On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations

In this paper we consider unbounded solutions of perturbed convolution Volterra summation equations. The equations studied are asymptotically sublinear, in the sense that the state--dependence in the summation is of smaller than linear order for large absolute values of the state. When the perturbation term is unbounded, it is elementary to show that solutions are also. The main results of the paper are mostly of the following form: the solution has an additional unboundedness property $U$ if and only if the perturbation has property $U$. Examples of property $U$ include monotone growth, monotone growth with fluctuation, fluctuation on $\mathbb{R}$ without growth, existence of time averages. We also study the connection between the times at which the perturbation and solution reach their running maximum, and the connection between the size of signed and unsigned running maxima of the solution and forcing term.


Introduction
In this paper we determine conditions under which the solutions of a forced Volterra summation equation of the form x(n + 1) = H(n + 1) + n j=0 k(n − j)f (x(j)), n ≥ 0.
(1.1) have bounded and unbounded solutions. It is assumed that f (x) = o(x) as |x| → ∞ and k is summable. These properties of f and k ensures the boundedness of the Volterra equation under moderate disturbances in the system: in fact, we show that x is unbounded if and only if the external force H is. Once that has been done, the bulk of the paper is devoted to exploring refinements of these unboundedness results. Generally speaking, we find that if H has an additional unboundedness property U , then x inherits property U . In many cases, the converse is also true. In this sense, our results are related to the theory of admissibility for Volterra summation equations (cf. Baker and Song [28,29], Reynolds [26], Reynolds and Győri [19,20], Győri, Horváth [17,18] and Awwad and Győri [16]) and Volterra integral equations (see e.g., [3,6] inspired by work of Perron [25] and Corduneanu [10]). In many of the discrete papers the theme of research is often (but certainly not exclusively) to consider the following problem where V is a linear Volterra operator and H is an R p -valued sequence. The solution x is a sequence in R p . The form of V is (V g)(n) := n j=0 K(n, j)g(j) =: (K ⋆ g)(n), n ≥ 0 for any g : Z + → R p , where K is fixed and K : Z + × Z + → R p×p . If the operator V (or equivalently, the matrix K) has the appropriate properties, and H is a sequence with a nice asymptotic property characterised by a sequence space N , x will lie in the space N . In such situations, it will be the case that g → K ⋆ g takes N to N and we say that the mapping has an admissibility property. Sometimes, it can even happen that properties of x may be enhanced.
Much effort has gone into investigating nice spaces N , such as bounded, convergent, periodic, or ℓ q spaces of sequences. Instead, the results in this paper have a rather different flavour, as the types of external force H are either highly irregular (for example stochastic) or are unbounded or growing. One consequence of this is that it becomes reasonable to track new types of property, such as the the size of the largest fluctuations to date (both positive, negative, and in absolute terms), the times at which sequences H and x reach their maximum value to date, growth rates and growth bounds, or indeed time averages of functions of the sequences (which may be finite even though those sequences are unbounded). Therefore, in a sense, our results are of greater applicability in economics or finance, rather than engineering, because disturbances to the system are less likely to be regular or bounded in applications in the former disciplines, while in engineering, we cannot expect good performance if disturbances are irregular or unbounded.
Incidentally, we note if f is linear, and H is in the class of stationary ARMA processes, then x is a stationary process provided the equation without noise is stable. The statistical behaviour of such linear models is well known, and therefore is expressly not the subject of this work; however, path properties are less well understood, and in a parallel work we explore the properties of (1.1) using the same framework as in the present paper. Once again, we observe the pattern of this paper that the unboundedness property U of the external force H transmits to the solution x, and indeed, owing to the linearity of the problem the connection with the above-cited works on admissibility is more tangible: indeed, to a certain degree our contribution in the linear case is merely to identify nice spaces, and then to check by direct calculation, or by appeal to the general theory, that admissibility properties hold. In this work and its linear counterpart, we have focused on the convolution equation with a view towards applications and statistical inference: in this we merely adopt the time-honoured ceteris paribus perspective of the economist in trying to keep the structure of the model time-independent (apart from possible external shocks), and the desire of the statistician to consider models with time-invariant statistical properties. These constraints, which are purely driven by applications, lead us to study the (asymptotically) autonomous convolution operator in (1.1). However, nothing prevents the results in this paper, as well as the convolutional linear case, being developed for non-convolution Volterra equations, nor indeed the extension of the analysis to deal with the general p-dimensional case. Notwithstanding this, we feel that a substantial challenge has already been met by analysing successfully the scalar case.
Some of our results require the sequence H to be stochastic, but not all do. However, most of our results are inspired by a unspoken assumption that H could be stochastic, and that interesting properties of random processes can be assumed about H. For this reason, we sometimes talk about H as though it could be stochastic, and motivate our results by appealing to intuition about stochastic processes: therefore we freely use terminology like "shock", "noise" and "stochastic process" when talking about H. In our precise mathematical results, though, H can be irregular or unbounded, but deterministic (i.e., H could be a chaotic sequence), and our arguments would still be valid. In this sense, our analysis asks how the system modelled by the Volterra equation adjusts to shocks with certain characteristics, irrespective of whether they are stochastic or not.
We first show that H is unbounded, the maxima of |H| and |x| grow at the same rate, so that lim n→∞ max 0≤j≤n |x(j)| max 1≤j≤n |H(j)| = 1, This shows that shocks to the system, or growth from an external source, are not amplified nor damped by the system. However, it does not yet show whether unbounded fluctuations or growth in H give rise to fluctuations or growth in x, but merely that the absolute size of the running maxima grow at the same rate. We also prove that the largest absolute fluctuations in H to-date cause those in x. We do this by studying the the times t x n and t H n at which of the largest absolute fluctuations in x and H up to time n occur, and show for example that |x(t x n )|/|H(t x n )| → 1 and |H(t x n )|/|H(t H n )| → 1 as n → ∞. Our analysis to prove these results, and almost all others, is embarrassingly elementary and hinges mostly on careful analysis of the running maximum of sequences. Indeed, it is not unreasonable to state that almost all the analysis involves little more than taking maxima on both sides of (1.1), or obvious rearrangements of (1.1) and parts thereof.
