the multiplicative process [@Gibrat1930; @mitzenmacher2003]. Starting from an initial value $X_0$, random variables in a multiplicative process follow an iterative formula $X_{i+1}=\exp(\nu_i) X_i$ or $\ln X_{i+1}= \ln X_i + \nu_i.$ If the $v_i$ has finite mean and variance, and is independent and identically distributed, then according to the central limit theorem, for large $i$, $\ln X_i$ will follow a normal distribution, which means $X_i$ will follow a log-normal distribution.

If we want to check whether gamblers follow multiplicative processes when they wager, we can first check the correlation between consecutive bets $(b_i, b_{i+1})$. Due to the large variances of the wager distributions, Pearson’s correlation coefficient may perform poorly. Instead, we adopt two rank-based correlation coefficients, Kendall’s Tau [@Kendall1990] $\tau_K$ and Spearman’s Rho [@Taylor1987] $\rho_S$. At the same time, we also check the mean and variance of the log-ratios $\ln(b_{i+1}/b_{i})$ between consecutive bets. These statistics can be found in Table \[table:log\_ratio\]. The results reveal that the values of consecutive bets exhibit a strong positive correlation, with all the correlation coefficients larger than $0.5$. It shows that players’ next bet values are largely dependent on their previous bet values. At the same time, the bet values are following gradual changes, rather than 