# Tagged Questions

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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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### 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 ...
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### 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?
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### 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 ...
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### 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. ...
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### 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 ...
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### 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. $g_{X_n}(s) = \mathbb E[s^{X_n}] = \sum_{k=0}^{\infty} s^k \mathbb P(X_n=k)$. ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...