Let x be a stochastic variable with a probability density function p(x) and a cumulative probability distribution function of P(x). Thus the probability of an observation being less than or equal to x is P(x). This function P(x) necessarily has an inverse function P^-1(z). Note that P(½)=x_median. Let x_max be defined to be the lowest value of x such that P(x)=1. Likewise x_min is the largest x such that P(x)=0. Note that x_max and x_min may or may not be finite.

The probability density function for the sample maximum of a sample of size n, q_n(x) is given by the probability of getting (n-1) observations which are less than or equal to x and one that is exactly x. The one observation that is exactly x can occur at any one of n places in the sample. Thus the probability density is

q_n(x) = n[P(x)]^n-1p(x)
 

Note that

q_n(x)dx = n[P(x)]^n-1p(x)dx = d[P(x)ⁿ]
 

The expected value of the sample maximum is

M_n = ∫_-∞^∞ xq_n(x)dx
 

Let z=[P(x)]ⁿ so x=P^-1(z^1/n). Then changing the variable of integration in the above expression to z results in

M_n = ∫₀¹ P^-1(z^1/n)dz
 

Now consider the limit of M_n as n increases without bound and note that the limit of a function of a variable is equal to the function of the limit of the the variable; i.e.,

lim_n→∞ M_n = ∫₀¹ P^-1(lim_n→∞z^1/n)dz
 

But for all z in the interval 0<z≤1, lim_n→∞z^1/n = 1. Therefore

lim_n→∞ M_n = ∫₀¹ P^-1(1)dz = P^-1(1)
which can be expressed as
lim_n→∞ M_n = x_max
 

Note that M₁ = x_mean.

The Characteristic Function for the Distribution of Sample Maximums

The characteristic function for sample maximums can be defined as

Φ_n(ω) = ∫_-∞^∞ exp(−iωx)q_n(x)dx
 

where i is the square root of −1.

From the previous work this reduces to

Φ_n(ω) = ∫_-∞^∞ exp(−iωx)d([P(x)]ⁿ)
 

Integration by parts can be applied to this expression to obtain

Φ_n(ω) = [exp(−iωx)[P(x)]ⁿ]_-∞^∞ − (−iω) ∫_-∞^∞ exp(−iωx)[P(x)]ⁿdx
 

Since lim_x→∞P(x)=1` and lim_x→−∞P(x)=0 the above reduces to

` Φ_n(ω) = iω ∫_-∞^∞ exp(−iωx)[P(x)]ⁿdx
 

The integral on the right is the characteristic function of the n-th power of P(x) which can be expressed as the the n-th convolution product of the characteristic function of P(x). The characteristic function of P(x) is the characteristic function of p(x) divided by iω.

(To be continued.)