Concentration inequalities for sums of random variables, each having power bounded tails

In this work we present concentration inequalities for the sum $S_n$ of independent integer-valued not necessary indentically distributed random variables, where each variable has tail function that can be bounded by some power function with exponent $-\alpha$. We show that when $0<\alpha\leq 1$, then the sum does not have finite expectation, however, with high probability we have that $|S_n|=O\left(n^{1/\alpha}\right)$. When $\alpha>1$, then the sum $S_n$ is concentrated around its mean. Since the r.vs. that constitute the sum has tails, which can be bounded by some power function, it follows that results of this paper are applicable to a wide range of different distributions, including the exponentially decaying ones.