It should be noted that the unboundedness of the sequences as described by these results does not make an assumption about whether H grows or fluctuates. We show that essentially monotone growth in the forcing term H produces monotone growth in x, and that x is asymptotic to H; if the growth in H is non-monotone, but a monotone trend about which H grows can be identified, x inherits this property also. We also study what happens when H has large positive or large negative fluctuations. The main result shows that the dominating large fluctuations (positive or negative) in H produce large positive or large negative fluctuations in x of the same order of magnitude as those in H. It is also shown that when the large positive and negative fluctuations of H are of the same order of magnitude, then x has both large positive and negative fluctuations, and these follow the asymptotic growth of the respective fluctuations in H.
Our first main results about absolute fluctuations show that max 1≤j≤n |x(j)| ∼ max 1≤j≤n |H(j)|, as n → ∞, so in order to understand the growth in the partial maximum of x, it is necessary to determine the growth rate of max j≤n |H(j)|. However, it is more straightforward, especially in the case when H is an independent and identically distributed (iid) sequence, to try to find a deterministic sequence a(n) which is increasing at the same rate as H in absolute value terms. This is true because the Borel-Cantelli lemmas will only yield upper bounds for the growth in the maximum, while they can give upper and lower bounds in the iid case when the auxiliary sequence a is introduced. In order to be of greater use for stochastic systems, we therefore prove that, for general sequences if lim sup n→∞ |H(n)|/a(n) = ρ ∈ [0, ∞] then lim sup n→∞ |x(n)|/a(n) = ρ. This general result does not employ stochastic arguments.
The final results examine the boundedness of time averages of the same function ϕ of H and x and how they are related even though the sequences H (and therefore x) are tacitly assumed to be unbounded (it is trivially the case that time averages of any well-behaved function of H and x will be finite if both sequences are bounded).
In particular, we show the equivalence of these "ϕ-moments" of H and x in the case where ϕ is an increasing and convex function. This covers important examples such as the finiteness of time averages, variances, skewness, and kurtosis for example (by taking ϕ(x) = x p for p = 1, 2, 3, 4). However, for "thin tailed" distributions, such as Gaussian distributions, we can consider non-power convex functions. The parameterised family ϕ(x) = e ax 2 for a > 0 is useful in the Gaussian case, for instance.

Mathematical Preliminaries
2.1. Notation and assumptions on data. We now give the equation we study and impose hypotheses on the data. Suppose and that k = (k(n)) n≥0 is a sequence with We find it useful to define Let (H(n)) n≥1 be a real sequence and let (x(n)) n≥0 be another real sequence uniquely defined by (2.5) We introduce the notation We also define The times to date at which these sequences reach their running maximums are also of interest, are denoted by t x n , t H n ∈ 0, .., n, and defined by We are generally interested in the behaviour of the solution x of (2.5) when x becomes large (in absolute value terms), as when this happens, the solution is undergoing a large fluctuation of growth. We assume that f is nonlinear, and are therefore particularly interested in the behaviour of f (x) for large |x|. We assume that the impact of the past is of smaller that linear order for large x; the other extreme would be to consider when f (x)/x → ±∞ as x → ±∞, which we do not do here.
The assumption that f is sublinear in the sense of (2.2) achieves this. One of the effects of this assumption (2.2) is that the equation (2.5) will be quite stable with respect to moderate disturbances. This is attractive, because this is not always the case if f is linear or obeys f (x)/x → ∞ as x → ∞.

2.2.
Time indexing in the Volterra equation. Before listing and discussing the main results, we stop to comment on the time indexing used in (2.5). Many authors choose to write or even study the equation (especially in the case that f (x) = x, and impose a solvability condition in order to ensure the existence of a solution of (2.11)). We prefer to express the equation in the form (2.5), however, for a technical reason related to the situation when H is a stochastic process, and also from the perspective of viewing (2.5) as modelling an economic system in which agents can observe the state of the system x up to the current time n, but cannot know the future values of the system {x(j) : j ≥ n + 1} with certainty, owing to the randomness in H.
The appropriate probabilistic formulation of (2.5) in the case that H is a stochastic process is the following, and we will adopt this formulation. Suppose that (Ω, F , (F (n)) n≥0 , P) is an extended probability triple. We suppose that (F (n)) n≥0 is a filtration (an increasing sequence of σ-algebras) with F ⊃ F (n) for each n ≥ 0 and F (n + 1) ⊃ F (n) for n ≥ 0. We suppose that H(n) is F (n)-measurable for n ≥ 1; the process H is then said to be adapted to the filtration (F (n)) n≥0 , or adapted for short.
Remembering that F (n) represents the information available about the system at time n, and granted the assumption (2.1) that f is continuous and k is deterministic, then in equation (2.5), x(n) is F (n)-measurable for each n ≥ 1, so x is adapted. Therefore, the value of x(n+1) is not known with certainty at time n, but is at time n + 1, as soon as H(n + 1) has been observed. In the formulation (2.10), however, if we still suppose that H(n) is F (n)-measurable, then x(n + 1) is known with certainty at time n. A process with this property is called previsible or predictable, and typically we would not wish to assume a priori in a discrete-time economic model that a publicly visible state of the system (such as a stock price, interest rate, or important economic indicator) could be predicted with certainty one timestep ahead by agents possessing only publicly available information. Therefore, for such economic models (2.5) is preferable to (2.10).
