如何更有效地获得相关离散随机变量之和的概率分布

Question

我希望你很好。我想知道您是否可以帮助我解决所附链接中提供的问题。在链接下面，我附加了一个R代码，它可以递归地解决所涉及的分布参数的特定值。但是，我意识到这种方法效率低下。非常感谢你的帮助。

How to obtain the probability distribution of a sum of dependent discrete random variables more efficiently

library(boot)     # The library boot is necessary to use the command inv.logit.

TMax <- 500       # In this R-code, I am using TMax instead of using T.
M <- 2000
beta0 <- 1
beta1 <- 0.5 
Prob_S <- function(k, r){        # In this R-code, I am using r instead of using t.
    if(r == 1){
        Aux <- dbinom(x = k, size = M, prob = inv.logit(beta0))
        }
    if(r %in% 2:TMax){
        Aux <- 0
        for(u in 0:k){
            Aux <- Aux + dbinom(x = k - u, size = M - u, 
                prob = inv.logit(beta0 + beta1 * u)) * Prob_S(u, r - 1)
            }
        }
    Aux
    }

m <- 300
P <- Prob_S(k = m, r = TMax)    # Computing P takes a loooong time.   :(