Probability and statistics

Actuarial statistics uses probability theory and statistical inference to quantify insurance claims, premiums, and the uncertainty of future losses. Aggregate claims model describes portfolios, reinsurance redistributes losses, and the Bühlmann model combines experience with population information.

A credibility estimate combines a risk's individual experience with population information. In the basic equal-exposure form, the credibility factor $Z$ weights the observed sample mean, while $1-Z$ weights a prior or collective expected value. The Bühlmann credibility premium is the best affine estimate under mean squared error; an exact Bayesian posterior mean can also have this form, as in the natural conjugate credibility identity.

Given a latent risk parameter $\Theta$, observations are independent and identically distributed random variables with conditional expectation $m(\Theta )$ and conditional variance $v(\Theta )$. The Bühlmann credibility premium estimates $m(\Theta )$ by an affine function of past observations under mean squared error. Its population parameters are $m=\mathbb{E}m(\Theta )$, expected process variance $v=\mathbb{E}v(\Theta )$, and variance of hypothetical means $a=\mathrm{Var}(m(\Theta ))$.

Conditional on a common latent variable $\Theta$, annual averages $X_j$ are independent, have common conditional expectation $\mu (\Theta )$, and have conditional variance ${\sigma }^2(\Theta )/m_j$ for known positive exposures $m_j$. Its structural parameters are $m_0=\mathbb{E}\mu (\Theta )$, expected process variance $v=\mathbb{E}{\sigma }^2(\Theta )$ and variance of hypothetical means $a=\mathrm{Var}(\mu (\Theta ))$. It generalizes the equal-exposure Bühlmann model.

For total exposure $W={\sum }_j{m}_j$ and weighted average ${\overline{X}}_w={\sum }_j{m}_j{X}_j/W$, the best affine mean squared error estimate of the conditional mean is $Z{\overline{X}}_w+(1-Z)m_0$, where $Z=Wa/(Wa+v)$. If the coefficients sum to $B$, their error is $a(1-B{)}^2+v{\sum }_j{b}_j^2/m_j$. Cauchy-Schwarz inequality minimizes the latter term at $b_j=B{m}_j/W$, and a one-dimensional quadratic minimization then gives $B=Z$.

With a Poisson-gamma conjugacy with unequal exposures model, $m_0=v=\alpha /\beta$ and $a=\alpha /{\beta }^2$, so the Bühlmann–Straub credibility factor is $W/(W+\beta )$. The credibility estimate simplifies to $(\alpha +{\sum }_j{Y}_j)/(\beta +W)$, exactly the posterior mean and hence the Bayes estimator under squared error loss. The agreement is exact because that posterior mean is already affine in the exposure-weighted data.

The experience weight is $Z=Wa/(Wa+v)$, where $W$ is total observed exposure, $a$ the variance of hypothetical means, and $v$ the expected process variance. With positive $a$, it is $W/(W+v/a)$. More exposure raises the weight; larger process noise lowers it. If $a=0$, the unknown conditional mean is constant and one can set $Z=0$.

The Bühlmann credibility premium is the best affine mean squared error estimate of a risk’s conditional claim expected value. In the Bühlmann model it is $Z\overline{X}+(1-Z)m$, where $Z=na/(na+v)$. The linear least-squares projection equations use $\mathrm{Var}(X_j)=a+v$ and $\mathrm{Cov}(X_i,X_j)=a$ for distinct years.

With fixed structural parameters and $K=v/a$, the Bühlmann credibility estimate has form $({\sum }_{i=1}^n{x}_i+K{m}_0)/(n+K)$. The next observation updates it by the displayed rule. Consequently the estimate decreases exactly when the new observation is less than the previous credibility estimate, with strict equality handled separately.

The credibility factor in the Bühlmann model is the weight $Z=na/(na+v)$ placed on the sample mean after $n$ observations. The remaining weight is placed on the population expected value. More experience or larger variance of hypothetical means increases $Z$; larger expected process variance decreases it.

The variance of hypothetical means in the Bühlmann model is the between-risk variance of the conditional claim expected value. It creates the shared covariance $a$ between different years of the same risk and determines how much individual experience should influence the premium.