The equation (2.11) shares with (2.5) advantageous adaptedness properties, provided (2.12) If (2.12) holds, then there exists an adapted process x satisfying (2.11). The process is unique if there is a unique solution to the nonlinear equation in (2.12) for each y ∈ R. This is certainly true for all |y| sufficiently large, under the sublinearity hypothesis (2.2). However, we slightly prefer the formulation (2.5) from a modelling perspective, as the summation term can represent the impact of agents on the system at time n + 1, based on actions they make using any subset of publicly available information up to time n. In (2.11), the value of the system at time n appears both in the summation term (which we view as including information causing the future value of the output x) and as "output" itself at time n. While this does not violate causality in the model, it does impose on the system the additional mathematical constraint (2.12) as well as its economic interpretation.
In the meta-model we describe, (2.12) amounts to agents instantaneously solving nonlinear equations which may involve the actions of other agents at the same instant. We can, and do, avoid such problems by studying (2.5) instead.
2.3. Motivation from economics. We do not have any particular economic model in mind in formulating this equation, but merely try to capture interesting dynamical effects which seem to us to arise in economics, although we mention three situations where equations of the type (2.5) may be germane. Our general question is: if we have a system which, although small, is relatively robust (in being able to handle moderate shocks), how does that system react to strong shocks or strong and persistent external forces? Do the shocks persist, or fade rapidly? How does the system adjust to persistent and possibly positive changes in the external environment? How does the memory that the system has of its own past effect the transmission of the external forces through the system over time? We are also interested in tracking quantities and time at which the solution reaches its maximum to date: such times and quantities are thought be investigators to be of psychological importance to agents. With these questions in mind, the structure of (2.5) becomes more apparent. The state of the (small) system at time n is x(n). The external force or shock at time n is H(n). Although the form of (2.5) does not preclude that H(n + 1) can be a function of x at past states, it is tacit in our formulation that H(n + 1) is independent of {x(j) : j ≤ n}. Therefore, while H influences x, x does not influence H. In this sense, we view x as modelling a "small" system: the external environment influences the system, but the influence of the system on the external environment is small (and modelled as being absent).
Another interpretation of H is that it models the effect of "news" or hard-tomodel external effects on the system. This is a common feature in autoregressive time series models, for instance: the system adjusts according to the previous values of the system, and is in addition, subjected to a stochastic shock which cannot be predicted with certainty based on the past states of the system.
The Volterra term has the following interpretation: as usual for Volterra equations, we take the view that all past terms have an effect on the system, but that terms in the distant past have a vanishing impact (so k ∈ ℓ 1 (N)). The sublinearity in f makes the system very robust to moderate shocks, as demonstrated by Theorem 1 below. This is true without making any assumption on the size of k: in contrast, in the linear case, restrictions on k would be necessary in order for solutions to remain bounded for all bounded H. Of course, if the state is an asset price or income, the system's smallness and sublinearity in f is a disadvantage: it is unable to grow unboundedly by its own means (or exhibit so-called endogeneous growth). We remark in passing that if one desires endogeneous growth in the unperturbed system, this can be achieved by considering the difference-summation equation If we still assume that k ∈ ℓ 1 (N), f is sublinear in the sense of (2.2), and f : (0, ∞) → (0, ∞) and k is non-negative, then all solutions of (2.13) with positive initial condition grow to infinity at a rate determined by f . A continuous analogue of (2.13) with these positivity assumptions is considered in [8]. Some existing economic models take the form of (2.5) or are closely related to it. In the classic dynamic linear multidimensional Leontief input-output model (see e.g, [23,24]), H is the final demand, and x the output, and the Volterra term is so-called intermediate demand. The sublinearity assumption means that the (onecommodity) economy exhibits diminishing marginal returns to scale. The presence of time lags signifies that production can take many time steps to enter the final demand. Early examples of nonlinear input-output models include [14,27].
We can think of the model in terms of an inefficient market for an asset, where new signals about the price arrive H(n + 1) which drive the price, but the agents use past information about the price to determine their demand, and this also has an impact on the price. The sublinearity in this instance suggests that the traders become conservative in their net demand, relative to the price level, when the market is far from equilibrium. Our results suggest that large shocks to the price transmit quickly to the system in that case, despite the fact that the traders may use a lot of information about the past of the system. Models of this type include [1,2,4,9].
Our model also takes inspiration from the important class of (linear) autoregressive models. The class of ARMA(p, q) models (see e.g., [13]), for instance, have the form where H is a stationary process which has non-trivial autocorrelation at q ∈ Z + time lags (i.e., Cov(H(n), H(n + q)) = 0 for all n ≥ 0), and trivial autocorrelations for time lags greater than q (i.e., Cov(H(n), H(n + k)) = 0 for all n ≥ 0 and k > q). In this case, the equation has bounded memory of the previous p values of the state.
