Moment Existence Across Normal and Tail Families

Why Moment Existence Depends on the Tail

A distribution can have a finite median and finite central quantiles while still lacking a finite mean or variance. The distinction is controlled by the probability assigned to very large values: a tail that decays exponentially can support moments of every positive order, whereas a power-law tail eventually makes sufficiently high moments diverge. Moment existence is therefore a useful way to compare distributions beyond their central behavior.

Throughout, \(r>0\) denotes a positive real order. For a real-valued random variable \(X\), the \(r\)th absolute moment is \(E[|X|^r]\). When \(X\geq 0\), this is also \(E[X^r]\), the positive-order raw moment. For a variable that can be negative, a noninteger power \(X^r\) need not be real-valued; in that case the absolute moment is the unambiguous quantity. In particular, if \(E[|X|^r]\) is finite, every integer raw moment of order at most \(r\) is well-defined and finite.

A Tail-Integral Criterion

For any nonnegative random variable \(Y\) and \(r>0\), the moment can be calculated from its survival probabilities \(P(Y>t)\). This representation makes the upper tail’s contribution explicit.

To prove the identity, for each fixed \(y\geq 0\) write \(y^r=\int_0^\infty r t^{r-1}\mathbf{1}_{\{t<y\}}\,dt\). Substitute \(y=Y\) and take expectations. The integrand is nonnegative, so Tonelli’s theorem permits exchanging expectation and integration without first assuming the moment is finite. Since \(E[\mathbf{1}_{\{t<Y\}}]=P(Y>t)\), the stated identity follows. \(\square\)

For \(Y=|X|\), the part of the integral over \(0<t<1\) is always finite: \(t^{r-1}P(|X|>t)\leq t^{r-1}\), and \(\int_0^1t^{r-1}\,dt=1/r\). Thus, for positive-order absolute moments, existence is decided by the behavior of \(P(|X|>t)\) as \(t\) becomes large.

Classification Across the Six Families

Use the standard parameterizations below: \(N(\mu,\sigma^2)\) has \(\sigma>0\); a logistic variable has location \(\mu\) and scale \(s>0\); a Laplace variable has location \(\mu\) and scale \(b>0\); a lognormal variable is \(X=e^Y\) for \(Y\sim N(\mu,\sigma^2)\); a Weibull variable has shape \(k>0\) and scale \(\lambda>0\); and a Pareto Type I variable has minimum \(x_m>0\) and tail index \(\alpha>0\).

For the normal, logistic, and Laplace families, the absolute-value tail is bounded by a constant times an exponentially decreasing function for all sufficiently large \(t\); the normal tail in fact has Gaussian decay. For example, if \(X\sim N(\mu,\sigma^2)\), then for \(t>|\mu|\), the Gaussian tail bound gives \(P(|X|>t)\leq 2\exp\{-(t-|\mu|)^2/(2\sigma^2)\}\). Multiplying this bound by \(t^{r-1}\) still gives an integrable function on the upper tail for every fixed \(r>0\). For a logistic variable the two tails decrease exponentially at rate \(1/s\), and for a Laplace variable at rate \(1/b\); the same tail-integral criterion applies.

The lognormal tail is heavier than an exponential tail, but it still permits every positive-order moment. To see the relevant tail behavior, write \(X=e^Y\). For \(t>1\), \(P(X>t)=P(Y>\log t)\), which has a Gaussian upper bound in \(\log t\). In the tail-integral criterion, the substitution \(t=e^u\) turns the upper-tail integrand into a constant multiple of \(e^{ru}P(Y>u)\). A Gaussian tail bound makes this integrable for every fixed \(r>0\): the negative quadratic term in \(u\) eventually dominates the linear term \(ru\).

The lognormal and Weibull moments can also be computed directly. If \(X=e^Y\) with \(Y\sim N(\mu,\sigma^2)\), completing the square in the normal density gives \(E[e^{rY}]=\exp(r\mu+r^2\sigma^2/2)\), finite for every real \(r\). For a Weibull variable, whose density on \(x>0\) is \(\frac{k}{\lambda}(x/\lambda)^{k-1}e^{-(x/\lambda)^k}\), substitute \(u=(x/\lambda)^k\) into the moment integral. This gives \(E[X^r]=\lambda^r\Gamma(1+r/k)\), finite whenever \(r>-k\), in particular for every \(r>0\).

The Sharp Threshold for Pareto Moments

A Pareto Type I variable is supported on \([x_m,\infty)\) and has survival function \(P(X>x)=(x_m/x)^\alpha\) for \(x\geq x_m\). Unlike the distributions above, its moment existence depends on the order.

Apply the tail-integral identity. The survival probability is \(1\) for \(0\leq t<x_m\), and is \((x_m/t)^\alpha\) for \(t\geq x_m\). Therefore \(E[X^r]=r\int_0^{x_m}t^{r-1}\,dt+r x_m^\alpha\int_{x_m}^\infty t^{r-\alpha-1}\,dt\). The first term is \(x_m^r\). In the second integral, the power integral converges exactly when \(r-\alpha-1<-1\), or \(r<\alpha\). In that case its value is \(x_m^{r-\alpha}/(\alpha-r)\), so \(E[X^r]=x_m^r+\frac{r x_m^r}{\alpha-r}=\frac{\alpha x_m^r}{\alpha-r}\). If \(r=\alpha\), the second integral is proportional to \(\int_{x_m}^\infty t^{-1}\,dt\), which diverges. If \(r>\alpha\), its integrand has exponent greater than \(-1\), so it also diverges. This proves both the formula and the threshold. \(\square\)

Reading the Classification Correctly

For the normal, logistic, and Laplace distributions, all positive-order absolute moments are finite, and hence all integer raw moments exist. The lognormal and Weibull distributions are supported on positive values and have finite moments of every positive order as well. For the Pareto law, positive-order moments exist only below \(\alpha\); since \(X\geq0\), its absolute moments have the same threshold. The tail index is therefore not just a description of tail shape: it directly determines the highest moment orders that are available.

A common mistake is to infer moment existence from the fact that a distribution has a well-defined center or from the existence of some lower-order moments. For the Pareto example, both the mean and variance are finite when \(\alpha=3\), but the third moment is not. Another mistake is to assume that every named continuous distribution has all moments. That is true for the families here with Gaussian, exponential, lognormal, or Weibull tails, but not for a power-law tail. The correction is to check the upper-tail integral at the particular order of interest.