Strong limit theorem for largest entry of large-dimensional random tensor

Suppose that [Formula: see text] are i.i.d. copies of random vector [Formula: see text]. Let [Formula: see text] then the random tensor product constructed by [Formula: see text] is defined by [Formula: see text] In this paper, we obtain the strong limit theorems of the largest entry of large-dimensional random tensor product [Formula: see text] under two high-dimensional settings the polynomial rate and the exponential rate. The conclusions are established under weaker moment condition than the exist papers and the relationship between [Formula: see text] and [Formula: see text] is more flexible.