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For $C(K)=\mathbb{E}(S-K{)}_{+}$ and $\mathbb{E}S<\infty$, split the static call integral at a fixed positive strike. The first moment bounds the integrand near zero; polynomial call decay controls the integral at infinity for $0<\epsilon <\delta$. The endpoint can fail, as shown by a Pareto distribution with survival exponent $1+\delta$.

The transformation of the uniform distribution gives, for $x\ge 1$,Thus $X_1$ has a Pareto distribution with lower endpoint $1$ and shape $2$. It has mean $2$, but no positive exponential moments, so the preceding version of the Cramér theorem is unavailable.

At $a=0$, the event in question is certain. For every fixed $a>0$ and all sufficiently large $n$, positivity of the summands gives the one-big-jump polynomial lower boundTaking logarithms and dividing by $n$ gives an upper bound $0$ and a lower bound $-2\log (na)/n\to 0$. HenceThis covers thresholds below, at and above the mean. Above the mean it means decay is slower than exponential speed $n$, not that the tail probability tends to $1$.

Multiplying the likelihood function by the Pareto distribution prior givesThe integral of this kernel is $m^{-(\alpha +n)}/(\alpha +n)$, so the normalized posterior distribution isThus it is $\mathrm{Pareto}(\alpha +n,m)$. This proves uniform-Pareto conjugacy: applying Bayes theorem preserves the family of prior distributions.

Here $m_i=\beta$ for every breed, so $S=0$ andThe Bayesian model evidence increases towards its supremum as $\alpha \to \infty$. ThusFor a Pareto distribution, $\mathbb{P}(\theta >\beta (1+\epsilon ))=(1+\epsilon {)}^{-\alpha }\to 0$. The limiting empirical prior, and each corresponding posterior distribution, collapses onto $\beta$.

The boundary result is coherent within the assumed model: the observations favour the smallest allowed upper limits. Nevertheless, reporting exact concentration on a prespecified bound is overconfident for finite data and ignores hyperparameter uncertainty. A proper hyperprior on $\alpha$, sensitivity analysis for $\beta$, or a scientifically justified restriction on concentration avoids treating this limit as certain biological knowledge.

Let $A_{\delta }={\sup }_{K\ge 0}{K}^{\delta }C(K)<\infty$. The call-price decay and moment threshold follows by splitting the preceding integral at one. Since $(S-K{)}_{+}\le S$, for $0<\epsilon <\delta$,For $K\ge 1$, the decay bound givesThe power payoff static call representation therefore yieldsThe $\epsilon =0$ case is the given finite first moment. The strict endpoint matters: a Pareto distribution with $\mathbb{P}(S>x)=x^{-(1+\delta )}$ for $x\ge 1$ has $C(K)=K^{-\delta }/\delta$ for $K\ge 1$, but its moment of order $1+\delta$ is infinite. Thus the stated decay condition does not generally imply the endpoint moment.

A Pareto distribution prior for the upper limit of independent uniform distributions remains Pareto after observing data. The likelihood adds the sample size to the shape and replaces the lower bound by the larger of the prior bound and sample maximum.

For positive observations and $m=\max (\beta ,{\max }_j{y}_j)$, the Bayesian model evidence follows by integrating the uniform distribution likelihood against a Pareto distribution prior. For independent groups with shared fixed hyperparameters, these evidences multiply. The resulting hyperparameter log likelihood is $I\log \alpha -{\sum }_i\log (\alpha +n_i)-\alpha {\sum }_i\log (m_i/\beta )$ up to a constant.