Appendix A. Mathematical derivations. A pdf version of this appendix is also available.
Mathematical Derivations
We start with the basic procedure used to compute stage durations; Kemeny, Snell and Knapp (1966) provides useful mathematical background, Cochran and Ellner (1992) and Caswell (2001) have a biological focus. Throughout we use definitions as in the main text.
Environmental variability determines the life history transition matrices 
for times 
. For an individual in stage 
at time 
, and any integer 
, define a random quantity 
such that 
if the individual is in state 
at time 
, otherwise 
. This is the indicator function of the transition 
from stage at time 1 to stage at time : in symbols, 
, where 
if event 
occurs and otherwise equals 0. Stage duration, i.e., the total time that an individual starting in stage will spend in stage during its life, is
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(A.1) |
Given an environmental sequence, the probability that 
is given, for 
, by the 
element of the matrix product
|
(A.2) |
For 
we know that 
if 
and is otherwise zero. Summing the 
over as in () yields the fundamental matrix in text equation (2). The main task of this appendix is to show how we average the fundamental matrix in text equation (2) over environments.
Our assumption that death is certain means that every possible life history transition matrix 
has a maximum column sum 
. Hence for all environments there is a 
whose column sums are also , and the series in text equation (2) is bounded above by the convergent series 
. Convergence is assured because the matrix 
is nonnegative and its spectral radius will be because the column sums are .
To compute the variance of the stage durations 
we need the expectation of
|
(A.3) |
In computing 
where the average lifetime 
, we need to evaluate a slightly more general double sum,
|
(A.4) |
Set 
above to get the double sum in (A.3). The term 
is nonzero only if the individual starts in stage , is in stage at time , and then is in stage 
at time 
. Thus
|
(A.5) |
It is useful to note that if we suppress the conditioning here, the product 
. Keep in mind that in every case below we first evaluate the expectation (A.5) and then perform the double sum in (A.4).
We establish results first for cycles starting in state 1. To establish text equation (6), write the fundamental matrix from text equation (2) for a cycle that starts in state 1 at time ,
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(A.6) |
For cycles starting in any state 
, use the relabelled indices 
in the text to rewrite the above expressions. The cycle matrix for a cycle starting in state is
|
(A.7) |
and the fundamental matrix for cycles starting in state is
|
(A.8) |
Expected stage durations are
|
(A.9) |
and expected total lifetimes are
|
(A.10) |
To compute variances, first evaluate (A.4) by considering transitions between times 1 and , followed by transitions between times and . We must keep track of the state of the cycle at time , so consider the transition 
from stage , state 1, time 1, to stage , state 
, time . Between time and time we care only about the final stage , not the final state of the cycle, so the second transition we consider is 
. At time 
the cycle is in some state with 
. State 
appears only at times 
for some integer 
and so we have the 
matrix elements
|
(A.11) |
On the other hand, state 
appears only if 
for some and so we have the matrix elements
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(A.12) |
Next consider transitions from stage , state at time to some future state at a time , with 
. Since we start in state at time , the probability of is given by the 
th term of the fundamental matrix 
for a cycle starting with state . For fixed and every we can sum over 
to get the 
matrix elements
|
(A.13) |
We want the expectation of (A.3),
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Since equation (A.13) depends only on , we need only sum 
over for each . In (A.11) and (A.12), summing over for fixed is to sum over ; the sum over 
via a geometric series yields the matrices 
of text equation (9). Now sum over the intermediate state to get
|
(A.14) |
From text equations (6) and (9) it is easy to see that
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(A.15) |
Use this and set in equation (A.14) to get text equation (10) for the variance of conditional on starting state 1. For the variance of we also sum over all the stages and in equation (A.14), yielding text equation (11).
For cycles starting in state , define the matrix
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(A.16) |
and for integers with 
define
|
(A.17) |
Then
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(A.18) |
and the variance of expected total lifetimes is
|
(A.19) |
Survivorship is given by column sums of the product matrices as indicated in text equation (13). Individuals aged 0 at time attain age 
at time 
and survivorship is determined by 
. Consider cycles that start with environmental state at time . Any age consists of some number of complete cycles plus a period that is shorter than a cycle; thus any age 
where 
are integers, and 
. For 
and 
we have 
and 
. If is between 1 and 
and we have
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(A.20) |
The cycle matrices 
are products of matrices each having all column sums , as seen in text equation (9). Hence we can bound above by the th power of a matrix whose spectral radius is , and thus will have a dominant eigenvalue 
. Distinct cycle matrices and 
differ only in the order of the matrices that constitute them and so must have equal eigenvalues.
At high age the number is large, so from (A.20) the behavior of 
depends on 
for large . For primitive cycle matrices and large , the Perron-Frobenius theorem (Seneta 1981) shows that
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(A.21) |
We normalize the right eigenvector 
so that its elements add to 1. Use (A.21) to obtain the asymptotic result in text equation (14). To get the factors in text equation (15), multiply by 
on the left in the above equation, put , and take logs to find 
, where 
is the first element of 
. When and 
the same procedure yields
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(A.22) |
The asymptotic behavior of text equation (15) follows.
