0
votes
1answer
10 views

What does the “empirical” autocovariance function represent?

My professor gave me the sequence ${X_n} = {1,5,5,1,5,5,...}$ and asked us to compute the empirical autocovariance function given below. $$\displaystyle \hat \rho(1) = \lim_{N \to \infty} \frac{1}{N} ...
0
votes
2answers
28 views

Is the autocovariance function of a sequence identically zero if the sequence is iid?

My professor gives the following definition for the autocovariance function. $$\rho(i,j) = Cov(X_i , X_j)$$$\\$If I have a sequence that is iid, when i compute $\rho(n,n+1)$ for $n \geq 0$, I found ...
1
vote
1answer
132 views

Finding the joint distribution of a random process with memory

I'm modeling a digital system as a random process and attempting to solve for the autocorrelation in order to arrive at the power spectral density of the process. The system is as follows: At any ...
0
votes
1answer
69 views

Autocovariance of a given stochastic process?

I need to find the autocovariance $C_{YY}(t,s)$ of the stochastic process $Y(t) = t^2 X(t) -2X'(t)$ where $C_{XX}(t,s) = e^{-t^2 -s^2}$ is given. Using known properties I can calculate the ...
0
votes
1answer
104 views

correlation between two different variables

I am studying stochastic processes and found the next problem: Let $A$ and $\Phi $ be two independent random variables such that $E(A) = 0$, $E(A^2) < \infty$, and $\Phi$ is uniformly distributed ...
3
votes
1answer
130 views

Autocorrelation of wrapped Wiener process

Let $\phi(t)$ be a Brownian Walk (Wiener Process), where $\phi\in[0,2\pi)$. As such we work with the variable $z(t)=e^{i\phi(t)}$. I would like to calculate $$E(z(t)z(t+\tau)).$$ This is equal to ...
1
vote
1answer
52 views

How can I show that $z_i =\cos(iw)$ where $w$ is uniform on $[0,2\pi]$ is a white noise process?

How can I show that $z_i =\cos(iw)$, where $w$ is uniform on $[0,2\pi]$ is a white noise process? So far, I have shown $E(z_i)=0$ by integrating. However, I need to show ...
2
votes
0answers
161 views

Correlated diffusion processes and covariance matrix

I'm really noob in maths topics so I hope you will excuse me if I use terms which aren't correct. I would like to simulate $n$ dimensional diffusion processes with $n$ noises. Each process has its ...
1
vote
0answers
115 views

Windowed Linear Correlation

$\DeclareMathOperator \Cov {Cov}$ $\DeclareMathOperator \Var {Var}$ $\DeclareMathOperator \E {E}$ Consider the following experiment: For $N\geq1$, consider $N$ black balls. Let us paint each black ...
2
votes
1answer
85 views

constructing “pseudonoise” sequences other than (2^n)-1? (low cyclical autocorrelation)

Pseudonoise LFSR sequences of length $N = 2^k-1$ have the nice property that their cyclical autocorrelation is $N$ when the sequence is lined up with itself, and $-1$ elsewhere. Is there a way to ...
0
votes
0answers
796 views

Pseudo-random binary sequence generated by shift register

Binary sequence generated by shift register with feedback have periodic properties. A simple 4-bit shift register shown in Fig (a). For the initial condition shown, it can be verified that the ...
5
votes
1answer
146 views

Correlations between neighboring Voronoi cells

For a sequence $X_1,X_2,X_3,\ldots$ of random variables, what it means to say $X_1$ is correlated with $X_2$ is unambiguous. It may be that the bigger $X_1$ is, the bigger $X_2$ is likely to be. If, ...
2
votes
1answer
179 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}} = ...