The expected process variance in the Bühlmann model averages within-risk conditional variance over the population of risks. It measures variation that repeated observations can average out and contributes $v/n$ to the variance of the sample mean.

The classical risk model has surplus $U_t=u+ct-{\sum }_{j=1}^{N_t}{X}_j$, where $N_t$ is a Poisson process of rate $\lambda$, independent claim sizes are positive, and premiums flow in at constant rate $c$. The relative safety loading is $\rho =c/(\lambda \mathbb{E}{X}_1)-1$. Ruin is the first time surplus becomes negative.

The probability that the insurance surplus in a classical risk model becomes negative at some finite time, starting from capital $u\ge 0$. Ultimate survival has probability $1-\psi (u)$. A finite-horizon ruin probability instead restricts the crossing time to a specified time interval.

For positive claims with finite variance and $c\le \lambda \mu$, the classical risk model has ruin probability one from any finite capital. At claim times, surplus increments are independent copies of $cT-X$ with mean $c/\lambda -\mu$. Negative mean sends their partial sums to minus infinity by the strong law of large numbers. At zero mean, the increments have finite nonzero variance. For every fixed $K$, the central limit theorem gives limiting probability $1/2$ of a partial sum below $-K$. The probability of unboundedness below is therefore at least $1/2$; as a tail event it has probability zero or one by the Kolmogorov zero-one law, and hence one.

Independent positively loaded portfolios with zero initial capitals have separate joint survival probability ${\prod }_i(1-{\lambda }_i{\mu }_i/c_i)$. Pool their claims and premium incomes to obtain zero-capital merged survival $1-({\sum }_i{\lambda }_i{\mu }_i)/({\sum }_i{c}_i)$. This is the premium-weighted average of the individual survival probabilities and is at least their product. The event of aggregate solvency permits transfers of surplus between the original portfolios.

The ultimate survival probability satisfies $\phi (u)=\phi (0)+(\lambda /c){\int }_0^u\phi (u-x)(1-F(x))dx$. Condition on the first arrival of the Poisson process, differentiate the resulting exponentially weighted integral, and integrate the convolution derivative equation from zero. Tonelli theorem converts the claim-density convolution to the tail convolution. The kernel mass is $\lambda \mu /c<1$, so this is a defective renewal equation.

In the classical risk model with premium rate $c>0$ and claim Poisson process rate $\lambda$, the first-claim decomposition for survival probability implies the displayed equation for ultimate survival probability. Change the first-claim integral to an integral over available capital and differentiate its lower limit. The bounded survival probability function convolved with the integrable claim probability density function is continuous, so this also establishes the needed differentiability. In particular ${\phi }^{\prime }(0)=(\lambda /c)\phi (0)$.

For a mixture distribution of exponential distributions, introduce one convolution state $A_j(u)={\int }_0^u\phi (z)e^{-{\beta }_j(u-z)}dz$ for each rate ${\beta }_j$. It satisfies $A_j^{\prime }=\phi -{\beta }_j{A}_j$. Together with the survival integro-differential equation these form a constant-coefficient first-order system; applying the differential operators $D+{\beta }_j$ eliminates the convolution states. For equal mixing weights at rates one and one half, with $r=\lambda /c$, the result is ${\phi }^{\prime \prime \prime }+(3/2-r){\phi }^{\prime \prime }+(1/2-3r/4){\phi }^{\prime }=0$. The original integral equation supplies initial conditions lost in elimination.

In a classical risk model, before the first claim at time $t$, available capital is $u+ct$. A claim of size $x\le u+ct$ leaves future survival probability $\phi (u+ct-x)$ by the Markov property. Integrating over the independent first-arrival exponential distribution and claim density gives $\phi (u)={\int }_0^{\infty }\lambda e^{-\lambda t}{\int }_0^{u+ct}\phi (u+ct-x)f(x)dxdt$.

In a classical risk model with positive relative safety loading, zero initial capital has ultimate survival probability $1-\lambda \mu /c$, where $\lambda$ is the claim-arrival rate, $\mu$ the claim expected value and $c$ the premium rate. It depends on the claim law through its mean alone. It is not the event of never receiving a claim: premium accumulates between arrivals.

