2
votes
0answers
53 views

Random walk with $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} < \infty$

Consider a random walk started at $S_0=0$, denoted $S_n = \sum_{k=1}^{n}X_k$, where $X_1$, $X_2$... are the i.i.d increments. If we have $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} ...
2
votes
1answer
101 views

$L^1$ convergence of a sequence of stochastic integrals and convergence of their quadratic variations

On a filtered probability space $(\Omega, \mathcal F, \mathcal F_t, \mathbb P)$ containing a Brownian motion $W_t$. Let $\sigma^n_t$ be a sequence of square intergable adapted processes and consider: ...
0
votes
0answers
47 views

Given an innovations sequence, calculate transformation matrix, A

Let $Y_1,Y_2,Y_3,X)^T$ be a zero mean random vector with correlation matrix, $$ \begin{pmatrix} 2 & 1 & 1 & 2 \\ 1 & 2 & 1 & 2 \\ 1 & 1 & 2 & 2 \\ 2 & 2 & 2 ...
1
vote
1answer
112 views

recurrent states in markov chain with poisson-like transition matrix

I am considering a Markov chain $X$ with state space $\mathbb{N}$ that has transition probabilities $p_{ij}=\begin{cases}1\mbox{ for }i=j=0\\e^{-i}\frac{i^{j}}{j!} \mbox{ otherwise }\end{cases}.$ I ...
1
vote
1answer
86 views

Expected value of a stochastic harmonic series

It doesn't seem straightforward to put this into mathematical notation, but I'll do my best to explain the setup. Consider a harmonic series of the following type. For the sake of argument, say we ...
3
votes
1answer
140 views

A Coupled Random Walk on the xy-Plane

Consider a point on the $xy$-plane whose position is updated in iterations. In each iteration, the point undergoes, with equal probability, either an $A$- or a $B$-update, defined as follows: ...
-1
votes
1answer
45 views

Can some explain to me why this property is true for left-continuous RW

Source Durrett, http://www.math.duke.edu/~rtd/EOSP/EOSP2E.pdf, page 170 example 5.12. I don't understand how his implications (i),(ii) imply that there exists an $\alpha \lt 0 \:s.t\: ...
1
vote
1answer
170 views

About a proof that elements in a certain $L_2$ convergent series are also in $L_\infty$

The problem I have is about convergence of series expansions of random fields (or stochastic processes, if you will), which don't converge in the norm I want, that is $L_\infty$, but in $L_2$. I have ...
1
vote
1answer
188 views

$X$ is a Geometric random variable find the expectation of $1/X$

Let $X$ be a geometric random variable with parameter $p$, find the expectation of $E[1/X]$. I need help simplifying the series.
3
votes
1answer
101 views

Limit of a probability regarding a random walk

Consider a random walk on the integers starting at 0, where in each step you move either 1 or 2 meters (back or forth alike). As soon as you reach either the $N$ or $N+1$ meter mark, you stop. What is ...
0
votes
2answers
54 views

Given a random sequence of input points does it always produce a random output, excluding f(x) = constant?

Assume I have $y = f(x) \ne \mathrm{constant} $ and $(x_1 , \ldots ,x_N)$ a sequence N random input points, is there a set of criteria or a theorem that tells me that the output sequence $(y_1, ...
2
votes
1answer
215 views

Finding Probability Generating function for $P\left\{ X > n+1\right\} $

I am trying to find probability generating function for $P\left\{ X > n+1\right\} $. Let X be a random variable assuming the values $0, 1, 2, ...$. The notation both for the distribution of $X$ ...
1
vote
1answer
105 views

MA process ACF proof - don't understand it

I've got the proof but I don't understand a small detail. As you know for an MA process: $X_n = \sum _{i=0} ^q \beta_i Z_{n-i}$ where $Z_n$ is WGN (pure Gaussian random process). Then the ACF is: ...
2
votes
2answers
213 views

Convergence of $\frac{a_n}{n}$ where $a_0=1$ and $a_n=a_{\frac{n}{2}}+a_{\frac{n}{3}}+a_{\frac{n}{6}}$

Given $a_0=1$ and:$$a_n=a_{\frac{n}{2}}+a_{\frac{n}{3}}+a_{\frac{n}{6}}$$Find convergence or divergence of $\frac{a_n}{n}$. Such a weird problem; I don't know how to attack it. I'm also fairly ...
2
votes
1answer
107 views

MA and AR process stationarity intuition

$y_{MA}$ = $ε_t$ + $ε_{t-1}$ <- stationary $y_{AR}$ = $ε_t$ + $y_{AR_{t-1}}$ $y_{AR_{t-1}}$ = $ε_t$ + $ε_{t-1}$ + $y_{AR_{t-2}}$ $y_{AR_{t-2}}$ = $ε_t$ + $ε_{t-1}$ + $ε_{t-2}$ + ...
40
votes
5answers
7k views

Convergence of $np(n)$ where $p(n)=\sum_{j=\lceil n/2\rceil}^{n-1} {p(j)\over j}$

Some years ago I was interested in the following Markov chain whose state space is the positive integers. The chain begins at state "1", and from state "n" the chain next jumps to a state uniformly ...