0
votes
0answers
8 views

Differentiability of random processes.

I know the appropriate criterions for mean-square differentiability of random processes. These criterions are connected with covariance function of a process. Are there any criterions for ...
0
votes
0answers
17 views

Sufficient conditions for Uniform Law of Large Numbers

I would need a Uniform Law of Large numbers for $f_T(\theta)$ over $\Theta$ when $f$ is the indicator function and, thus, not continuous over $\Theta$. Do you know about any sufficient conditions?
2
votes
1answer
37 views

Three questions about ucp convergence

We say that a sequence of processes $X^n$ converges to a process $X$ uniformly on compacts in probability if for all $\epsilon >0, t>0$ $$P[\sup_{s\le t}|X^n_s-X_s|>\epsilon]\to 0 $$ for ...
2
votes
0answers
92 views

When does almost sure convergence of stochastic integral imply $L^2$ convergence?

Consider a probability space $(\Omega, \mathcal{F}, P)$ equipped with a Brownian motion $W$. Let $(\xi_n)_{n=1}^\infty$ be a sequence of adapted $\mathcal{F}(t)$-progressively measurable processes. ...
1
vote
1answer
38 views

questions on mean-square convergence for a AR(P) example

In the following example related to AR(P) process, I have two questions I marked these two questions with two different colors. Question 1) (I marked with yellow), why that sum is mean-square ...
2
votes
1answer
65 views

Convergence of Random Variables to $- \infty$

The next exercise is taken from Alan Karr: If $X_1, X_2,...$ are independent with: $$P(X_k=k^2)=\frac{1}{k^2}$$ $$P(X_k=-1)=1-\frac{1}{k^2}$$ Prove that $\sum_{k=1}^{n} X_k \xrightarrow{a.s.} ...
1
vote
0answers
42 views

Infinite discounted sum of weakly dependent Normal random variables

Say I have the expected value of a sum of weakly dependent Normal random variables of the form $\mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right]$, where $0<a<1$. I was wondering under what ...
0
votes
1answer
101 views

Using characteristic function to deduce convergence of Bernoulli random variables

Let $Y_1, Y_2,...$ be a sequence of independent Bernoulli(0.5) random variables and $X_n = \sum_{i=1}^{n} Y_i 2^{-i}$ I need to use the characteristic function to deduce that $X_n$ converges in ...
0
votes
1answer
118 views

Large deviation of Bernoulli random variables and applying Chernoff bound

Let $X_1,X_2,..,X_n$ be i.i.d Bernoulli random variables with $P(X_1)=0.005$ and let $S_n:=X_1+...+X_n$. I need to: Evaluate the exact value of $P(S_{100} \geq 4)$ Use the Chernoff bound to estimate ...
0
votes
1answer
75 views

On the quadratic variation

I understand that the Quadratic Variation of Brownian Motion $B_t$ is $[B_t,B_t]=t$ and I know that the equality is under the meaning of $\mathcal{L}^2$ convergence. Yet I saw in some book saying that ...
1
vote
1answer
67 views

Does $\sup_{t \leq T} |M_{n_k}(t)-M_{m_k}(t)|\to 0$ imply $\lim_k M_{n_k}(t)$ exists and is continuous?

This came up in proving that $\mathcal{M}^2_c$ is a complete metric space using the invariant metric induced by $$ ||M|| = \sum_k \frac{||M(k)||_2\wedge 1}{2^n}. $$ Suppose $M_n(t)$ is a sequence of ...
2
votes
1answer
109 views

Dominated Convergence Thm (DCT) for Double Sequences

By a version of the Dominated Convergence Theorem (Thm 25.12 in Billingsley 86) $ E(X_n)\rightarrow E(X) $, if $X_n \overset{p}{\rightarrow} X$ and $X_n$ is uniformly integrable sequence of random ...
1
vote
1answer
217 views

Convergence in distribution of Gaussian processes

Assume given a sequence $(W_n)$ of Gaussian processes indexed by, say, $\mathbb{R}^p$, with mean zero and covariance function $R_n$. This means that for each $n$, the finite-dimensional distributions ...
2
votes
3answers
163 views