Here the matrices are independently identically distributed. Distinguish the environmental state at and the subsequent environmental sequence 
for times 
. Given 
the fundamental matrix in text equation (2) is now
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(A.23) |
Independence means that for
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(A.24) |
Taking the expectation over all in (A.23) using (A.24) and then summing the geometric series yields text equation (28).
We now show how to compute the expectation of equation (A.4). First set in that double sum to find
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(A.25) |
Here 
if and is otherwise 0, and simply expresses the fact that we start in life history stage at time . For all the transitions 
, 
and so on the environment is in state at time 1, so the probability of transition 
for an is just the term of 
. Summing over yields the second term in the product in (A.25).
Next set 
to find
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(A.26) |
The first expectation in the product is just the probability of the transition 
and is determined by the fixed initial state . Subsequent transitions 
do not depend on the environment at time 1, because environments vary independently over time. Therefore the relevant probabilities are terms in 
(text equation (19)) rather than in .
Next consider 
to find
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(A.27) |
The first term in the product gives the average over environments of the probability of the transition , and the second term is the same as in (A.26). Adding over all yields text equations (21) and (22).
For a given environmental sequence , survivorship at age is defined as the sum of the first column of the matrix 
. Average survivorship follows from the use of equation (A.24) to yield text equation (23). The asymptotic behavior depends on the analog of equation (A.21) applied to 
, written here as
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(A.28) |
Following the arguments used for cycles, we obtain text equations (24) and (25) with
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(A.29) |
To examine the variance of 
, recall that survivorship is the sum of the first column of 
, so we require the variance of the column sum. For any matrix 
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(A.30) |
where 
indicates a Kronecker product. Hence we consider the average over environments of the Kronecker product 
. Because environments are independent,
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(A.31) |
using the matrix defined in text equation (26). Combining this with equation (A.30), define the 
vector
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(A.32) |
Letting 
be the first element of this vector, the environmentally driven variance in survivorship, conditional on starting environment is
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(A.33) |
Asymptotic behavior is deduced as for the average survivorship and leads to text equation (27).
Along environmental sequence let the environmental state at time 
be 
. The fundamental matrix for this sequence is
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(A.34) |
Consider the term
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(A.35) |
To average this over all environments except the initial state 
we use a device from the theory of stochastic demography (Cohen 1977, Tuljapurkar 1982). Using the environmental transition probabilities in text equation (28) we have
In terms of the block matrix 
of text equation (29), we can write
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(A.36) |
using the matrices 
defined in Table 1. Sum the geometric series to get the fundamental matrix in text equation (30).
We now show how to compute the expected value of the double sum over 
in equation (A.4). First set to find
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(A.37) |
The above term is arrived at by the same reasoning as used for equation(A.26) above. Next set to find
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(A.38) |
where
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(A.39) |
The first term in the product is just the probability of the transition as determined by the fixed initial state . Subsequent transitions depend on the environments 
at times 2, 3, etc., and we average over all these environments. Using 
and the arrays 
from Table 1, we see that matrix 
sums to the closed form 
which is the matrix 
listed in Table 4.
Next consider . Here we must keep track of the environmental state 
which determines the life history transition matrix 
that tells us about stage transitions between time 
and time . The value of is some integer with 
; let’s fix 
. For the transition 
we have the matrix element
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(A.40) |
Conditional on the environmental sequence, the probability of a subsequent transition 
is the element of 
; the probability of transition 
is the element of 
; and so on. Therefore
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(A.41) |
This is immediately recognizable as 
in equation (). Hence for each the sum over depends only on . Thus we can do the double sum by summing the probabilities of transitions over for each , then use () and finally sum over . The sum over is
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Using matrix and letting stand for for all , this sum is
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(A.42) |
which is also listed in Table 4. Combining this with the factor from equation (A.41), we have
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(A.43) |
Combining equations (A.37), (A.38) and (A.43), and noting that
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(A.44) |
leads to text equations (32), (33).
For survivorship we use equation (A.36) and text equation (34), and note that for large
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(A.45) |
The limit for large yields text equation (35). To obtain text equation (36), note that 
is a row vector of size 
. Using a subscript 1 to indicate the first element of this vector, the asymptotic level 
.
To analyze the variance of survivorship we consider, as in the case of temporally uncorrelated random variability, . The steps used above to compute the average extend in a straightforward way to calculating the averages of the necessary Kronecker products for the variance; see Tuljapurkar (1982) for a related analysis in the case of population growth. First define a matrix of size 
made of 
blocks each 
that has a nonzero block only in position ,
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(A.46) |
Define another matrix of blocks,
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(A.47) |
Using these, define the matrix
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(A.48) |
Letting be the first element of this vector, the environmentally driven variance in survivorship is
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(A.49) |
Note that this is the variance conditional on the starting environment being state . The asymptotics in text equation (38) follow in the same way as for the average survivorship.