0
votes
1answer
17 views

Formally proving $\sum_{k=1}^{\infty}P\left(-k<X\leq-k+1\right)=P\left(X\leq0\right)$?

$\sum_{k=1}^{\infty}P\left(-k<X\leq-k+1\right)=P\left(X\leq0\right)$ This fact seems pretty obvious but how would I formally prove it, is there a painless way?
0
votes
2answers
57 views

How to show that $\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$

Let $x_n$ be a sequence of reals. Show that $$\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$$ Since the weak convergence is equivalent to pointwise convergence of characteristic functions ...
1
vote
1answer
21 views

In an irreducible, aperiodic, null-recurrent Markov chain holds $\sum_n p_{ij}^{(n)} = \infty$

My lecture notes state the following theorem: Theorem 2. Let $(X_n)$ be an irreducible, aperiodic, null-recurrent Markov chain. Then $$\forall i,j \in S : p_{ij}^{(n)}\to 0 \text{ and } \sum_n ...
1
vote
1answer
26 views

Question on a variation of Borel Cantelli Lemma

In this question, what is the purpose of the summation? If the limit of the sequence is zero, the corresponding series is convergent. Does the desired conclusion not then follow from BC1?
0
votes
3answers
279 views

What is the probability that A will win…

Two players are rolling two dices, if they get 6 Player A wins, if they get 7, player B wins, else they rolling the two dices again... What is the probability that A will win? I'd like to get any ...
0
votes
1answer
15 views

Determining the number of values required to find the probabilities of a set of values

Im trying to estimate when a random function with a set number of events will trend to its probability values and a ballpark on how much data I need to collect will help. Is there an equation that ...
0
votes
0answers
15 views

action of transition operator on function

Let $P$ be the transition operator of a markov chain with discrete time and discrete state space $X$. The action of the transition operator on a function $X \to \mathbb{R}$ is defined by $Pf(x) = ...
1
vote
0answers
18 views

Geometric Series and other Sequences

I am currently taking a class that assumes background with the mentioned. The class is "Intro to Probability." I however am not quite comfortable with them and wish to re learn the basics and ...
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
1answer
87 views

Convergence of the series of identically distributed dependent random variables

Let $a_1$, $a_2$, $\ldots$ be identically distributed, positive, not necessarily independent random variables. Consider the series $$\sum^{\infty}_{n=1} a_n$$ Is it true that the series diverges ...
1
vote
3answers
34 views

A question about sum of probability… if $P(X\ge n)-P(X=n)=P(X>n)$

If we know that $P(X\ge n)=(1-p)^{n-2}$ (This is not the main subject of the question, so I wont explain about it, hope this OK, but in sort: we get it because $P(X\ge n)=\sum_{k=n}^\infty ...
3
votes
2answers
61 views

A problem on random series

Suppose $X_1, X_2,\dots, $ be an independent sequence of random variables and $E[X_n] = 0 \forall n$ and $\sum_{n=1}^{\infty} \operatorname{Var}(X_n) < \infty$. I need to prove that ...
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 ...
1
vote
1answer
26 views

Why is this set an event?

As a part of a setup for another problem, my text remarks that it can be used without a proof that if $X_1, X_2, \ldots$ are random variables then $$C:=\{ \omega\in\Omega \ | \ \sum X_n(\omega) \ ...
1
vote
1answer
40 views

How does one show that the series converges almost surely?

Let $X_1, X_2, \ldots:\Omega\to \mathbb R$ be random variables. Define $C:=\{ \omega \ | \ \sum X_n(\omega) \text{ converges} \}$. There is such $q\in(0,1)$ that for all $n\in \mathbb N: P\{ |X_n| ...
2
votes
0answers
62 views

Random walk with $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} < \infty$

Consider a random walk started at $S_0=0$, denoted $S_n = \sum_{k=1}^{n}X_k$, where $X_1$, $X_2$... are the i.i.d increments. If we have $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} ...
0
votes
0answers
28 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
0answers
67 views

Sequence of probabilities with monotone function

Let $\{X_k\}_{k=1}^{\infty}$ be a sequence of i.i.d. random variables with finite support $S = \{ 1, 2, ..., N\}$. Let $P$ be the corresponding probability measure. For all $k \geq 1$, define $A_k := ...
3
votes
2answers
58 views

A different type of Law of Large Numbers

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of i.i.d. real random variables (with finite variance if needed). For $k>0$ fixed, I'm interested in estimating the quantity $$ ...
1
vote
1answer
37 views

