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let $ (\xi_i)_{i \ge 1} $ be identical independent random variable, taking value in $ (1,3]$

can we show:

$P( \exists N \in \mathbb{N}, \text{ s.t. } \forall k \ge 0, \prod_{i=1}^{N}\xi_{i+kN} > 6N ) =1 ?$

i.e, almost surely, for each consecutive block with length $ N$, its product is greater than its length?

I think law of large number may give clue for this problem:

the main problem is that $ \xi_i $ can be closed to 1, but we can prove $ \mathbb{E}\xi_i >1$, then roughly speaking $ \xi_i \sim \mathbb{E}\xi_i$, which may imply $\prod_{i=1}^{N}\xi_{i+kN}$ has exponential growth.

this is my idea, but do not know how to give a formal proof. Thanks!

Let $ (\xi_i)_{i \ge 1} $ be independent identically distributed random variables, taking values in $ (1,3]$.

Can we show:

$P( \exists N \in \mathbb{N}, \text{ s.t. } \forall k \ge 0, \prod_{i=1}^{N}\xi_{i+kN} > 6N ) =1 ?$

i.e, almost surely, for each consecutive block with length $ N$, its product is greater than its length?

I think law of large number may give clue for this problem:

the main problem is that $ \xi_i $ can be closed to 1, but we can prove $ \mathbb{E}\xi_i >1$, then roughly speaking $ \xi_i \sim \mathbb{E}\xi_i$, which may imply $\prod_{i=1}^{N}\xi_{i+kN}$ has exponential growth.

this is my idea, but do not know how to give a formal proof. Thanks!

let $ (\xi_i)_{I \ge 1} $ be identical independent random variable, taking value in $ (1,3]$

can we show:

$P( \exists N \in \mathbb{N}, \text{ s.t. } \forall k \ge 0, \prod_{i=1}^{N}\xi_{i+kN} > 6N ) =1 ?$

i.e, almost surely, for each consecutive block with length $ N$, its product is greater than its length?

I think law of large number may give clue for this problem, but do not know how to prove it. Thanks!

let $ (\xi_i)_{i \ge 1} $ be identical independent random variable, taking value in $ (1,3]$

can we show:

$P( \exists N \in \mathbb{N}, \text{ s.t. } \forall k \ge 0, \prod_{i=1}^{N}\xi_{i+kN} > 6N ) =1 ?$

i.e, almost surely, for each consecutive block with length $ N$, its product is greater than its length?

I think law of large number may give clue for this problem:

the main problem is that $ \xi_i $ can be closed to 1, but we can prove $ \mathbb{E}\xi_i >1$, then roughly speaking $ \xi_i \sim \mathbb{E}\xi_i$, which may imply $\prod_{i=1}^{N}\xi_{i+kN}$ has exponential growth.

this is my idea, but do not know how to give a formal proof. Thanks!

let $ (\xi_i)_{I \ge 1} $ be identical independent random variable, taking value in $ (1,3]$

can we show:

$P( \exists N \in \mathbb{N}, \text{ s.t. } \forall k \ge 0, \prod_{i=1}^{N}\xi_{i+kN} > N ) =1 ?$

i.e, almost surely, for each consecutive block with length $ N$, its product is greater than its length?

I think law of large number may give clue for this problem, but do not know how to prove it. Thanks!

let $ (\xi_i)_{I \ge 1} $ be identical independent random variable, taking value in $ (1,3]$

can we show:

$P( \exists N \in \mathbb{N}, \text{ s.t. } \forall k \ge 0, \prod_{i=1}^{N}\xi_{i+kN} > 6N ) =1 ?$

i.e, almost surely, for each consecutive block with length $ N$, its product is greater than its length?

I think law of large number may give clue for this problem, but do not know how to prove it. Thanks!