The adjustment coefficient is a positive root of $\lambda (M_X(r)-1)=cr$ in the classical risk model. When the moment-generating function is finite at $R$, the process $\exp (R[{\sum }_{j=1}^{N_t}{X}_j-ct])$ is a continuous-time martingale. It yields the Lundberg inequality and, under the relevant tilted integrability, the Cramér–Lundberg ruin asymptotic.

If each claim includes an independent expense $A$, replace its payment law by the convolution of independent random variables $X+A$. Its moment-generating function is $M_X(r)M_A(r)$ and its expected value is $\mathbb{E}X+\mathbb{E}A$. Keeping the relative safety loading $\theta$ fixed therefore changes the premium rate as well. The new coefficient solves $M_X(R)M_A(R)-1=(1+\theta )(\mathbb{E}X+\mathbb{E}A)R$, within the common finite-transform domain.

For positive claims with finite nonzero expected value $\mu$, the function $(M_X(r)-1)/r$ is continuous and strictly increasing on the positive finite-transform domain, starting at $\mu$. If $M_X$ diverges at a finite upper endpoint, or is finite for all positive arguments, the secant slope tends to infinity. In the latter case use $M_X(r)\ge \mathbb{P}(X\ge a)e^{ar}$ for some $a>0$ of positive tail probability. Every target $(1+\theta )\mu$ with positive relative safety loading therefore has exactly one positive root.

For a classical risk model with adjustment coefficient $R>0$, the ultimate ruin probability from capital $u\ge 0$ satisfies $\psi (u)\le e^{-Ru}$. Stop the exponential continuous-time martingale at ruin or a finite horizon, bound its value on the ruin event, and then increase the horizon.

In the classical risk model with positive relative safety loading $\rho$ and adjustment coefficient $R$, tilting the ruin defective renewal equation gives a proper renewal equation. The key renewal theorem yields $e^{Ru}\psi (u)\to \rho /[R{\int }_0^{\infty }x{e}^{Rx}{f}_I(x)dx]$. The constant is positive if the denominator is finite and zero if it is infinite; the claim-size density provides the nonarithmetic hypothesis.

When the adjustment coefficient lies inside the finite domain of the claim moment-generating function, the tilted kernel has finite mean $(\lambda M^{\prime }(R)-c)/(cR)$. The forcing integral is $(c-\lambda \mu )/(cR)$. Their ratio is the constant $C$ in $\psi (u)\sim C{e}^{-Ru}$, by the key renewal theorem. An extra exponential moment controls the forcing tails and proves direct Riemann integrability.

If the expected claim outflow per unit time is $\lambda \mu$, the relative safety loading is $\rho =c/(\lambda \mu )-1$. Positive loading means expected premium income exceeds expected claim outflow. This is the net profit condition in the classical risk model.

Reinsurance transfers part of an insurer’s claim liability to another insurer. If aggregate claims are $S$, a retained payout $g(S)$ with $0\le g(S)\le S$ leaves the reinsurer with $S-g(S)$. Quota share reinsurance retains a fixed fraction, whereas aggregate stop loss reinsurance retains losses only up to a fixed aggregate threshold.

Per-claim reinsurance under which the insurer retains each claim up to a level $M$ and the reinsurer pays its positive part $(x-M{)}_{+}$. The annual retained amount is a sum of capped claims, so it is generally different from the cap of the annual sum in aggregate stop loss reinsurance.

For positive claim sizes with finite second moment in a compound Poisson distribution aggregate of parameter $\lambda$, the two excess of loss reinsurance payouts are $\min (X,M)$ and $(X-M{)}_{+}$. The derivative of their total aggregate variance is $2\lambda (M\overline{F}(M)-\mathbb{E}[(X-M{)}_{+}])$. Thus the displayed condition characterizes stationarity. When the tail probability is positive it says that the retention equals the mean residual life. Global minimality requires an additional sign or comparison argument.

For exponential distribution claims of mean $\mu$ and compound Poisson distribution count parameter $\lambda$, the total party variance under excess of loss reinsurance is $g(M)=2\lambda {\mu }^2(1-(M/\mu )e^{-M/\mu })$. Its derivative is $2\lambda e^{-M/\mu }(M-\mu )$, so $M=\mu$ is the unique global minimum, with value $2\lambda {\mu }^2(1-e^{-1})$.

