Moments compress a distribution into integrals of powers. Low moments locate and scale a law, while high moments amplify rare observations. Kurtosis belongs to this second group: it is a standardized fourth moment, and its mathematical force comes from tail mass rather than from the visual height of a density at its mode.
Raw, central, and absolute moments
The first distinction is between moments around the origin and moments around the mean.
Definition. Let \(X\) be real-valued. If \(\mathbb E|X|^r\lt\infty\), the \(r\)-th raw moment is \(\mathbb E X^r\) when meaningful. If \(\mu=\mathbb E X\) exists, the \(r\)-th central moment is \[ m_r=\mathbb E[(X-\mu)^r], \] and the \(r\)-th absolute central moment is \(\mathbb E|X-\mu|^r\).
Proposition. On a probability space, if \(0\lt p\le q\) and \(\mathbb E|X|^q\lt\infty\), then \(\mathbb E|X|^p\lt\infty\) and \[ \|X\|_p\le \|X\|_q . \]
Proof. If \(p\lt q\), Jensen's inequality applied to the concave function \(u\mapsto u^{p/q}\) gives \(\mathbb E|X|^p=\mathbb E[(|X|^q)^{p/q}]\le(\mathbb E|X|^q)^{p/q}\). Taking \(p\)-th roots gives the stated norm inequality. The case \(p=q\) is immediate.
Tail integrals for powers
Power moments can be read directly from tail probabilities. This is the cleanest reason that high moments are tail quantities.
Theorem. For \(Y\ge0\) and \(r\gt0\), \[ \mathbb E Y^r=r\int_0^\infty t^{r-1}P(Y\gt t)\,\mathrm{d}t. \] In particular, \[ \mathbb E|X-\mu|^4=4\int_0^\infty t^3P(|X-\mu|\gt t)\,\mathrm{d}t. \]
Proof. For each \(y\ge0\), \(y^r=\int_0^y r t^{r-1}\,\mathrm{d}t=\int_0^\infty r t^{r-1}1_{\{t\lt y\}}\,\mathrm{d}t\). Substituting \(Y(\omega)\) and applying Tonelli's theorem gives \[ \mathbb E Y^r=\int_0^\infty r t^{r-1}P(Y\gt t)\,\mathrm{d}t . \]
Remark. The weight \(t^3\) in the fourth-moment formula means that a small amount of probability far from the mean can dominate the value of the moment.
Standardized moments and excess kurtosis
Standardization removes units and makes distributions comparable after location-scale changes.
Definition. Suppose \(\mu=\mathbb E X\), \(0\lt\sigma^2=\operatorname{Var}(X)\lt\infty\), and \(\mathbb E|X-\mu|^4\lt\infty\). The standardized fourth moment is \[ \beta_2=\mathbb E\left[\left(\frac{X-\mu}{\sigma}\right)^4\right] =\frac{\mathbb E[(X-\mu)^4]}{\sigma^4}. \] The excess kurtosis is \(\gamma_2=\beta_2-3\).
Proposition. Kurtosis is invariant under nondegenerate affine changes. If \(Y=aX+b\) with \(a\ne0\), then \(Y\) and \(X\) have the same \(\beta_2\) and the same excess kurtosis whenever these quantities exist.
Proof. The mean of \(Y\) is \(a\mu+b\), its standard deviation is \(|a|\sigma\), and \(Y-\mathbb EY=a(X-\mu)\). Hence \[ \mathbb E\left[\left(\frac{Y-\mathbb EY}{\operatorname{sd}(Y)}\right)^4\right] =\mathbb E\left[\left(\frac{a(X-\mu)}{|a|\sigma}\right)^4\right] =\beta_2 . \]
Why kurtosis is tail-sensitive
A distribution can have fixed variance and still have arbitrarily large kurtosis if a tiny amount of mass is placed very far away.
Proposition. For every \(K\gt0\), there exists a mean-zero, variance-one random variable with excess kurtosis greater than \(K\).
Proof. Let \(0\lt p\lt1\), set \(a=p^{-1/2}\), and define \(X_p\) by \[ P(X_p=a)=p/2,\qquad P(X_p=-a)=p/2,\qquad P(X_p=0)=1-p . \] Then \(\mathbb EX_p=0\), \(\mathbb EX_p^2=pa^2=1\), and \(\mathbb EX_p^4=pa^4=1/p\). Thus the excess kurtosis is \(1/p-3\), which tends to infinity as \(p\downarrow0\).
Remark. This example also has a large atom at the center, but the explosion is not caused by central height alone. The fourth power of the distant values is what drives the calculation.
Canonical distributions
The standard examples separate bounded, exponential, Gaussian, and polynomial tail behavior.
Proposition. The following excess kurtoses hold when the stated moments exist: \[ \begin{array}{c|c} \text{law} & \gamma_2\\ \hline N(0,1) & 0\\ \mathrm{Unif}[-1,1] & -6/5\\ \mathrm{Laplace}(0,b) & 3\\ t_\nu,\ \nu\gt4 & 6/(\nu-4)\\ \mathrm{Pareto}(\alpha),\ \alpha\gt4 & \dfrac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)} \end{array} \]
Proof. For a standard normal variable, integration by parts or differentiation of the moment generating function gives \(\mathbb EX^4=3\). For \(U\sim\mathrm{Unif}[-1,1]\), \(\mathbb EU^2=1/3\) and \(\mathbb EU^4=1/5\), so \(\beta_2=(1/5)/(1/3)^2=9/5\). For the Laplace density \((2b)^{-1}e^{-|x|/b}\), the even moments are \(\mathbb EX^{2k}=(2k)!b^{2k}\), giving variance \(2b^2\), fourth moment \(24b^4\), and \(\beta_2=6\). The Student and Pareto formulas follow by evaluating their standard moment formulas: \(t_\nu\) has finite fourth moment only for \(\nu\gt4\), and a type-I Pareto variable has raw moment \(\alpha x_m^r/(\alpha-r)\) for \(r\lt\alpha\), from which the displayed central standardized expression follows.
Not a measure of peakedness
The density near the mode can suggest a sharp or flat center, but kurtosis is defined by a fourth moment. It is therefore a tail-weighted integral, not a direct measurement of the height of a peak.
Remark. The Laplace distribution has both a sharper central cusp than the normal law and heavier exponential tails; its excess kurtosis is \(3\). The uniform law has no central peak and bounded tails; its excess kurtosis is negative. These examples are useful visually, but the definition decides the matter: kurtosis is governed by the standardized fourth central moment.
Proposition. Sample kurtosis is highly sensitive to extreme observations. The population functional \(P\mapsto \mathbb E_P[(X-\mu_P)^4]/\sigma_P^4-3\) has an unbounded influence function at any distribution with positive finite variance and finite fourth moment.
Proof. Contaminate \(P\) by \(P_\varepsilon=(1-\varepsilon)P+\varepsilon\delta_z\). The derivative at \(\varepsilon=0\) contains a leading term proportional to \((z-\mu)^4/\sigma^4\), together with lower-order corrections from the perturbed mean and variance. As \(|z|\to\infty\), this leading term is unbounded, so the influence function is unbounded.