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(Looking at the question and notation used more closely, the formulation seems to be problematic in couple places.) General Fact Let $W$ be standard Brownian motion with respect to filtration $( \mathscr F_t )_{t \in [0,T]}$. Consider $(L_t)_{t \in [0,T]}$ defined by $$\frac{dL_t}{L_t} = \psi_t dL_t, \; L_0 = 1.$$ In general, $L_t = e^{\int_0^t \psi_s ... 2 So we have$S_n \thicksim^{iid} \ ?\bar{s} = \frac{1}{n}\sum{s_n}$Exponential Suppose$S_n$follows the exponential distribution. $$f(s|\beta) = \frac{1}{\beta} e^{-\frac{s}{\beta}} \quad , \quad 0 \leq s < \infty \quad , \quad \beta > 0$$ Take the simple bivariate case. Say$Z = \frac{S_1 + S_2}{2}$and$W = S_1$. So$S_2 = 2Z - W$and$S_1 ...