Vtyjkyuk

Definition of a block of events:

Let $(Omega, mathscr F, mathbb P)$ be a probability space. Consider the collection of events $H_n_n ge 1$. Define a block of $H_n_n ge 1$ to be an event of the form $$A_f(n) = bigcap_i=1^f(g(n)) H_n+i$$

Now consider the following examples of independence of a collection of blocks which would lead to succeeding/subsequent (correct my grammar!) questions of independence of something I call sub-events:

On independence of a collection of blocks:

Let $(Omega, mathscr F, mathbb P)$ be a probability space.

Let $H_n_n ge 1$ be independent coin tossing, i.e. $H_n = $nth coin is heads$$, and let $$A_n = bigcap_i=1^g(n) H_n+i$$

represent a block of coin tossing, where $g(n) = left lfloor log_2 n right rfloor$

We can show that

1. The collection of blocks $$A_f(n)_n ge 1,$$ where $f(n) = n$, is not independent (see here).




2. The collection of blocks $$B_n = A_f(n)_n ge 1,$$ where $f(n) = left lfloor nlog_2 n^2 right rfloor$, is independent (see here).




3. The collection of blocks $$C_n = A_f(n)_n ge 1,$$ where $f(n) = left lfloor log_2 n right rfloor$, is not independent (see here). Note that:

$$C_n = H_n+1 cap H_n+2 cap dots cap H_n+ left lfloor log_2 left lfloor log_2 n right rfloor right rfloor = bigcap_i=1^left lfloor log_2 left lfloor log_2 n right rfloor right rfloor H_n+i = A_left lfloor log_2 n right rfloor$$

4. Now, collection of blocks $A_f(n)_n ge 1$ is a collection of independent events if $A_f(n)_n ge 1$ is a collection of pairwise independent events, i.e. if $$f(n) + g(f(n)) < f(n+1) + 1,$$ then $A_f(n)_n ge 1$ is independent (see here).

Definition of a sub-event, a $sigma$-algebra that is generated by events and that is a sub-$sigma$-algebra of a $sigma$-algebra generated by events:

Let $(Omega, mathscr F, mathbb P)$ be a probability space. Consider the 2 collections of events $H_n_n ge 1$ and $B_n_n ge 1$. I define the collection $B_n_n ge 1$ to sub-event of $H_n_n ge 1$ if $B_n_n ge 1$'s $sigma$-algebra is a subset of the $H_n_n ge 1$'s $sigma$-algebra, i.e. $$sigma(B_1, B_2, ...) subseteq sigma(H_1, H_2, ...).$$

Proposition: A collection of blocks of $B_n_n ge 1$ is a sub-event of $H_n_n ge 1$.

Questions on independence of events in a sub-event:

Let $B_n_n ge 1$ be a sub-event of the collection of independent events $H_n_n ge 1$. I believe we do not necessarily have that $B_n_n ge 1$ is a collection of independent events. Otherwise, I wasted 50 reputation points on this bounty.

1. What conditions are there to say that a sub-event $B_n_n ge 1$ is a collection of independent events?

For example, can we say similar to (4) above that a sub-event $B_n_n ge 1$ is a collection of independent events if $B_n_n ge 1$ is a collection of pairwise independent events? Or if $B_n_n ge 1$ is a collection of pairwise independent events, and the events independent to $B_n$ form an algebra for any $n$? I'm thinking that for blocks $A_f(n)$, perhaps it can be shown that if the blocks are pairwise independent, i.e. $f(n) + g(f(n)) < f(n+1) + 1,$ then the events independent to the blocks $A_f(n)$ form an algebra/a $sigma$-algebra? Then we somehow extend this to sub-event.

2. What conditions are there to say that a sub-event $B_n_n ge 1$ is a collection of $k$-wise independent events?

Possibly related questions:

Kolmogorov 0-1 Law Converse?

Choosing the correct subsequence of events s.t. sum of probabilities of events diverge

$sigma$-algebra of independent $sigma$-algebras is independent