# Tagged Questions

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### kullback liebler divergence for correlated processes

Suppose $X_n^{(1)}=\lambda_1 X_{n-1}^{(1)}+\mu_1+\epsilon_n^{(1)}$ and $X_{n}^{(2)}=\lambda_2X_{n-1}^{(2)}+\mu_2+\epsilon_n^{(2)}$ where $|\lambda_i|<1$ for $i=1,2$ and $\epsilon_n^{(i)}$ are ...
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### Derivation of Black-Scholes equation by riskless portfolio

The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the ...
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### Distribution of number of Poisson arrivals in interval

$X_1$ and $X_2$ are both Poisson processes. $N$ is the number of arrivals of $X_1$ in between two subsequent arrivals of $X_2$. Derive the probability density $f_N(n)$ of $N$. I wanted to start from ...
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### Supply the transition matrix for these (possible) Markov chains

Reading Grimmet, Stirzaker: Probability and Random Processes, which unfortunately doesn't have solutions. Trying to make sure I understand Markov chains. A die is rolled repeatedly. Which of these ...
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### Conditional expectation for Poisson process

Let $X(t)$ be a Poisson process with rate $\lambda = 6$ describing arrivals per hour of customers at a bank. Let the probability of a customer being male be $2/3$. Suppose 10 males has arrived during ...
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### Invariant mesure of a reflected random walk

Let $(X_n), n \geq 0$ be a Reflected Random Walk defined by: $X_0 = 0$ and: $X_{n+1}=\max( 0 , X_n + \xi )$ $\xi$ is a random variable such that $P(\xi=a)=\theta$ and $P(\xi=-b)=1-\theta$ for a ...
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### Reflected random walk

Suppose that $X_n$ is a reflected (in 0) random walk with parameter $\theta$. So $X_{n+1}-X_n = 1$ with probability $\theta$ , and -1 with probability $1-\theta$ when $X_n \geq 1$, if $X_n=0$ then ...
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### An example of stochastic process

I use the following definition for a stochastic process. Let $(\Omega, \mathcal F, P)$ be a probability space, $(E, \mathcal E)$ be a measurable space, and $T$ be a non-empty set. A collection ...
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### Autocorrelation of a Wiener Process proof

Given a Wiener process X, how do I prove this? $R_x(s,t) = E[X(s)X(t)] = min(s,t)$ There seems to be a trick with dividing to two cases of $s<t$ and $s>t$, but I can't figure out how this ...
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### Approximating the probability of an event by finite-dimensional distributions

Let $(X(t))_{t\ge 0}$ be a stochastic process on $\mathbb{R}^d$, say an Ito diffusion (with continuous sample paths). Let $A\subset \mathbb{R}^d$ be a measurable set and $t>0$. Does the following ...
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### Simulation Lévy process

I need to simulate a Lévy process from its characteristic triple $(\gamma,\Sigma,\nu)$ where $\nu$ is the Lévy measure. I know that I can simulate it by summing a brownian motion and a compound ...
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### Proving that a process has the Markov property

Let $X_t=xe^{ct+aB_t}$ where $B_t$ is one dimensional Brownian motion. How would I prove this is a Markov process using the expectation definition of a Markov process, i.e., ...
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### Calculating probability of a time-series probability crossing a threshold

(Please feel free to suggest a better title -- I'm still not sure what to call this in the first place.) I'm having trouble getting my head wrapped around a time-series stochastics problem I've run ...
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### Find the probability $P[ x(t) \le 1]$ where $x(t)$ is a filtered Poisson process (rect pulses)

I can't understand the following question: "The random process x(t) is defined as $$x(t) = \sum_{n=- \infty}^{+\infty} rect(\frac{t-\tau_{n}}{T}) \quad ,\quad t \ \epsilon \ (R)$$ where {$\tau_{n}$} ...
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### Conditional expectation of the sum of two random variables

I've got some difficulties in calculating the conditional expectation of the sum of two RV. I am not sure if I correctly formalized the scenario I am looking at. So I am trying to describe it first: ...
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### CDF for non-homogeneous Poisson process [duplicate]

I am trying to understand the inverse transform method for simulating random processes and have managed to completely confuse myself. Consider a Poisson process whose conditional intensity is ...
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### Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
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### Deducing an optimal gambling strategy (using martingales).

Apologies in advance for the length, I tried being precise. Suppose a game where in each turn you can gamble a certain amount of money on the result of a fair coin toss. If the coin comes out tails ...
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### How to compute cumulative intensity process integral?

I am faced with a basic question about counting process and its intensity process used in survival analysis. It is actually the textbook example from Aalen's Survival and Event history analysis book. ...
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### Card Shuffling and Convergence in Probability

There are $4n$ cards, and we denote the set of cards with number $4k,k \in \{1,2,\ldots,n\}$ as $S$. The we shuffle the whole cards randomly, which means that each permutation will happen with the ...
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### Covariance between real and imaginary parts of Fourier transform of a stationary time series

Since Fourier transform of a random stationary time series(in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary parts of ...
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### How to make sense out of this: Ergodic theorem for Markov chains

We had the ergodic theorem for Markov chains, stating that: For a state space $S \subset \mathbb{N}$ and all functions $f \in L^1$ (meaning that $\sum_{s \in S} |f(s)|\pi(s) < \infty$) and an ...
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### law of iterated logarithm

Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the ...
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### Does irregularly transformed stationary process preserves stationarity?

I would like to apply the following theorem in a probably unusual way. Let $Z_t=f(Z_{t-1},Z_{t-2},\dots,Z_{t-M})+\varepsilon_t, t=1,2,\dots$ be a stationary and ergodic Markov chain as well as ...
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### How to prove that convergence in MGF implies Convergence in Distribution?

I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ?
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### Return time Markov chain

I have been wondering about this for quite a while now that I found in a textbook in the proof that an irreducible positive recurrent markov chain $(X_n)$ has a stationary distribution Let $t_i$ ...
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### Need help with a basic exercise about Markov chains

Suppose $\left\{ X_{n}\right\} _{n=1}^{\infty}$ is a Markov Chain taking real values. Are the following Markov Chains? $$Y_{n}=\sum_{i=1}^{n}X_{i} , Z_{n}=\left(X_{n},X_{n-1}\right)$$ Edit1 I ...
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### Zombie outbreak on a $k$-regular graph

Suppose we have a zombie outbreak on a connected $k$-regular graph of order $n$. There are $n_0$ initially infected zombie nodes, and each turn, each zombie infects its neighbors with probability ...