## 1. Introduction

[2] Even if one accepts that orbital forcing is the “pacemaker of the ice ages” [*Hays et al.*, 1976], over the range ≈30 yrs to ≈30 kyrs, there is no doubt that most of the variance in paleotemperature records is associated with the continuous spectral “background” [*Lovejoy and Schertzer*, 1986; *Wunsch*, 2003] (for a recent spectrum see Figure S1 in Text S1 in the auxiliary material). This strongly suggests that other internal and/or external mechanisms are needed to explain the multidecadal, multicentennial and multimillenial variability. The discussion of these issues has been strongly tinted by the development of GCM's and their response to various external climate forcings. However, if the amplification factors are large – as they must be – then it will be hard to distinguish nominally external forcing paradigms from purely internal ones.

[3] The usual approach to evaluating climate forcings is via the climate sensitivity (*λ*) defined as the equilibrium change in a quantity, (here the temperature) per unit of radiative forcing. Sensitivities (*λ*) are commonly estimated with the help of (deterministic) numerical models; the usual example being the doubling of CO_{2}. The change in conditions (compositional in this example) simultaneously leads to changes in the typical mean global temperature (Δ*T*) and to the earth's radiative equilibrium from which the radiative forcing (Δ*R*_{F}) is determined by:

This definition of climate sensitivity is convenient for numerical experiments with strong anthropogenic forcings. In this case, the response is relatively regular (smooth) so that the estimate *λ* = Δ*R*_{F}(Δ*t*)/Δ*T*(Δ*t*) is well defined, insensitive to Δ*t.* However, for natural forcings, it has several shortcomings. First, GCM outputs fluctuate over a wide range of Δ*t*so that – except for very small time scales comparable to the model integration time steps - fluctuations Δ*T*(Δ*t*) (and presumably) Δ*R*_{F} (Δ*t*) typically have nontrivial scaling behaviours Δ*T*(Δ*t*) ≈ and Δ*R*_{F}(Δ*t*) ≈ implying *λ*(Δ*t*) ≈ with *H*_{λ} = *H*_{T} − *H*_{R} generally noninteger. Second, the usual definition of climate sensitivity is only valid if there is a causal link: the fluctuations Δ*T* and Δ*R*_{F} must have the same underlying cause such as a change in solar output. Strictly speaking, it therefore cannot be used empirically since in the real world there is only a single realization of climate. From the climate record, we can only measure correlations, not causality. In addition to the causality assumption, empirical estimates of *λ* must rely on model outputs in order to estimate Δ*R*_{F} [e.g., *Harvey*, 1988; *Claquin et al.*, 2003; *Chylek and Lohmann*, 2008; *Ganopolski and Schneider von Deimling*, 2008].

[4] As a consequence of these difficulties, *λ*has not been systematically explored as a function scale and it mostly known from models - not empirically. We therefore give a new stochastic definition of climate sensitivity which allows us to empirically estimate it for any physical forcing process whose consequent radiative forcing can be determined.