0
votes
1answer
28 views

Corollary of Kolmogorov zero-one law

Here is another corollary of the theorem: Kolmogorov zero-one law given in my textbook (Probability path). How can I apply the said theorem given that if $X_n$ are independent random variables, ...
0
votes
1answer
19 views

Notation in sequences of random variables

I read in statistics books that a sequence of random variables are often written as ${X_n}$. But in all the theorems it just says $X_n$. Why is that? And does ${X_n}$ symbolize the WHOLE sequence ...
1
vote
1answer
43 views

Prove that the series $\sum\limits_{n=0}^{\infty}X_n$ converges almost surely

I'm trying to solve the following Problem: Let $(X_n)_{n\ge 1}$ be a sequence of real valued random variables defined on some probability space $(\Omega, \mathcal{A},P)$. Assume that there ...
1
vote
0answers
57 views

Sum of poisson random variables

Let $N, X_1, \dots , X_n$ be independent random variables. $N \sim P(\lambda) \quad (\text{Poisson distribution})$, while $X_k \sim B(p)$ (Bernoulli) Let us consider the "random" sum $S = X_1 + ...
0
votes
0answers
31 views

Expected Value With Signum

So I am trying to show convergence of a filter, and in order for it to converge, I need the following condition to hold: $ E\{ \mathbf{s} x sgn(\mathbf{h}^{T} \mathbf{s} x) \} \; \alpha \; E \{ ...
3
votes
4answers
147 views

Does this sum converge or diverge?

Does the infinite sum $\large{\sum_{n=1}^\infty \frac{1}{n^{x_{\small{n}}}}}$ converge if $x_n$ is a random variable (generated within each term) that takes values between $0$ and $2$ with equal ...
1
vote
0answers
123 views

Limit of sequence of integral related i.i.d. observations

Let $X_1,\dots,X_n$ be i.i.d. random variables, each uniformly distributed on $[0,1]$. Let $\hat F_n$ be their modified empirical distribution function, i.e., $$ \hat ...
2
votes
0answers
60 views

weak convergence and composition

Assume $X_n$ is a sequence of random variables defined on a common probability space and $X_n$ converges weakly (in distribution) to $X$ as $n \to \infty$. Assume $u_n$ is a sequence of integer valued ...
2
votes
0answers
69 views

A nice sequence of random variables

Let $f:U\mapsto \mathbb{R}^k$ with $U\subset \mathbb{R}$ be a smooth injective function. Suppose that $\sqrt n(Y_n- Y)\to N(0,\Omega)$ in distribution with $Y=f(X)$. Define $X_n$ by ...
0
votes
0answers
34 views

Robbins-Siegmund like theorem for a nonlinear system

Recently I came to know about this theorem due to Robbins and Siegmund which states the following: Let us have on a probability space $(\Omega, \mathcal{F},P)$, a filtration $\{\mathcal{F}_n\}$ and ...
1
vote
1answer
69 views

Is there a sequence of i.i.d. random variables that is eventually monotonically decreasing?

Here is the problem I'm struggling with: Let $(X_n)$ be is a sequence of independent and identically distributed random variables. What is the probability that the sequence is monotonically ...
3
votes
1answer
92 views

Three series of Kolmogorov

Let $X_n\geqslant 0$ be a sequence of independent random variables. The following are equivalent: $i) \sum_{n=1}^{\infty}{ X_n} <\infty$ a.s $ii)$ $\sum_{n=1}^{\infty}{ \mathbb P(X_n>1)} ...
0
votes
0answers
63 views

Given an innovations sequence, calculate transformation matrix, A

Let $Y_1,Y_2,Y_3,X)^T$ be a zero mean random vector with correlation matrix, $$ \begin{pmatrix} 2 & 1 & 1 & 2 \\ 1 & 2 & 1 & 2 \\ 1 & 1 & 2 & 2 \\ 2 & 2 & 2 ...
3
votes
0answers
80 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}\sim $Bernoulli$(1/2)$ taking values in $\{0,2\}$. Let us define $X_n$ for $n>1$ as below$$X_{n+1}=a_nX_n+a_{n-1}X_{n-1},\ n\geq1 $$ Then it follows that ...
1
vote
1answer
40 views

Uniform convergence of a series through induction principle

I'm working on this paper. Can you please explain me the following passage of the proof that the series (2.9) converges uniformly? Given: $$ |\delta_{k+1}| \le ...
1
vote
1answer
688 views

Sums of independent random variables converging almost surely

I am working through Achim Klenke's text entitled "Probability Theory", and I came across the following interesting exercise: Let $(X_i)_{i\in\mathbb{N}}$ be independent, square-integrable random ...
0
votes
1answer
281 views

Convergence of sequence of Bernoulli random variables

I'm stuck with this problem. Let $X_1, X_2, ...$ be a sequence of independent Bernoulli random variables. Show that if $$\sum_{i=1}^n \frac{p_i}{n} \to l \; \text{ as } \; n \to \infty$$ then ...