On the moments of the Ulam–Kac adder

Abstract Let left brace upper U left parenthesis n right parenthesis right brace Subscript n greater than or equals 0 { U ( n ) } n ≥ 0 $\{U(n)\}_{n \geq 0}$ be a sequence of independent random variables such that U ( n ) is distributed uniformly on StartSet 0 comma 1 comma 2 comma ellipsis comma n EndSet { 0 , 1 , 2 , … , n } $\{0, 1, 2, \dots, n\}$ . The Ulam–Kac adder is the history-dependent random sequence defined by upper X Subscript n plus 1 Baseline equals upper X Subscript n Baseline plus upper X Subscript upper U left parenthesis n right parenthesis X n + 1 = X n + X U ( n ) $X_{n + 1} = X_{n} + X_{U(n)}$ with the initial condition upper X 0 equals 1 X 0 = 1 $X_0 = 1$ . We show that for each m greater than or equals 1 m ≥ 1 $m \geq 1$ , log double struck upper E left bracket upper X Subscript n Superscript m Baseline right bracket divided by StartRoot n EndRoot log ⁡ E [ X n m ] / n