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If samples $X_1, X_2, ... X_t$ are picked independently and identically from the discrete uniform distribution $[1,2, ..., P]$, what is the joint distribution of the last $k$ order statistics and last $k$ samples: $$p_{X_{t-k},X_{t-k+1},...X_{t},X_{(t-k)},X_{(t-k+1)},...,X_{(t)}}(x_{t-k},x_{t-k+1},...x_{t},x_{(t-k)},x_{(t-k+1)},...,x_{(t)})?$$

where $X_{(k)}$ is the $k$-th order statistic.

If we were to define the cumulative sums

$$r_0 = 0, r_1 = \frac{1}{P}, r_2 = \frac{1}{P} + \frac{1}{P}, \cdots r_P = \frac{P}{P} = 1$$

Then, $$ P(X_{(j)} \leq x_i) = \sum_{k=j}^{t} \binom{t}{k} r_i^k (1-r_i)^{t-k}$$

and

$$ P(X_{(j)}=x_i) = \sum_{k=j}^{t} \binom{t}{k} \bigg[r_i^k(1-r_i)^{t-k} - r_{i-1}^{k} (1- r_{i-1})^{t-k}\bigg]$$

There is non-Markov behavior due to the fact that in discrete cases ties between observations are possible. Hence, I don't see an easy way to describe the joint distribution explicitly.

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