All Questions
33 questions
12
votes
1
answer
330
views
Convergence of an implicitly defined sequence of random variables
Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
- 171
9
votes
1
answer
556
views
Berry-Esseen bound for martingale sequence with varying and dependent variances
Let $(X_{1},\ldots,X_{k},\ldots)$ be a martingale difference sequence, i.e.
$$
E[X_{k}|\mathcal{F}_{k-1}] = 0
$$
where $\mathcal{F}_{k-1}$ is the $\sigma$-algebra filtration at $k-1$.
Let $\sigma_{...
- 719
6
votes
1
answer
374
views
Large deviation for Brownian path on $[0,\infty)$
It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path.
If we equip the space of continuous function starting from $0$, ...
- 123
5
votes
0
answers
1k
views
Asymptotic behavior of row sums in 2-d array of random variables
Set-up. Let $f : \mathbb{N} \to \mathbb{N}$ be increasing. For each $m \in [0,1]$, consider an infinite two-dimensional array of random variables, where row $n$ has $f(n)$ variables:
$B^m_{1,1}$ $B^...
- 61
4
votes
1
answer
537
views
Convergence of random variables with hypergeometric distribution
This is a very interesting conjecture of large scale property of hypergeometric distribution.
Let $a>1$ be a integer constant, $N\in\mathbb{N_+}$, for any $x<N-1$, consider $N+(a-1)x$ balls in ...
- 71
3
votes
1
answer
829
views
The weak convergence of finite dimensional distribution of Gaussian process does not imply the weak convergence in $C[0,1]$
In the study of weak convergence in $C[0,1]$, a common example is always being considered: $$X_{n}(t)=nt1_{[0,1/n]}(t)+(2-nt)1_{(1/n,2/n]}(t).$$ This example serves a counter-example to show that the ...
user157884
3
votes
1
answer
125
views
Convergence of function of stochastic processes
Let $X_t$ be a fixed cadlag semi-martingale and $J_n$ be a fixed sequence of functions from $\mathbb{R}^d$ to $\mathbb{R}$ which are twice continuously differentiable. If $J_n$ converge pointwise to ...
- 5,405
3
votes
1
answer
182
views
Superlinear Convergence of a Markov Chain
Suppose that we have a Markov process $\{Z_t\}_{t=0}^\infty$, where $Z_t \geq 0$ for any $t$. Assume that, conditioning on $Z_t = z_t$, we have
$
\mathbb{E}\{Z_{t+1}|Z_t = z_t\} \leq \kappa z_t^2
$. ...
- 1,127
3
votes
1
answer
177
views
Convergence of SDEs
Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...
- 5,405
3
votes
0
answers
136
views
An integral involving Levy process with no positive jumps
Let $L_t$ be a Levy process that has no positive jumps, but is not strictly decreasing, i.e
$$
L_t = \gamma t + \sigma B_t + J_t,
$$
where $B_t$ is a Brownian motion, $J_t$ is a pure jump process with ...
- 103
2
votes
1
answer
120
views
Approximation of a stationary process by a sequence of ergodic and stationary sequence of stochastic processes
Let $X = [X_t : t \in \mathbb{Z}] \sim P$ and $Y = [Y_t : t \in \mathbb{Z}]\sim Q$ be two stochastic processes. Let's define the Mallows metric. Let $\mathcal{M}_m$ be the random vectors $(X,Y)$ ...
- 135
2
votes
1
answer
184
views
A question about convergence of stochastic processes converging to a random walk
Consider the following random walk $(y_t)_{t \in \mathbb Z_+}$:
$$y_t = y_{t-1} + u_t,\quad (u_t)_{t \in \mathbb Z_+} \overset{iid}{\sim} N(0,1), \quad (t \in \mathbb Z_+)$$
where $y_0, u_1, u_2,...$ ...
- 13
2
votes
1
answer
124
views
Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables
This is related to one of my previous questions here.
Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
- 399
2
votes
1
answer
315
views
Linear or quadratic combinations of i.i.d. random variables [closed]
I already posted this question here https://math.stackexchange.com/questions/769920/law-of-large-numbers-for-linear-quadratic-combinations-of-i-i-d-random-variab but I received no answers.
Let $(X_i)...
- 223
2
votes
0
answers
175
views
Representing a continuous time-inhomogeneous Markov chain by a stochastic integral
I am interested in the following mean-field model introduced in the reference below:
There are $N$ particles. At each instant of time, a particle's state is a particular value taken from the finite ...
- 31
2
votes
0
answers
207
views
markov processes and ergodic theory
For an ergodic Markov Chain
$$
\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]
$$
where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...
- 169
2
votes
0
answers
199
views
CLT for a Markov Renewal Process
Suppose $(X,T)=\{(X_n,T_n)\}_{n\geq0}$ is a Markov renewal process, where $X$ is a finite-state, discrete-time Markov chain with state space $\{1,2,...,R\}$. $T$ is the additive component, more ...