However so-called AR(∞) models are also considered, in which the entire history of the process is important. One motivation to do this is to introduce so-called long range dependence or long memory into the system. Classic papers finding evidence for slow decay in correlations in tree-ring data series, wheat market prices, stock market and foreign exchange returns are Baillie [11] and Ding and Granger [15]). Mathematical models in economics based on AR(∞) processes have been developed. For instance, Kirman and Teyssière [21,22] develop a time series model which arises from a market composed trend following and value investors which possesses long memory characteristics in the differenced log returns of price processes associated with these models. Appleby and Krol [7] analyse the long memory properties of a linear stochastic Volterra equation in both continuous and discrete time, with conditions for both subexponential rates of decay and arbitrarily slow decay rates in the autocovariance function being characterised in terms of the decay of the kernel of the Volterra equation. A continuous-time infinite history financial market model is discussed in Anh et al. [1,2], which generalises the classic Black-Scholes model, and exhibits long memory properties. All these papers study equations closely related to the classic AR(∞) model: If one chooses to subsume the history of the process up to time n = 0 in the forcing term, and further assume (for example) that {x(n) : n ≤ 0} is bounded, then the sequenceH (n + 1) := H(n + 1) is well-defined and we have x(n + 1) =H(n + 1) + n j=0 k(n − j)x(j), n ≥ 0, which is in the form of (2.5) with f (x) = x. Furthermore, if the history of x is bounded,H(n)−H(n) is bounded, so the unboundedness properties of the adjusted perturbationH and the original perturbation H are the same.
We remark that stationarity in H in these linear models does not necessarily entail stationarity in x: in the case of the ARMA(p, q) model for example, it relies on the ℓ 1 -stability of the resolvent r(n + 1) = n j=n−p k(n − j)r(j), n ≥ 0; r(0) = 1; r(n) = 0 n < 0, which is equivalent to all the zeros of the polynomial equation k(l)z p−l lying in {z ∈ C : |z| < 1}. Although we have not proven it in this paper, we conjecture that stationarity in H in (2.5) implies asymptotic stationarity in x in (2.5). Such a result would be in line with other results we observe here, namely that an unboundedness property U in H is inherited by x.

Main Results
In this section we list and discuss the main results of the paper. Proofs are largely postponed to the end.

Bounded and unbounded solutions. Our first main result shows that if
H is a bounded sequence, then so is x, but that if H is unbounded, x must be also.

3.2.
Growth rates in the partial maximum. Theorem 1 shows that solutions of (2.5) are bounded if and only if H is bounded. We have already noted (for growth arising from dynamic input-output models, or for unbounded shocks that would result in a time series model if H were a stationary process) that for the applications we have mentioned, it is more natural to consider unbounded H. In this case lim n→∞ H * (n) = +∞ and therefore lim n→∞ x * (n) = +∞.
It is now a natural question to ask: if H * (n) → ∞ as n → ∞, how rapidly will x * (n) → ∞ as n → ∞? Our first result in this section shows that both maxima grow at the same rate. We also study the relationship between the times at which |x| and |H| reach their running maxima.
In advance of proving Theorem 2, we now provide an interpretation of its conclusions.
If we suppose that H fluctuates such that we see that part (i) implies that the order of magnitude of the large fluctuations in x is precisely that of the large fluctuations in H. The first limit in part (ii)(a) states that at the time to date (for large times) at which H reaches its maximum, x is of the same order. Moreover, the second limit says that if one considers the epoch {0, .., n}, the largest fluctuation of x is of the same order as the magnitude of x at the time of the largest fluctuation of H. In other words, a fluctuation of the order of the biggest fluctuation in x is "caused" at the time of the largest fluctuation in H, so, the largest fluctuations in H transmit rapidly into the largest fluctuations in x.
Turning to the first limit in part (b), we see that on the epoch {0, .., n}, if the largest fluctuation in x is recorded, the level of H at that time is of the order of the largest fluctuation in x. Furthermore, the level of H at that time is asymptotic to the largest fluctuation in H over the epoch {0, ..n}. This means that if the largest fluctuation to date in the process x is observed at a specific time, then this is caused by a large fluctuation in H at that time and this fluctuation in H is of the order of the largest fluctuation in H recorded to date. To summarise briefly, if we observe the largest fluctuation to date in x, it has essentially been caused by the largest fluctuation in H to date, which occurred at that time.

3.3.
Growth Rates. Theorem 2 shows that if H is unbounded, then so is x, and their absolute maxima grow at the same rate. However, what we do not know at this point is whether growth in H will produce growth in x, and whether fluctuations in H will produce fluctuations in x. In this section, we show that "regular" growth in H (in a sense that we make precise) gives rise to regular growth in x, and indeed that such regular growth in x is possible only if H grows regularly. It is clear that for every y in B a there is a bounded sequence Λ a y (which is unique up to asymptotic equality) such that We are interested in the case when Λ a H(n) does not tend to a limit as n → ∞: if the limit is trivial, then a does not describe the rate of growth of H very well and the situation is of less interest; if the limit is non-trivial, we are in the situation covered by Theorem 3.
and that x is the solution of (2.5). Let (a(n)) n≥1 be an increasing and positive sequence such that a(n) → ∞ as n → ∞. Then the following are equivalent (a) H ∈ B a , and Λ a H defined by (3.1) is asymptotically non-null; The interpretation of the implication (a) implies (b) of Theorem 4 is clear: if the external force grows at a rate a, modulo a non-trivial and non-constant bounded multiplicative factor Λ a H, then the solution grows at the same rate a, multiplied by the factor Λ a H. Therefore, regular growth in H (with fluctuations about a trend growth rate) are reflected in x, and the character of the fluctuations about the trend is the same for the output x. Conversely, if we observe growth modified by a multiplicative fluctuation in the output x, this must have been caused by the same pattern of growth in the forcing term H.
Example 5. Let a > 0 and suppose that H(n) = e an π(n), where π is N -periodic with max i=0,...,N −1 π(i) = π > 0 and min i=0,...,N −1 π(i) = π ∈ (0, π). Thus H exhibits exponential growth with a periodic component, and as such is a crude model for growth with periodic booms and recessions in the world economy. The small system, whose output is influenced by H, is modelled by x. In the above notation, we can take a(n) = e an and Λ a H = π. Then by Theorem 4 lim n→∞ x(n) e an − π(n) = 0, so we see that x inherits the main properties of the growth path of H: in economic terms, the booms and recessions in the outside system propagate rapidly into the small system.