Proof of Levy's zero-one law

Let $(\Omega, \mathcal{F},\mathbb P)$ be a probability space and let $X$ be a random variable in $L^1$. Let $(\mathcal{F}_k)_k$ be any filtration, and define $\mathcal{F}_{\infty}$ to be the minimal ...
0
votes
1answer
44 views

Order of convergence of a sum

Let $(X_t)_{t\geq 0},\;X_0=0$, be a positive stochastic process such that \begin{align*} \mathbb{E}\left[\sum_{n=1}^{\infty}X_t^n\right]=\sum_{n=1}^{\infty}\mathbb{E}[X_t^n]<\infty. \end{align*} ...
0
votes
1answer
74 views

Brownian motion and convergence in probability of step functions

For positive $a$ and Brownian motion $B$, I want to compute $\int_0^a g(s)dB_s$ where $g \in L^2$ and $g$ is a step function if there exists partition $0=t_0 < ... < t_n = a$ such that $g = ...
2
votes
1answer
102 views

Weak convergence and generating function

Let $X_n$ be a sequence of $\mathbb N_0$ valued random variables and denote by $g_{X_n}$ their generating function, i.e. $g_{X_n}(s) = \mathbb E[s^{X_n}] = \sum_{k=0}^{\infty} s^k \mathbb P(X_n=k)$. ...
3
votes
1answer
141 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
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
84 views

Convergence to Brownian motion integral

Let $X_i$ be i.i.d with $\mathbb{E}(X_i) = 0$ and $Var(X_i) =1, \, S_n = \sum_{i=1}^n X_i$. I would like to show that $\sum_{i=1}^n \frac{f(S_i/\sqrt{n})}{n}$ converges to $\int_0^1 f(B_t)dt$ in ...
1
vote
1answer
74 views

Asymptotic Behaviour of a Continuous Square Integrable Martingale

Consider $\left( M^\alpha _t\right)_{t \geq 0} \in \mathcal M_c ^2$ such that $$ \mathbb E \left\{ \sup_{0\leq s \leq t} \left |M^\alpha _s \right| ^2\right \} \leq C_\alpha t^{1-\alpha} $$ How to ...
3
votes
1answer
222 views

Biased Random Walk Converging to a Brownian Motion with drift (Donsker's Theorem)

Fix $N$ and suppose $\{X_n\}_{k=1}^{N}$ are i.i.d steps that are $\pm 1$ with equal probability. Then $S_n = \sum_{k\leq n} X_k $ is a simple random walk, and (with the right scaling) we know that the ...
2
votes
0answers
73 views

Ergodic/stochastic convergence

I do have a problem with my homework, and to be honest I am simply lacking any idea on how to begin- maybe someone could give me a tip. First off, here is the assignment: The whole assignment deals ...
1
vote
0answers
168 views

Constructing Ito integral for adapted process

I am trying to construct Ito integral for adapted process. However, I am stuck at some point. Let $X^n(t)$ be a sequence of simple processes convergent in probability to the process $X(t)$. Then the ...
5
votes
2answers
178 views

Convergence to the stable law

I am reading the book Kolmogorov A.N., Gnedenko B.V. Limit distributions for sums of independent random variables. From the general theory there it is known that if $X_i$ are symmetric i.i.d r.v ...
2
votes
2answers
238 views

What does it mean for MCMC to converge?

I know that a Markov Chain is a discrete random process where the current state decides the next and in a random walk, the probability that we move from node u to v is 1/N(u). An MCMC sample will ...
1
vote
1answer
87 views

If almost all sample paths of a process converge to a constant, is it okay to say the process itself converges to a constant?

Suppose that $(a_n)$ is a sequence of reals and $(e_n)$ is a sequence of iid r.v.s such that $\Pr(e_n=\pm1)=1/2$. It is well known that $\sum a_ne_n$ converges a.s. to some limit r.v. iff $\sum a_n^2 ...
3
votes
1answer
198 views

Rate of Convergence: Competing Poisson Processes

I am analyzing one of the properties of the Poisson Process, that of Competing Processes: Suppose $N_1(t), t \geq 0$ and $N_2(t), t \geq 0$ are independente Poisson processes with respective rates ...