An alternative proof of the asymptotic formula for the Fourier coefficients of the elliptic modular $j$-function

In 1997, Báez-Duarte gave a probabilistic proof of the asymptotic formula for the partition function, which had originally been proved by Hardy-Ramanujan. Based on the probabilistic approach, this paper proves an asymptotic formula for the coefficients of the elliptic modular $j$-function using various expressions in terms of modular functions having simple infinite products.