Signed fluctuations and their magnitudes.
We have just seen that Theorem 2, while useful, does not distinguish between growth or fluctuations in solutions of (2.5). Theorem 3 demonstrates that regular growth in H gives rise to regular growth in x at the same rate as H. The question at hand now is to refine, in a similar manner, Theorem 2, in order to capture the large fluctuations in solutions of (2.5). It is reasonable to suppose that such fluctuations in x must result from large fluctuations in H, and in parallel with Theorem 3, it is also reasonable to try to connect the sizes of the large fluctuations in x to those in H.
We have used the term fluctuation loosely above, but now we want to try to capture it mathematically. We are assuming that H * (n) → ∞ as n → ∞, but in order to describe a fluctuation in H, we do not want to have H(n) → ∞ or H(n) → −∞ as n → ∞, or more generally, we do not want the limit of H to exist. Roughly speaking, we could have two types of fluctuation in H: the first type, which we emphasise here, is that H fluctuates without bound to plus and minus infinity. The second is that H has an infinite limsup but finite liminf (or negative infinite liminf and finite limsup).
Considering the first situation a little more, we should distinguish between the sizes of large positive and large negative fluctuations in H. To this end, we introduce the monotone sequences We see that H * + records the magnitude of the large positive fluctuations, while H * − records the magnitude of the large negative fluctuations. Clearly the overall maximum of these magnitudes is just H * , or H * (n) = max(H * + (n), H * − (n)). We expect that fluctuations in H will cause fluctuations in x, so we make make the corresponding definitions for x as well. These are We are now in a position to state and prove our main result, Theorem 6 below. It is useful to assume that there is λ ∈ [0, ∞] such that . (3.4) The existence of this limit helps us to decide whether the large negative or large positive fluctuations dominate.
If the large positive fluctuations in H dominate asymptotically the large negative fluctuations (in the sense that λ ∈ [0, 1) in (3.4)) then x experiences a large positive fluctuation of the same order as the large positive fluctuation in H, and this also captures growth rate of the partial maximum of |x|; in other words, if x experiences a large negative fluctuation, it will be dominated by the large positive fluctuation. This is the subject of part (i) in Theorem 6.
Symmetrically, if the large negative fluctuations in H dominate asymptotically the large positive fluctuations (in the sense that λ ∈ (1, ∞] in (3.4)), then x experiences a large negative fluctuation of the same order as the large negative fluctuation in H, and this also captures growth rate of the partial maximum of |x|; in other words, if x experiences a large positive fluctuation, it will be dominated by the large negative fluctuation. This is the subject of part (ii) in Theorem 6.
Finally, if the growth rates of the the large positive and large negative fluctuations in H are the same (in the sense that λ = 1 in (3.4)), then x experiences both large positive and large negative fluctuations, the growth rate of both are the same, and moreover equal to the growth rates of the fluctuations in H. This is the subject of part (iii) in Theorem 6.
We note an asymmetry here in parts (i) and (ii) between assumptions on H and conclusions concerning x. If λ ∈ (0, ∞), we have both H * + (n) and H * − (n) → ∞ as n → ∞. However, part (i) only yields x * We note that part (ii) does not allow us to conclude that lim inf n→∞ x(n) = −∞ under the condition that lim inf n→∞ H(n) = −∞. We now show that by strengthening hypothesis on f and H that it is possible to conclude more. However, we also show that lim inf n→∞ H(n) = −∞ does not necessarily imply that lim inf n→∞ x(n) = −∞.
We now make our additional assumptions on f : we assume that We invoke symmetry in f to see what the impact is when there is asymmetry in the growth of H * ± (n) as n → ∞. We deal only with the case in which the positive fluctuations dominate the negative ones: an analogous result can be obtained by the same means if the negative fluctuations are dominant (i.e., when λ = +∞).
Theorem 8. Suppose f obeys (2.1) and (2.2), k obeys (2.3) and that x is the solution of (2.5). Suppose further that there is φ such that f and φ obey (3.5) and (3.6). Suppose H * (n) → ∞ as n → ∞, and that H and λ obey (3.4), and that H * ± and x * ± are defined by (3.2) and (3.3). and If λ 2 = +∞, then (3.7) and (3.8) hold. (iii) If λ = 0 and λ 2 ∈ (0, ∞), then lim n→∞ x * + (n) = +∞ and If, in addition, λ 2 > ∞ j=0 |k(j)|, then (3.7) holds and (iv) If λ = 0 and λ 2 = 0, then lim n→∞ x * + (n) = +∞ and We see (roughly) that if the negative fluctuations are not too small relative to the positive fluctuations, then it is still the case that the size of the negative fluctuations of H determine the size of the negative fluctuations of x. However, if the negative fluctuations become too small relative to the positive fluctuations, it need not be the case that x * − (n) is of the same order of magnitude as H * − (n), as the following example shows. Indeed, it can even happen that the negative fluctuations of x are bounded, even though those of H are unbounded. We also see that the estimates in part (ii) and (iii) are better than the crude estimate x * − (n) = o(H * + (n)) as n → ∞ that is supplied by part (ii) of Theorem 7; notice also how part (ii) of Theorem 8 can be thought of as a limiting case of part (iii) when λ 2 → ∞. Example 9. Let µ + > µ − where µ + > 0 and µ − ≥ 0. Let α ∈ (0, 1). Let n ≥ 0. Define y(n) = (n + 1) µ+ for n an even integer and y(n) = −(n + 1) µ− for n an odd integer. Let (k(n)) n≥0 be a non-negative summable sequence and suppose f (x) = sgn(x)|x| α for x ∈ R. Define for n ≥ 0 H(n + 1) := y(n + 1) − n j=0 k(n − j)f (y(j)).