For a positive claim and $r>0$, the tail integral formula for moments gives the displayed expression. In particular the first moment is ${\int }_0^M\overline{F}(x)dx$ and the raw second moment is $2{\int }_0^{M}x\overline{F}(x)dx$. If the claim law has a density, capping adds an atom of a measure at $M$ with mass $\mathbb{P}(X\ge M)$.

With aggregate retention $M\ge 0$, the direct insurer pays $\min (S,M)$ and the reinsurer pays $(S-M{)}_{+}$, where $+$ denotes the positive part. The threshold applies to the whole annual loss; applying a threshold to individual claims is a different contract.

If $S$ has exponential distribution with expected value $\mu$, the tail integral formula for moments gives $\mathbb{E}\min (S,M)=\mu (1-e^{-m})$ and $\mathrm{Var}(\min (S,M))={\mu }^2(1-2m{e}^{-m}-e^{-2m})$, where $m=M/\mu$. At matching retained expected value, the excess variance under quota share reinsurance is $2{\mu }^2{e}^{-m}(m-1+e^{-m})\ge 0$.

Among retained payouts $g(S)$ with $0\le g(x)\le x$ and the same expected value as $\min (S,M)$, aggregate stop loss reinsurance minimizes the variance. Pointwise $(g(x)-M{)}^2\ge (\min (x,M)-M{)}^2$, and subtracting the identical squared distance of their common expected value from $M$ proves the claim. Equality requires equal payouts almost surely.

Quota share reinsurance retains a fixed proportion $\alpha \in [0,1]$ of each claim. The annual retained aggregate is $\alpha S$, whose expected value and variance are $\alpha \mathbb{E}S$ and ${\alpha }^2\mathrm{Var}(S)$.

Suppose $S=S_I+S_R$ has finite positive variance $v$, and a matching quota share reinsurance contract exists with fraction ${\alpha }^{\ast }=\sqrt{\mathrm{Var}(S_I)/v}\in [0,1]$. The Cauchy-Schwarz inequality bounds $\mathrm{Cov}(S,S_I)\le {\alpha }^{\ast }v$. Expanding $\mathrm{Var}(S-S_I)$ then proves that the matching quota share minimizes the other party's variance, and hence the sum of party variances. Claimwise retentions $0\le h(x)\le x$ in a compound Poisson distribution automatically satisfy the required variance range, because $\mathrm{Var}(S_I)=\lambda \mathbb{E}[h(X{)}^2]\le \lambda \mathbb{E}[X^2]$.

For any aggregate with positive finite variance $v$, quota share reinsurance with retained fraction $\alpha$ has total party variance $v({\alpha }^2+(1-\alpha {)}^2)=v/2+2v(\alpha -1/2{)}^2$. Equal sharing uniquely minimizes this objective. The two party totals remain dependent; adding their individual variances is a different objective from the fixed variance of their sum.

An aggregate claims model separates the number $N$ of claims from their sizes $X_j$, with total $S={\sum }_{j=1}^N{X}_j$ and the empty sum equal to zero. Under independent and identically distributed random variables for claim sizes, independent of $N$, conditional expectation gives tractable moments and transforms.

The probability distribution of the nonnegative integer number of claims over a specified accounting period. Its probability generating function determines the count probabilities. A Poisson distribution gives equal count expected value and variance; a Poisson mixture allows heterogeneity in the underlying risk intensity.

The nonnegative integer laws whose probabilities satisfy this recurrence for all $n\ge 1$. Their probability generating function solves $(1-az)G^{\prime }(z)=(a+b)G(z)$. This is the $(a,b,0)$ count class used in the Panjer aggregate algorithm; the count recurrence itself should be distinguished from that aggregate recursion. It includes the Poisson distribution, binomial distribution, and negative binomial distribution with suitable parameters.

