqslant0}$ be a one-dimensional standard Brownian motion defined on a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal ...
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22 views

A “Fourier Phase” for (stationary) random processes?

Let $X_t$ be a real w.s.s. random process. Its spectrum is given by $S(f)=\mathcal{F}R_X(\tau)(f)$ where $R_X$ is the process autocorrelation. As $X_t$ is real, the spectrum will be real and ...
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18 views

Local time for reflected random walk

Say I have a process starting from 0, and last for 100 steps, each step either moves up or down by one unit, within the boundary -10 and 10. My understanding is that since this is a martingale, the ...
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1answer
23 views

M/G/1 queueing problem

I need to prove that in the M/G/1 queueing system with Poisson arrivals with parameter lambda and exponential service time with parameter mu, that q_k = (lambda/(lambda+mu))^k (mu/(lambda+mu)).
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Prove that this space of stochastic processes is complete

See page 17 on http://www.stat.cmu.edu/~cshalizi/754/notes/lecture-19.pdf We define $\mathcal{QM}(T)$ to be the space of all non-anticipating processes $X$ such that the norm $||X||_{{QM}(T)}$ ...
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34 views

Markov chains: Condtitional independence implies independence?

In one proof, I encountered the following reasoning: $$P(T_1=n,T_2=m\mid X_0=j)=P(T_1=n\mid X_0=j)P(T_2=m\mid X_0=j)$$ Where $T$s are waiting times between returns to a state, $X_0$ is the state at ...
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21 views

Does the expected spreading of sample paths imply increase in variance?

Consider a sample-continuous stochastic process $\left\{ X_t \right\}_{t \in T}$ s.t. each $X_t$ is real-valued and $$\int_\Omega | X_t(\omega) | ^p \, \mathrm{d} P(\omega)< \infty$$ for all $1 ...
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1answer
40 views

Variance and expectation of the stochastic intergal [closed]

Compute the unconditional expected value and variance, and describe, as far as possible, the distribution of the random variable $Y_{t} = \int^{t}_{0} W_{s} ds $ with the hint below $\int^{t}_{0} ...
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27 views

Conditional distribution on arrival time (Poisson process)

Suppose that $\{N_t: t\geq 0\}$ is a Poisson process of rate $\lambda$ and $T_1< T_2< \dotsb\ $ are its arrival times (i.e. $T_i := \min \{t\geq 0 : N_t \geq i\} $). What is the conditional ...
3
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37 views

the 2D fractional Gaussian noise as derived from the 2D fractional Brownian motion

Let $X_n$ be a 1D discrete fBm. Then, its 1st order difference, $W_n=X_n-X_{n-1}$ is fractional Gaussian noise (fGn). This case is simple. But what happens in 2D? Let $Y(m,n)$ be a 2D fBm, then we ...
0
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1answer
52 views

Relation between Hermite polynomials and Brownian motion (on martingale property) [duplicate]

Let us define Hermite polynomials as $H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n} e^{-x^2/2}.$ One can prove that $e^{\theta x-\frac{1}{2} \theta^2}=\sum_{n=0}^{\infty} \frac{1}{n!}H_n(x)\theta^n \quad ...