Then x(n) = y(n) for n ≥ 0 is a solution of (2.5). f obeys all the hypothesis of Theorem 8.
Now some remarks are in order. First, we see that part (iii) of Theorem 8 is not necessarily sharp, because we know that x * − (n) ∼ (1 + n) µ− as n → ∞, while the upper bound furnished by the theorem grows strictly faster. Second, the rather precise estimates on the asymptotic behaviour of H * ± (n) as n → ∞ that we possess (i.e., H * − (n) ∼ ∞ j=0 k(2j) · n µ+α as n → ∞ and H * + (n) ∼ n µ+ ) are not sufficient to be able to predict the exact rate x * − (n) ∼ (1 + n) µ− as n → ∞: the parameter µ − does not appear in k, f nor in the leading order asymptotic behaviour in H * ± . Third, since µ − = 0 is an admissible parameter value, it is the case that H * − (n) → ∞ as n → ∞ does not necessarily imply that x * − (n) → ∞ as n → ∞. Fourth, it is the case that · n µ+α → 0 as n → ∞, so in general, we see that it is not necessarily the case that x * − (n) is of the same order of magnitude as H * − (n) as n → ∞. Fifth, even though the upper bound in part (iii) is not sharp, the Theorem does seem to identify, in this example at least, that there is a change in the asymptotic behaviour of x * − when µ − moves from being greater than αµ + to being less than αµ + .
3.5. Bounds on the fluctuations in terms of an auxiliary sequence. In applications, especially when H is a stochastic process, it may be possible to prove by independent methods that there are increasing deterministic sequences which give precise bounds on the fluctuations of H. In the important case where H is a sequence of independent and identically distributed random variables, it is possible to prove, by means of the Borel-Cantelli lemmas, that there exist sequences a + and a − , which have very similar (but non-identical) asymptotic behaviour such that lim sup n→∞ |H(n)| a + (n) = 0, lim sup n→∞ |H(n)| a − (n) = +∞, a.s.
An example of a case where this holds is when each H(n) has the power law density g(x) ∼ c α |x| −α and α > 1. In this case one can take for instance a + (n) = n 1/(α−1)+ǫ and a − (n) = n 1/(α−1)−ǫ for any ǫ > 0 sufficiently small. In some cases one can even show that a single sequence determines the asymptotic behaviour, so it is possible to show that lim sup n→∞ |H(n)| a(n) = 1, a.s.
An example for which this is true is a zero mean Gaussian white noise sequence, in which a(n) = σ √ 2 log n, where σ 2 is the variance of the white noise process. We give details of the calculations in the next subsection.
These examples show that the auxiliary sequence a may exactly estimate the fluctuations of H, or systematically over-or underestimate it. Therefore, it makes sense to formulate a result in which lim sup n→∞ |H(n)|/a(n) can be zero, finite but non-zero, or infinite, and attempt therefrom to determine the asymptotic behaviour of |x|. The following result shows, once again, the close coupling of the asymptotic behaviour of H and x.
Theorem 10. Suppose f obeys (2.1) and (2.2), and k obeys (2.3). Let x be the solution x of (2.5). Suppose that (a(n)) n≥1 is an increasing sequence such that with a(n) → ∞ as n → ∞. Then the following are equivalent: In the case when ρ ∈ (0, ∞), large fluctuations of both H and x are described by the increasing sequence ρa. If however, a sequence a does not exist (or cannot readily be found) for which this holds, a very easy corollary of Theorem 10 gives upper and lower bounds on the fluctuations of x in terms of those of H.
Theorem 11. Suppose f obeys (2.1) and (2.2), and k obeys (2.3). Suppose also that there exist increasing sequences (a − (n)) n≥1 and (a + (n)) n≥1 with a ± (n) → ∞ as n → ∞ such that Then the solution x of (2.5) obeys Proof. Take a(n) = a + (n) and note that ρ = 0 in Theorem 10. Applying Theorem 10 gives the first limit in the conclusion of the result. The second limit is obtained by taking a(n) = a − (n), in which case ρ = +∞, and Theorem 10 can be applied again.
It is equally reasonable to formulate results for the size of the positive and negative fluctuations in terms of auxiliary sequences. This result parallels Theorem 7. Rather than being comprehensive at the expense of repetition, we have considered the case when the positive fluctuations dominate the negative ones. Other results in this direction can be readily formulated and proven as desired using the same methods of proof: this result can be thought of as being representative. Applications of this result to Gaussian and heavy-tailed distributions are given in the next subsection.
Theorem 12. Suppose f obeys (2.1) and (2.2), k obeys (2.3) and that x is the solution of (2.5). Suppose also that there exist increasing sequences (a − (n)) n≥1 and (a + (n)) n≥1 with a ± (n) → ∞ as n → ∞ such that  3.6. Applications to stochastic processes. Let H(n) be a sequence of independent and identically distributed random variables each with distribution function F . For simplicity suppose that the distribution is continuous and supported on all of R (so that the random variables are unbounded and can take arbitrarily large positive and negative values). What follows is all well-known, but we record our conclusions to assist stating applying our results, which we do momentarily.
Since each H has distribution function F we have Since the events {|H(n)| > Ka(n)} are independent, we have that from the Borel-Cantelli Lemma that Therefore, for all K > 0 such that S(a, K) < +∞ we have that there is an a.s. event Ω + K such that lim sup n→∞ |H(n)| a(n) ≤ K, on Ω + K .
