Concentration of capital: The product form of the law of large numbers in \(L_1\) (Q5956484)

Let \(X_1, X_2, \ldots\) be i.i.d. copies of a non-negative real random variable \(X\) having continuous distribution and such that \(EX(\text{ln } X)^3 < \infty \). Let \(c_n\) denote the unique real number such that NEWLINE\[NEWLINEEX_1X_2 \ldots X_n 1_{X_1X_2 \ldots X_n \geq c_n} = \frac{1}{2} EX_1 X_2 \ldots X_n. NEWLINE\]NEWLINE Under these conditions upper and lower bounds for \(c_n\) are derived. The author refers to a gambler or an investor whose capital is developing in a multiplicative form. If \(E (X_1) = 1\), then the median fortune \(c_n\) is shown to be approximately equal to \(\exp (n E l_nX)\).