1
vote
1answer
49 views

locally linearize a CDF

I have a sequence of discrete CDF's that converge to continuous CDF. Assume we call it $F_n(x)$. If say at some point, say $R$, $F_n$ is differentiable, then we can write $F_n(R+\xi) \approx ...
1
vote
0answers
82 views

Entropy Rate of a sequence of Random Variables with Distributions related to Primes

Let us consider a stochastic process $\mathcal{X}=\{X_i\}_{i \in \mathbb{N} }$ where $X_i$'s are independent and $X_i$ is distributed as $$X_i=p_k \ \mbox{w. p.}\frac{p_k}{\sum_{l=1}^{i}p_l},\ 1\leq ...
0
votes
1answer
154 views

p-lim inf definition (limit inferior in probability)

For an arbitrary sequence of real-valued random variables $\{Z_n\}_1^\infty$ , we define limit inferior in probability as follow : $$ p-\liminf_{n\to \infty} Z_n \equiv \sup \{ \beta|\lim_{n\to ...
1
vote
2answers
172 views

Entropy expression optimization with Langrange multipliers

I have recently encountered variants of the following expression: \begin{equation} S = H(a,b,c,d)-H(a+b,c+d) \end{equation} where $H$ is the Shannon entropy function, that is $H(X)=\sum_{x\in X}-x\log ...
2
votes
2answers
100 views

Numbers which encode other numbers infinite times.

Let $g(x,n)$ be a function which chops off the first n digits of the binary decimal expansion of x. eg $g(0.1010111,2)=0.10111$ Is there a function $f(x)$ from the reals to the reals, such that for ...