Independent claim Poisson processes of rates ${\lambda }_i$ merge into a Poisson process of rate $\lambda ={\sum }_i{\lambda }_i$, by the Superposition theorem for Poisson point processes. The merged claim law is a mixture distribution of the individual claim laws with weights ${\lambda }_i/\lambda$. Equivalently, multiplication of the individual compound-Poisson transforms produces $\exp (\lambda [{\sum }_i({\lambda }_i/\lambda )M_i(r)-1])$.

For independent claim sizes with common expected value $m$ and variance $v$, independent of the nonnegative integer count $N$, the aggregate satisfies $\mathbb{E}S=m\mathbb{E}N$ and $\mathrm{Var}(S)=v\mathbb{E}N+m^2\mathrm{Var}(N)$. Its moment-generating function is $G_N(M_X(t))$ wherever finite. These identities follow from the law of total expectation and law of total variance.

For a geometric distribution count on the positive integers, independent of independent and identically distributed random variables $X_i$, the moment-generating function of $S={\sum }_{i=1}^N{X}_i$ is the displayed expression. Condition on the count and sum a geometric series. For real $t$, the finite domain requires $M_X(t)<\infty$ and $(1-p)M_X(t)<1$. Positive summands always permit $t\le 0$, even when no positive exponential moment exists; this gives a Laplace transform of a nonnegative random variable. The aggregate has no mass at zero when all summands are strictly positive.

Let the summands have gamma distribution with shape two and rate $\beta$, and let the independent positive-support geometric distribution count have parameter $p$. Put $a=\beta (1-\sqrt{1-p})$ and $b=\beta (1+\sqrt{1-p})$. The aggregate moment-generating function factors as $ab/((a-t)(b-t))$, so it is the sum of two independent exponential distributions of rates $a,b$. Its probability density function is $ab(e^{-as}-e^{-bs})/(b-a)$ for $s>0$, nonnegative and integrating to one. This contrasts with the geometric sum of exponential variables, which is itself exponential.

The law of a sum of independent identically distributed claims with an independent Poisson distribution count of parameter $\lambda$. If the claim moment-generating function is $M_X$, the aggregate transform is $\exp (\lambda (M_X(r)-1))$ wherever finite. It has an atom at zero when the claims are positive. This is the fixed-time law of a Compound Poisson process.

Applying a measurable per-claim retention $g$ to independent claim sizes preserves the compound Poisson distribution form. The transformed severity is the pushforward measure of the original severity law. With count parameter $\lambda$, the retained aggregate has expected value $\lambda \mathbb{E}g(X)$ and variance $\lambda \mathbb{E}[g(X{)}^2]$. Zero retained marks may be removed by Poisson thinning.

For positive claim sizes, a random sum of independent claims is zero exactly when its count is zero. Its law is a mixture distribution of a zero atom of mass $p=\mathbb{P}(N=0)$ and, with weight $1-p$, a random sum whose count has the zero-truncated claim-count distribution. An independent Bernoulli random variable multiplying that positive component gives the same law. This is distinct from arbitrarily inserting additional zeros into an otherwise unchanged count law.

If $\Lambda$ has gamma distribution with shape $2$ and rate $p/q$, $N\mid \Lambda$ has Poisson distribution with intensity $\Lambda$, and the independent claim sizes have exponential distribution of mean $\mu$, where $p+q=1$, the aggregate law is $p^2{\delta }_0+2pq\mathrm{Exp}(p/\mu )+q^2\mathrm{Gamma}(2,\text{rate }p/\mu )$. This follows by expanding its moment-generating function as $[p+qp/(p-\mu t){]}^2$.

A quantile-quantile plot compares ordered observations with quantiles of a reference distribution, or with quantiles of another sample. Approximate linearity supports agreement up to location and scale. A regression normal Q-Q plot compares ordered standardized regression residuals with standard normal distribution quantiles.

Change-point detection estimates locations where a sequence's distribution or mean changes. A piecewise-constant mean has a sparse vector of adjacent differences; one-dimensional total variation denoising exploits this sparsity without fixing the number of changes in advance.

For an Inhomogeneous Poisson process whose known positive intensity changes from ${\lambda }_1$ to ${\lambda }_2$ at an unknown time $\theta \in (0,T)$, the ordered arrival-time likelihood function isUnder a uniform prior, each interval between successive arrivals has an exponential posterior density with rate in the exponent ${\lambda }_2-{\lambda }_1$. Its integrated weights allow exact interval selection followed by a truncated exponential draw; when the two rates coincide, the Bayesian posterior is uniform and the data contain no change-time information.