On the other hand, for all K > 0 such that S(a, K) = +∞ we have that there is an a.s. event Ω − K such that lim sup It can be seen therefore that it may be possible for a well-chosen sequence a and number K sequence Ka(n) for which S(a, K) is either finite or infinite. This will then generate upper and lower bounds on the growth of |H(n)|, and thereby, by then applying Theorem 10, allow conclusions about the growth of the fluctuations of x to be deduced.
In the first example, we are able to find a sequence a for which Λ a |H| ∈ (0, ∞).
Therefore, there are a.s. events Ω ± ǫ such that Next we consider the case of a symmetric heavy tailed distribution with power law decay in the tails. In this case, we find sequences a + and a − such that a − = o(a + ) and

Example 14.
Suppose that H(n) are independently and identically distributed random variables such that there is α > 0 and finite c 1 , c 2 > 0 for which Suppose that a + and a − are sequences such that Then we see that S(K, a + ) < +∞ for all K > 0 while S(K, a − ) = +∞ for all K > 0. Therefore we have for all K > 0 lim sup n→∞ |H(n)| a + (n) ≤ K, on Ω + K .
On the other hand, for all K > 0 we have that there is an a.s. event Ω − K such that Consider the event Ω − = ∩ K∈Z + Ω + K . Then Ω − is an almost sure event and we have To show we can get a + and a − close, notice that for every ǫ > 0 sufficiently small we can take a ± (n) to be a ±ǫ (n) = n 1/α±ǫ .
It is now standard to get limits independent of the small parameter ǫ, and we show now how this can be done. First, from the existence of the sequences a ±ǫ we may conclude from that there are a.s. events Ω − ǫ and Ω + ǫ such that Finally, take This is an a.s. event, and we have lim sup n→∞ log |x(n)| log n = 1 α , on Ω * . |x(j)| p < +∞, so that the p-th moment of x is finite if and only if the p-the moment of H is. The equivalence of the finiteness of the ϕ-moments also holds in the more general case that ϕ is a regularly varying function at infinity.
In order to make our discussion precise, we recall the definition of convexity of a real function, and a discrete variant of an important inequality relating to convex functions, namely Jensen's inequality.
Lemma 1 (Jensen's Inequality). If 0 ≤ a 1 , a 2 , ...a n are such that n i=1 a i = 1 and if ϕ is a convex function, then We state next our main result: its proof is in the last section of the paper.  However, when applying Jensen's inequality to estimate the sums, it is necessary to impose a slightly stronger summability hypothesis on H in order to get the finiteness of the "ϕ"-moment of x. In the case when ϕ(x) = x p (or more generally when ϕ is a convex and regularly varying function at infinity (see e.g. [12])) we can forego this slight restriction, and show that the existence of the ϕ-moments of H and x are equivalent. This result is of particular interest if H is a stationary stochastic process, for it shows that the only way in which x will have a finite p-th moment is if H does also. This also enables us to make predictions about so-called moment explosion: if, for some p, Example 19. In Example 13, we have that the density of the normal is given by Take ϕ(x) = e ax 2 for a > 0. Then ϕ ′ (x) = 2axϕ(x) and ϕ ′′ (x) = 2axϕ ′ (x) + 2aϕ(x) > 0. Hence ϕ is increasing and convex. Moreover for any η > 0 we have The integral is finite if a(1 + η) < 1/(2σ 2 ) and infinite if a(1 + η) ≥ 1/(2σ 2 ). Thus for a < 1/(2σ 2 ) we can choose η > 0 sufficiently small such that a(1 + η) < 1/(2σ 2 ), and so by the strong law Our analysis is not sufficiently refined to conclude what the situation is if a = 1/(2σ 2 ).
3.9. Further work. Scrutiny of the proofs that follow shows that the analysis presented here for (2.5) works with trivial modification for the continuous-time integral equation It is still assumed that f obeys (2.1) and (2.2). We assume now that k is in L 1 (0, ∞). For continuous solutions, we ask that k and H are continuous, and to guarantee uniqueness of a continuous solution of (3.11), we can assume that f is locally Lipschitz continuous. Then direct analogues of all the main results apply. We have not focussed on nonconvolution equations, but it is easy to see that the proofs of all results (with the possible exception of Theorem 16) go through with cosmetic changes for the Volterra summation equation x(n + 1) = H(n + 1) + n j=0 k(n, j)f (x(j)), n ≥ 0; x(0) = ξ, (3.12) where k : Z + × Z + → R is such that sup n≥0 n j=0 |k(n, j)| < +∞, and f once again obeys (2.1) and (2.2). The corresponding nonconvolution integral equation can also be analysed successfully, once sup t≥0 t 0 |k(t, s)| ds < +∞. We have remarked already that many interesting results of a similar character to those presented here can be obtained for the linear Volterra summation equation and the corresponding linear integral equation There are two chief differences in the nature of the results: first, the manner in which the kernel k fades is important in the linear case, in stark contrast to the situation here. The reader will have seen throughout how small a role k plays in the nature of the solution x, whose properties are inherited rather directly from H: there is no "long memory" or hysteresis effect present in (2.5), which can contrast markedly with the situation in (3.13) in the case when k fades slowly. Second, the type of "nice" unbounded space we consider in the linear case tends to be slightly more restrictive than that we consider here, mainly because the Volterra term in (3.13) can be of the same order as x(n + 1) when the latter is large. On the other hand, the corresponding Volterra term in (2.5) is of smaller order when x(n + 1) is large: this makes the analysis considerably easier, and therefore weaker hypotheses on the data suffice to make good progress in the sublinear case. Analysis of systems in R p requires more thought. From the perspective of applications, it is of evident interest to study not only max 0≤j≤n x(j) (where · is a norm on R p ), but also the running maximum of the i-th component of the system max 0≤j≤n |x i (j)| (i = 1, . . . , p). Such an analysis likely requires a more delicate analysis of the maxima than the confines of this paper allow.