Exchange of sequences of probability variables

Suppose I have two sequences of positive random variables $(X_n)$ and $(Y_n)$ and the following is true: for all $\epsilon>0$ and $\delta>0$, there exists $n_0$ such that for all $n\geq n_0$, ...
1
vote
1answer
29 views

Convergence of the limit of a sum of $|P_{nk}-P_{k}|$, where $P_{nk}$ and $P_{k}$ are sequences of nonnegative numbers summing to 1

Let $(P_{nk})_{k \geq 1}$, $n=1,2,\cdots$ and $(P_{k})_{k \geq 1}$ be a sequence of nonnegative numbers satisfying $\sum_{k=1}^{\infty}P_{nk}=1$ and $\sum_{k=1}^{\infty}P_{k}=1$, and let $\lim_{n ...
0
votes
1answer
40 views

Convergence of a series of independent r.v's iff $\sum_{n=1}^{\infty}a_{n}<\infty$ for $a_{n} \in (0, 1/3)$

Suppose $(X_{n})_{n\geq1}$ is a sequence of independent random variables with $P(X_{n}=2)=a_{n}$; $P(X_{n}=n^{\beta})=a_{n}$; and $P(X_{n}=a_{n})=1-2a_{n}$ with $a_{n}\in (0, \frac{1}{3})$ for all n, ...
1
vote
0answers
72 views

Expectation involving a maximum of a sequence of i.i.d. Gaussians

Let $X_1,\ldots,X_n$ be a sequence of i.i.d. standard Gaussian random variables. Denote the maximum of this sequence by $M_n$. I am interested in evaluating the following expectation: ...
1
vote
1answer
201 views

The normal approximation of Poisson distribution

(I've read the related questions here but found no satisfying answer, as I would prefer a rigorous proof for this because this is a homework problem) Prove: If $X_\alpha$ follows the Poisson ...
1
vote
1answer
89 views

Probability of index at which sequence stops decreasing

Let $X_1,X_2, \dots $ be a sequence of independent and identically distributed continuous random variables. Let $N \ge 2$ be such that $X_1 \ge X_2 \ge X_{N-1} < X_N$. That is, $N$ is the ...
1
vote
0answers
71 views

Probability of a sequence

For all $k = 1, 2, ...$, consider $f_k: \mathbb{R}^k \rightarrow \{1, 2, ..., C\}$, for some given positive integer $C$, with the following properties. For all $k$, if $(y_1, ..., y_k)$ is a ...
0
votes
0answers
58 views

Asymptotic (in)dependence of a maximum of an i.i.d. sequence of Gaussian random variables on a single random variable in this sequence

Suppose that I have a sequence of $n$ i.i.d. standard Gaussian random variables $X_1,\ldots,X_n$ where $X_i\sim\mathcal{N}(0,1)$. Denote the maximum of this sequence by $M_n=\max(X_1,\ldots,X_n)$. I ...
5
votes
2answers
272 views

Convergence of a sequence involving the maximum of i.i.d. Gaussian random variables

It's well known that, for a sequence of $n$ i.i.d. standard Gaussian random variables $X_1,\ldots,X_n$, where $X_\max=\max(X_1,\ldots,X_n)$, the following convergence result holds: ...
0
votes
0answers
34 views

Limits of expectation of a function

Suppose that $g(x,y)$ is a smooth function with respect to $x$ and $y$, and that it is bounded on the domain of interest: $a\leq g(x,y)\leq b$, with $a$ and $b$ being real constants. Now let $(X)_n$ ...
0
votes
1answer
174 views

The meaning of almost surely convergence

Consider the space of sequences of tosses of a fair coin. In particular let $X_{nH}$ be the number of times that a coin lands on heads in a sequence of $n$ coin flips. Consider statement $S$ below. ...
1
vote
0answers
112 views

Maximum of a sequence of almost-identical independent normal random variables

Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about ...
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)} ...
1
vote
2answers
100 views

Probability of a sorted sequence

Now I tried tackling this question from different sizes and perspectives (and already asked a couple of questions here and there), but perhaps only now can I formulate it well and ask you (since I ...
1
vote
1answer
108 views

Almost sure convergence of generalized random harmonic series

Problem. Prove that $$\sum_{n=1}^\infty \frac{1}{n} \cos(2 \pi \cdot2^{n^2} x)$$ converges for almost every $x \in [0, 1]$ (with Lebesgue measure). (Credits to B. Tsirelson.) My partial solution. I ...
3
votes
1answer
68 views

distribution of $\cos(\omega_0 n)$ where n are integers?