- 41
1
vote
2
answers
169
views
Asymptotic properties of weighted random walks / infinite convolutions of random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. real-random variables. Let further $0<c<1$. I'm interested in the asymptotic properties of
$$
\sum_{k=1}^n c^k X_k.
$$
I can prove that this ...
1
vote
1
answer
412
views
Almost sure convergence of the supremum over a class of random variables
Let $\mathcal{X}_n=\{ X_{n,\lambda}, \lambda \in \Lambda\}$ be a collection of random variables (defined on the same probability space) indexed by a deterministic index $\lambda$ over an index space $\...
- 195
1
vote
2
answers
194
views
Continuity of the densities of a stochastic process
Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ an interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\mathbb{R}^d)$ for ...
- 463
1
vote
1
answer
107
views
Convergence of discretized process when its predictable part converges to infinite variation process
This question seems to be related to Theorem IX.7.28 in J. Jacod and A. Shiryaev's Limit theorems for stochastic processes (2013), and it is very important to prove asymptotic properties of my ...
- 284
1
vote
0
answers
55
views
Limit process of a sequence of Gaussian variables on mesh grid going to zero
Consider the interval $[0,1]$ and a partition $\mathscr{P}_n = \{ [t_i,t_{i+1}), \, i=1,\ldots,N_n \, : \, 0=t_0 < \ldots < t_{N_n} = 1\}$. Suppose that for all $i$ and $t \in [t_i,t_{i+1})$, we ...
- 141
1
vote
0
answers
36
views
Uniform distribution as argument for copula likelihood
I am reading a well-known paper about copulas by Chen and Fan (2006). Specifically, Proposition 4.2 (see attached), in which all the arguments are uniform $U_{t-1}, U_t$. However, when the copula is ...
- 33
1
vote
0
answers
57
views
Convergence of stochastic linear recurrences
Suppose that $\zeta_t$ is a univariate, stationary stochastic process ($t\in\mathbb{N}^+$).
Let $x_0\in\mathbb{R}^n$, and let $f:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$ be a continuously ...
- 183
1
vote
0
answers
61
views
Convergence of empirical measure to Mc-Kean Vlasov equation for mean-field model with jumps
I am interested in the following mean-field model introduced in the reference below:
There are $N$ particles. At each instant of time, a particle's state is a particular value taken from the finite ...
- 31
1
vote
1
answer
335
views
Finding a connection between two types of convergence
Please, help me find connections between two types of convergence:
Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences:
1) $X_n \...
- 31
0
votes
2
answers
182
views
Show that the set of strictly stationary, mean zero and finite variance stochastic processes is closed (or not)
Let $\mathcal{P}$ be the set of real-valued and strictly stationary processes with expectation zero and finite variance, i.e.:
\begin{equation}
\mathcal{P}:=\left\{ X = (X_t)_{t \in \mathbb{Z}} \, ...
- 135
0
votes
1
answer
72
views
Exceedance distribution of Levy process
Consider a Levy process $L(t)$ with linear drift $-1$, no Brownian motion component, and Poisson jumps at rate 2 with size Uniform($0, 1$), and with $L(0)=0$. This process has zero mean drift.
Let $\...
- 294
0
votes
1
answer
169
views
Understanding the approximation of a random sum of random processes
I want to understand an approximation of a compound Poisson distribution in this paper.
First, let's set the environment. Consider $\mathcal{P}$ the class of distributions of real-valued and strictly ...
- 135
0
votes
1
answer
450
views
A complex question related to a certain convergence of Lévy measures
Consider the sequence of stochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and:
\begin{equation}\label{I}\tag{SP}
X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...
- 13
0
votes
0
answers
74
views
Convergence of stochastic process $X_n$
Consider the discrete time random process $X_n,n\in \mathbb N$, with
$$X_{n+1}=(1-K)\cdot X_n+K\cdot\frac{G_n}{c}\cdot X_n$$
where $G_n$ is a random variable with expectation $\mathbb E[G_n\mid X_n]=\...
- 5
0
votes
0
answers
123
views
Tightness of a uniformly bounded sequence of functions integrated with respect to a semimartingale
I am reading this paper by Jacod, Jakubowski and Mémin. In the proof of Theorem 1.3 the authors define, for each $n\geq1$ the function $\phi_n$ by
$\quad\phi_n(s)=i+1-ns,\quad\text{if } \frac{i}{n}&...
- 101
-1
votes
1
answer
396
views
Convergence of Radon-Nikodým derivative
Imagine we have a sequence of finite measures $\nu_n << \mu_n$ (on the torus $\mathbb{T}^2\subseteq \mathbb{R}^2$) converging weakly to some measures $\nu << \mu$. Do we automatically have ...
- 183