Proof of Theorem 1
The proof is elementary, but several useful estimates are developed which we employ in later proofs. Therefore, we give more intermediate details than are strictly necessary for present purposes. The proof takes its inspiration in part from [5,Lemma 5.3].

Proof of Theorem 2
Suppose that ǫ ∈ (0, 1) is so small that ǫ|k| 1 < 1, then there is F (ǫ) > 0 such that f obeys (4.1). As in the proof of Theorem 1, we have the estimate (4.4). Therefore, as H * (n) → ∞ as n → ∞, we get This gives the required upper estimate in part (i). To get the lower estimate (i.e., to obtain a lower bound on lim inf n→+∞ x * (n)/H * (n)), we start by recalling the estimate (4.8) which rearranges to give Since H * (n) → ∞ as n → ∞ by hypothesis, taking the limit inferior as n → ∞ yields and now letting ǫ → 0 + yields Combining this with (5.1) gives part (i). We next prove part (iii) of the result. By definition of x and S in (4.5), if t x n ≥ 1, we have x(t x n ) = H(t x n ) + S(t x n − 1). Thus by (4.6), |x(t x n )| ≤ |H(t x n )| + F (ǫ) |k| 1 + ǫ |k| 1 x * (t x n − 1). Now, by the monotonicity of x * and the definition of t x n we have since t x n ≤ n and |x(t x n )| = max 0≤j≤n |x(j)|. Therefore, If H * (n) → ∞ as n → ∞, we have max 0≤j≤n |x(j)| → ∞ as n → ∞ and so |x(t x n )| → ∞ as n → ∞. The above inequality then implies that |H(t x n )| → ∞ as n → ∞. Rearranging and taking limits as before yields We now get a lower estimate for the limit. First, rearranging (2.5) at the time t x n and taking the triangle inequality and the estimate (4.1) gives Hence |H(t x n )| ≤ |x(t x n )| (1 + ǫ |k| 1 ) + F (ǫ) |k| 1 . Rearranging this inequality and taking limits gives Letting ǫ → 0 + and combining with (5.2) yields |x(t x n )|/|H(t x n )| → 1 as n → ∞, completing the proof of the first limit in part (iii).
We now prove the second limit in part (iii), namely lim n→+∞ |H(t x n )| / H(t H n ) = 1. Notice that part (i) of this Theorem gives x * (n)/H * (n) → 1 as n → ∞. By definition, x * (n) = |x(t x n )| and H * (n) = H(t H n ) . It has just been shown that |x(t x n )|/|H(t x n )| → 1 as n → ∞. Therefore, as n → ∞, This proves the second limit in part (iii). Finally, we prove part (ii). By assumption, H * (n) → ∞ as n → ∞. Since and thus by (5.3), we have lim inf n→+∞ |x(t H n )|/|H(t H n )| ≥ 1−ǫ |k| 1 . Letting ǫ → 0 + yields lim inf n→+∞ |x(t H n )|/|H(t H n )| ≥ 1 and therefore combining this with (5.4) yields the desired, first limit lim n→+∞ x(t H n ) / H(t H n ) = 1 in part (ii). We now prove the second limit in part (ii). We have from part (i) that x * (n)/H * (n) → 1 as n → ∞, and by the first part of (ii) we get lim n→∞ H(t H n ) / x(t H n ) = 1. Therefore, as claimed.
6. Proof of Theorems 3 and 4 6.1. A preparatory lemma. Before we prove our main result, we need the following preparatory lemma.
Moreover as H is asymptotic toH we have x(n)/H(n) → 1 as n → ∞, so x is asymptotic to the increasing sequenceH. This proves statement (b). Conversely, suppose x(n) → ∞ and x is asymptotic to an increasing sequence (x(n)) n≥1 . Then by Lemma 2, it follows that x * (n) ∼x(n) and so, as n → ∞, we also have x * (n) ∼ x(n). Therefore, since x * (n) → ∞ as n → ∞, we have H * (n) → ∞ as n → ∞. Hence x * (n)/H * (n) → 1 as n → ∞. Since S obeys (4.6), we have |S(n)|/x * (n) → 0 as n → ∞. Hence, since x * (n) ≤ x * (n + 1) we have But since x(n) = H(n) + S(n − 1), we have H(n)/x(n) → 1 as n → ∞, which proves part of the desired conclusion. Recall that x is asymptotic to the increasing sequencex, so H is asymptotic to the increasing sequencex which itself tends to infinity. Therefore H(n) → ∞ as n → ∞. Hence H enjoys all the properties listed in statement (a).
To prove the forward implication in (b), note that |H(n)| ≤ max 1≤j≤n |H(j)|. To prove the reverse implication, suppose otherwise, i.e., that Then there is N > 0 such that for all n ≥ N , |H(n)| ≤ (λ + 1)a(n). Hence for n ≥ N we may use the monotonicity of a to get Since a(n) → ∞ as n → ∞, we have +∞ = lim sup n→∞ H * (n)/a(n) ≤ λ+1 < +∞, a contradiction. To prove part (c), note that because |H(n)| ≤ H * (n), the second statement in (c) proves the first. Suppose the first statement is true. Then for every ǫ > 0 there is an N (ǫ) > 0 such that |H(n)| < ǫa(n) for all n ≥ N (ǫ). Thus for n ≥ N (ǫ) we may use the monotonicity of a to get Since a(n) → ∞ as n → ∞, we have lim sup n→∞ H * (n)/a(n) ≤ ǫ, and letting ǫ → 0 completes the proof.