Measurement error is the difference between a recorded variable and the quantity it is intended to measure. It can attenuate associations, create misclassification, and bias nonlinear estimators even when the error has mean zero.

Regression toward the mean is the tendency for units selected because of an extreme noisy measurement to have less extreme subsequent measurements even without an intervention. It follows because the selection captures both a persistent component and an unusually extreme error component.

A sensitivity analysis examines how conclusions change when assumptions, tuning choices, unidentified parameters, or plausible data perturbations are varied.

Specify distributions of measurements separately by observation pattern and vary assumptions about unseen values that the observed data do not identify. For example, shift the imputed binary-response log odds for nonresponders away from its missing at random prediction and refit the substantive analysis. This investigates missing not at random departures rather than testing them from observed data.

The power spectrum describes how the variance of a random field or stationary process is distributed over spatial wavenumber or temporal frequency.

A random field is statistically homogeneous when all of its joint distributions are invariant under a common spatial translation of their arguments. Its momentum-space correlation functions consequently contain a Dirac delta function imposing total-momentum conservation.

A random field is statistically isotropic when all of its joint distributions are invariant under a common spatial rotation of their arguments. A statistically isotropic scalar power spectrum depends only on the magnitude of its wavevector.

A time series is a sequence of observations indexed by time. Time-series analysis models temporal dependence for description, inference, and forecasting.

The best mean-square linear predictor is orthogonal projection onto the span of the available centered observations, plus the known mean. Orthogonal innovations give a convenient basis of that span. Prediction at a future horizon must use only actually observed variables; it need not be a one-step predictor.

Starting observations at a finite time, subtract the best linear predictor based on the available earlier observations. The resulting residual is orthogonal to their linear span. The unit-triangular relation between observations and residuals makes the residuals an orthogonal basis of the same finite observation space. These differ from infinite-past innovations until the initialization effect disappears.

In a centered MA(1) process, $X_{T+2}$ is uncorrelated with every observation through time $T$. Its best linear predictor from that information is therefore zero, with error variance ${\sigma }^2(1+{\theta }^2)$. One-step innovation formulas using $X_{T+1}$ cannot be substituted when that observation is unavailable.

Starting at time one, the orthogonal residuals satisfy $U_t=X_t-{\lambda }_t{U}_{t-1}$, with ${\lambda }_t={\gamma }_1/{\phi }_{t-1}$ and ${\phi }_t={\gamma }_0-{\gamma }_1{\lambda }_t$. Earlier innovation directions have zero covariance with the new observation. Explicitly, with $S_m={\sum }_{j=0}^m{\theta }^{2j}$, ${\phi }_t={\sigma }^2{S}_t/S_{t-1}$ and ${\lambda }_t=\theta S_{t-2}/S_{t-1}$ for $t\ge 2$. These formulas follow by induction from $S_t=(1+{\theta }^2)S_{t-1}-{\theta }^2{S}_{t-2}$, with $S_0=1$ and $S_1=1+{\theta }^2$. This includes $\theta =\pm 1$, where ${\phi }_t={\sigma }^2(t+1)/t$.

For an MA(1) process, the recursion ${\lambda }_t=\theta /(1+{\theta }^2-\theta {\lambda }_{t-1})$ has candidate limits $\theta$ and $1/\theta$. The limit in $[-1,1]$ is $\theta$ when $|\theta |\le 1$ and $1/\theta$ otherwise. The limiting innovation variance is ${\sigma }^2\max (1,{\theta }^2)$, expressing the same covariance law through an invertible reciprocal representation.

The best mean-square linear predictor from a semi-infinite past is the orthogonal projection onto the closed linear span of past observations in the random-variable Hilbert space. For a causal invertible ARMA representation, past observations and past innovations generate the same closed span. The next linear innovation is orthogonal to it and is therefore the prediction error.

