Wald's equation explained

In probability theory, Wald's equation, Wald's identity^[1] or Wald's lemma^[2] is an important identity that simplifies the calculation of the expected value of the sum of a random number of random quantities. In its simplest form, it relates the expectation of a sum of randomly many finite-mean, independent and identically distributed random variables to the expected number of terms in the sum and the random variables' common expectation under the condition that the number of terms in the sum is independent of the summands.

The equation is named after the mathematician Abraham Wald. An identity for the second moment is given by the Blackwell–Girshick equation.^[3]

Basic version

Let be a sequence of real-valued, independent and identically distributed random variables and let be an integer-valued random variable that is independent of the sequence . Suppose that and the have finite expectations. Then

\operatorname{E}[X _{1+...+X} _{N]=\operatorname{E}[N]} \operatorname{E}[X _{1].}

Example

Roll a six-sided dice. Take the number on the die (call it) and roll that number of six-sided dice to get the numbers, and add up their values. By Wald's equation, the resulting value on average is

\operatorname{E}[N] \operatorname{E}[X] = (1+2+3+4+5+6)/(6 ⋅ (1+2+3+4+5+6)/(6)) = (441)/(36) = (49)/(4) = 12 . 25 .

General version

Let be an infinite sequence of real-valued random variables and let be a nonnegative integer-valued random variable.

Assume that:

. are all integrable (finite-mean) random variables,

. for every natural number, and

. the infinite series satisfies

\sum ^{inf ty\operatorname{E} l [|X} _{n|} 1 _{\{N\ge} }\bigr]<\infty.

Then the random sums

S _{N:=\sum} ^{NX} ^{n,} T _{N:=\sum} ^{N\operatorname{E}[X} ^{n]} are integrable and\operatorname{E}[S _{N]=\operatorname{E}[T} _{N].} If, in addition,

. all have the same expectation, and

. has finite expectation,

then

\operatorname{E}[S _{N]=\operatorname{E}[N]} \operatorname{E}[X _{1].}

Remark: Usually, the name Wald's equation refers to this last equality.

Discussion of assumptions

Clearly, assumption is needed to formulate assumption and Wald's equation. Assumption controls the amount of dependence allowed between the sequence and the number of terms; see the counterexample below for the necessity. Note that assumption is satisfied when is a stopping time for a sequence of independent random variables . Assumption is of more technical nature, implying absolute convergence and therefore allowing arbitrary rearrangement of an infinite series in the proof.

If assumption is satisfied, then assumption can be strengthened to the simpler condition

. there exists a real constant such that for all natural numbers .

Then all the assumptions,, and, hence also are satisfied. In particular, the conditions and are satisfied if

. the random variables all have the same distribution.

Note that the random variables of the sequence don't need to be independent.

The interesting point is to admit some dependence between the random number of terms and the sequence . A standard version is to assume,, and the existence of a filtration such that

. is a stopping time with respect to the filtration, and

. and are independent for every .

Then implies that the event is in, hence by independent of . This implies, and together with it implies .

For convenience (see the proof below using the optional stopping theorem) and to specify the relation of the sequence and the filtration, the following additional assumption is often imposed:

. the sequence is adapted to the filtration, meaning the is -measurable for every .

Note that and together imply that the random variables are independent.

Application

An application is in actuarial science when considering the total claim amount follows a compound Poisson process

S _{N=\sum} ^{NX} ^{n}

within a certain time period, say one year, arising from a random number of individual insurance claims, whose sizes are described by the random variables . Under the above assumptions, Wald's equation can be used to calculate the expected total claim amount when information about the average claim number per year and the average claim size is available. Under stronger assumptions and with more information about the underlying distributions, Panjer's recursion can be used to calculate the distribution of .

Examples

Example with dependent terms

Let be an integrable, -valued random variable, which is independent of the integrable, real-valued random variable with . Define for all . Then assumptions,,, and with

Notes and References

1. Book: Jacques . Janssen. Raimondo . Manca . 10.1007/0-387-29548-8_2 . Renewal Theory . Applied Semi-Markov Processes . limited . 45–104 . 2006 . 0-387-29547-X . Springer.
2. Thomas Bruss . F. . Robertson . J. B. . 'Wald's Lemma' for Sums of Order Statistics of i.i.d. Random Variables . Advances in Applied Probability . 23 . 3 . 612–623 . 10.2307/1427625 . 1427625. 1991 . 120678340 .
3. Blackwell . D. . Girshick . M. A. . On functions of sequences of independent chance vectors with applications to the problem of the 'random walk' in k dimensions . Ann. Math. Statist. . 17 . 310–317 . 10.1214/aoms/1177730943 . 1946. 3 . free .