Mathematics Stack Exchange
Mathematics Stack Exchange

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.

16128 questions
0
votes
1 answer

Random Walk Problem

I understand the basic concept of random walk and how to solve them but when it comes to more complex ones like the one below I don't understand how they're calculated. The books and online resources I've looked into have not really clarified any of…
Kurapika
  • 175
0
votes
0 answers

Maximum of sum of iid random variable with two outcomes, threshold crossing

Let $S_n=\sum_{i=1}^n X_i$ be the sum of n iid random variables, and $M_N$ be $\max_{n\leq N} S_n$. I know that given it is a simple random walk, i.e. $X_i=-1$ with probability $p$ and $-1$ with probability $1-p$, the probability of crossing a…
JT09
  • 1
0
votes
0 answers

White Noise Process

Suppose $w_{t}$ is a normal white noise process. Is $z_{t} = w_{t}*w_{t-1}$ stationary? Is my reasoning correct? $Ew_{t}w_{t-1}w_{t+h}w_{t+h-1} = 0 $ for all $h$ implying that the series is stationary?
phil12
  • 1,547
0
votes
1 answer

proving stochastic process is independent

Guys can anyone help me with this question? On a probability space let be filtration $F = (F_n)_{n \in N_0}$ and a real valued adaptive stochastic process $(X_n)_{n \in N_0}$ for all the Borelsets $ A \in B(R) $ we have $P[X_{n+1} \in A | F_n ] = P…
Mathgamer
  • 45
0
votes
1 answer

Determining Stationarity of Sine Process

Suppose $X_{t} = B\sin(\omega t) + w_{t}$ and $\omega$ is between $(0,\pi)$, $B$ has mean 0 and variance 1 $w_{t}$ is $N(0,1)$ and $w_{t}$ is independent of $B$. Show that $X_{t}$ is weakly stationary. Attempt: Normally, this series would not be…
phil12
  • 1,547
0
votes
1 answer

Stochastic processes-Brownian Motion

I hope someone can help me with the following exercise... Show that $ \int_0^t s \, dB_s =tB_t-\int_0^t B_s \, ds $, for each $t>0$, where $B$ is a Brownian motion. Thanks in advance!!!
mari
  • 21
  • 3
0
votes
1 answer

Time Average Mean of X(t)=A, where A is a r.v. Ergodic vs. non-ergodic.

Time average of a sample function is defined as: $$\bar{x} = \langle~x(t)~\rangle = \lim \limits_{T \to \infty}\frac{1}{2T} \int \limits_{-T}^{T} x(t) ~ dt$$ This is how I see it: A few sample functions of X(t)=A, would be: $$x_1(t)…
pico
  • 941
0
votes
2 answers

Autocorrelation of telegraph signal with Poisson Distribution,,,

In the problem below, I'm not understanding how they came up with the formula for X(t) used to calculate the autocorrelation $E\Big[X(t)X(t+\tau)\Big]$. All they say about X(t) is that it has equal probability of switch polarity from +A to -A and…
pico
  • 941
0
votes
3 answers

Stochastic order

I need help with the following exercise: $X$ and $Y$ are random variables in $\mathbb{R}$ with the distribution functions $F_X$ and $F_Y$, so $X$ is stochastically smaller or equal to $Y$, i.e. $(X\leq_{st}Y)$ if $F_X(t)\geq F_Y(t)$ holds for all…
Tino
  • 137
  • 7
0
votes
1 answer

Simplify the following transfer function

This question is related to Weiner filter. However my doubt is not about the Wiener Filter itself but the calculations from one of the exercise. I was given: $$H(f) =…
Jim
  • 31
0
votes
0 answers

Find $S_Y(f)$ with given $H(f)$ and $S_X(f)$

I am a bit confused about some calculations of an example from a textbook: Given: $$S_X(f)=N_0/2$$ $$S_Y(f)=|H(f)|^2 S_X(f) $$ with $$H(f) = e^{-j\pi fT}T\frac{sin(\pi Tf)}{\pi Tf}$$ From the textbook, the $S_Y(f)$ is: $$S_Y(f)= T^2 [\frac{sin(\pi…
Jim
  • 31
0
votes
0 answers

Branching process: Prove that $g_Z(t) = g(tg_Z(t))$ (generative function)

Let $(Z_t)_{t \in \mathbb{N}_0}$ be a subcritical branching process such that $Z_0 = 1$ and $\mu = E[Z_1] \le 1.$ Furthermore, let $Z := \sum_{k = 0}^{\infty} Z_t$ be the number of overall successors. Show that $$g_Z(t) = g(tg_Z(t))$$ with $g$…
Borol
  • 491
0
votes
1 answer

Probability generating function of $Z_t$ (stochastic branching processes)

I want to find the PGF of $Z_t$ where $Z_t$ is the number of individuals in generation $t$ (in Galton-Watson process). The offspring distribution for this Galton-Watson tree is given by $X \overset{d}{=}Geometric(p)$, i.e. $Pr(X = k) = p(1-p)^{k}$…
Math is like Friday
  • 147
0
votes
2 answers

Questions about a birth and death process

My question has a queue M/M/1/2, that is, a system with exponential interarrival and service times, one server and having a room only for 2 customers (including the one in service, and another that is waiting). Let the number of customers in the…
Jessie
  • 87
0
votes
2 answers

A symbol in stochastic process

Let $\{ξ_i\}_{i=1}^n$ be i.i.d. random variable with $Eξ_i = 0$, $Dξ_i = σ^2 > 0$ and let $η_n = \frac{1}{\sigma\sqrt{n}}\sum_{i=1}^{n}ξ_i$. Prove/disprove existence of $(P) \lim_{n\to\infty} η_n$. What is the meaning of $(P)$ in the above…
Dasheng Wang
  • 899
1 2 3
…
46 47