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There I will list the inequalities and asymptotic theorems on random walks, that are currently known to me:
Notation that will be used in the list:
$\{X_n\}_{n = 1}^\infty$ are i.i.d. random variables....

All subgroups of a cyclic group (including the proper ones) are cyclic. And there is the following classification of non-cyclic finite groups, such that all their proper subgroups are cyclic:
A ...

One of the most classical examples is the word «algebra», which denotes not only a branch of mathematics, but also the following mathematical objects:
In linear algebra an algebra is a vector space, ...

Uncorrelatedness implies independence
This statement is indeed false. Suppose $X \sim U[-1, 1]$. Then $Cov(X, X^2) = EX^3 - EXEX^2 = 0$, but $X$ and $X^2$ are clearly not independent. However, that ...

The derived subgroup of a finite group equals to the set of all its commutators
or equivalently
A product of two commutators in a finite group is always a commutator
This mistake is very ...

You have already done the case, when $i = 1$.
Now suppose it i true for $i$. Let's prove it for $i+1$. Suppose $H \triangleleft G$ has Fitting length $i+1$. That means $\frac{H}{U_iH}$ is nilpotent. ...

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