# Tagged Questions

26 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, ...
17 views

### How can I find the expected value of a random variable with terms that increase until infinity?

Here is the question A company buys a policy to insure its revenue in the event of major snow storms that shut down business. The policy pays nothing for the first such snowstorm of the year and ...
42 views

### Dependent Expectation in Random Numbers Illustrated by Prime Repetition in Pi

When approximating Pi, appending each numerical digit as you refine, what is the first repetition of a four-digit prime number? For instance the first repetition of any one-digit number in the ...
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?
59 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 ...
26 views

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 ...
91 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 ...
35 views

41 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, ...
76 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: ...
216 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 ...
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 ...
72 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 ...
61 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 ...
286 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: ...
35 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 $(X)_n$ ...
177 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. ...
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 ...