Assume we have the sequence $\,x[n]=\cos(\omega_0 n),$ where $n$ are integers. If we suppose these are realizations of a random variable, what would be the p.d.f. of that random variable?
2
votes
4answers
2k views

Mutually Exclusive Events (or not)

Suppose that P(A) = 0.42, P(B) = 0.38 and P(A U B) = 0.70. Are A and B mutually exclusive? Explain your answer. Now from what I gather, mutually exclusive events are those that are not dependent upon ...
0
votes
0answers
75 views

a.s. convergence of exponential function

Suppose $Ω=[0,1]$, $\mathcal F=\mathcal B\left([0,1]\right)$, $\mathbb P=\lambda$, and consider $$X_n=e^n\chi_{A_n}, \qquad A_n=\left[0,\frac 1n\right].$$ I'd like to show that the sequence ...
1
vote
1answer
98 views

Upper bound on a probability generating function with a finite first moment

If $X$ is discrete random variable taking values in non-negative integers $\{0,1,\ldots\}$, its probability generating function is defined as follows: $$G(z)=\mathbb{E}(z^X)=\sum_{x=0}^\infty ...
3
votes
2answers
293 views

Convergence of random variables (Durrett: Probability Theory and Examples)

I was working out some problems from Rick Durrett's Probability theory and Examples (2010 edition), when I came across a very unusual question(reproduced here ad-verbatim): If $X_n$ is ANY sequence ...
0
votes
1answer
117 views

Calculating probabilities in genetic sequences

I am working with certain recurring sequences in genetics and try to calculate certain probabilities: Let for instance $$\langle g_i\rangle :=\{1,1,1,6,1,1,1,6,...,1,1,1,6\}$$ and $$\langle ...
2
votes
1answer
86 views

Behaviour of Two Coupled Sequences Towards a Stable Distribution

The following question arises from research that I am doing in swarm intelligence. The relationships given come from geometric considerations which, I believe, should not be relevant for this problem. ...
2
votes
1answer
126 views

Almost sure convergence of sequences

Let $(a_n)$ be a sequence of positive real numbers such that $\sum_{n \geqslant 1} a_n < \infty $ and $$\sum_{n \geqslant 1} \Pr\left(\left|X_{n+1} - X_n\right| > a_n\right) < \infty $$ Why ...
13
votes
2answers
614 views

On the set of the sub-sums of a given series

Choose a sequence $(x_n)_{n\in\mathbb N}$ of nonnegative real numbers with finite sum $x=\sum\limits_{n\in\mathbb N}x_n$ and consider the set $X=\{x_I\mid I\subseteq \mathbb N\}$ where, for every ...
3
votes
1answer
109 views

Convergence of a positive series with an application to sums of Poisson random variables

Main question: Let $a_n \geq 0$ and $b_n = \sum_{k=1}^n a_k \uparrow \infty$. Is it true that $$ \sum_{n=1}^\infty \frac{a_n}{b_n^2} < \infty \,? $$ I strongly suspect there is either a short, ...
1
vote
1answer
243 views

$X$ is a Geometric random variable find the expectation of $1/X$

Let $X$ be a geometric random variable with parameter $p$, find the expectation of $E[1/X]$. I need help simplifying the series.
2
votes
0answers
83 views

Identify the smallest $c$ such that $P(|X_n| \ge c \sqrt{\ln n} \text{ i.o.}) = 0$ for normally distributed $X_n$

The problem is to show that $P(|X_n| \ge c \sqrt{\ln n}\text{ i.o.}) = 0$ for standard normal $X_n$ that are not necessarily independent. Also, identify the smallest such $c$. I am thinking that the ...
2
votes
2answers
425 views

What is the limit of a sequence of events? Probability

Here's a question I'm sturglling with: Show that for an increasing sequence of events $$A_1\subset A_2\subset A_3\subset ...$$ the next equation holds ...
5
votes
1answer
973 views

Another question on almost sure and convergence in probability

Convergence in probability implies convergence on a subsequence almost surely. But this means we fix a subsequence, such that $X_{n_k}$ converges for almost every $\omega$, right? The subsequence we ...
1
vote
1answer
674 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 ...