A state-space model separates a latent transition equation $S_t=F{S}_{t-1}+Z_t$ from an observation equation $Y_t=H{S}_t+W_t$. A complete specification includes the initial-state law and its relation to the noise sequences, not just the two equations. Linear second-order models specify covariance structure; Gaussian versions additionally prescribe joint Gaussian laws.

For $|\varphi |<1$, initialize the state by $S_0={\sum }_{j\ge 0}{\varphi }^j{Z}_{-j}$, with variance ${\sigma }_z^2/(1-{\varphi }^2)$. It is orthogonal to future state noise and to observation noise orthogonal to all state noise. Under a Gaussian specification, take the initial state Gaussian with this variance and independent of future noises. Starting at zero instead gives a transient model.

An anticausal noise representation uses future rather than present and past innovations. For a two-sided AR(1) equation with $|\varphi |>1$, the stationary solution is $X_t=-{\sum }_{j\ge 1}{\varphi }^{-j}{Z}_{t+j}$. Thus stationary existence alone does not require causality.

A linear process is an L2-convergent white-noise filter $X_t=\mu +{\sum }_{j\in \mathbb{Z}}{\psi }_j{\epsilon }_{t-j}$, with square-summable coefficients. It is weakly stationary, and its autocovariance is ${\sigma }_{\epsilon }^2{\sum }_j{\psi }_j{\psi }_{j+h}$. A causal representation restricts the coefficients to $j\ge 0$; it is then an infinite moving-average representation.

When the transfer function is $(1+\theta z)/((1-rz)(1-sz))$ with distinct $r,s$ of modulus less than one, partial fractions give coefficients ${\psi }_j=a{r}^j+d{s}^j$, where $a+d=1$ and $-as-dr=\theta$. White-noise orthogonality gives covariance at lag $h\ge 0$ as ${\sigma }^2[a^2{r}^h/(1-r^2)+d^2{s}^h/(1-s^2)+ad(r^h+s^h)/(1-rs)]$. This combines stable recursion with a closed geometric-sum covariance.

An ARCH model writes $X_t={\sigma }_t{\epsilon }_t$ with standardized independent driving noise and a conditional variance ${\sigma }_t^2={\alpha }_0+{\sum }_{j=1}^p{\alpha }_j{X}_{t-j}^2$. Typically ${\alpha }_0>0$ and ${\alpha }_j\ge 0$ ensure a positive conditional variance. It allows a changing conditional scale even when $X$ is a stationary process.

With symmetric driving noise whose sign is independent of its magnitude, changing the sign of one driving variable changes the corresponding $X_t$ but leaves future conditional scales unchanged. Consequently $\mathrm{Cov}(X_t,f(X_{t+h}))=0$ for $h>0$ and square-integrable $f(X_{t+h})$. This conclusion uses symmetry, beyond the martingale difference sequence property.

For ${\alpha }_0>0$ and $0<{\alpha }_2<1$, the model $X_t=\sqrt{{\alpha }_0+{\alpha }_2{X}_{t-2}^2}{\epsilon }_t$ splits into independent even-time and odd-time chains in its stationary causal solution. With standard normal noise, $\mathbb{E}{X}_t^2={\alpha }_0/(1-{\alpha }_2)$, and its fourth moment is finite exactly when $3{\alpha }_2^2<1$.

The stationary squared process has the positive series $X_t^2={\alpha }_0{\sum }_{k\ge 0}{\alpha }_2^k{\prod }_{j=0}^k{\epsilon }_{t-2j}^2$. Even and odd observations therefore use disjoint families of independent random variables. This explains zero lag-one covariance of the squares despite dependence at lag two.

Volatility clustering means that large absolute observations tend to be followed by further large absolute observations, and small ones by small ones. In an ARCH process, persistence is in the conditional variance rather than necessarily in the signed observations.

Seasonality is systematic repetition at a calendar period. It can appear in a deterministic mean, in a periodic variance, or in dependence across seasons. A seasonal difference operator removes a fixed periodic mean but seasonal differencing does not remove periodic variance. A periodic-looking path alone does not prove a nonstationary process.

A weakly stationary process can have very slowly decaying autocovariance, for example $\gamma (h)\sim C{h}^{-\beta }$ with $C>0$ and $0<\beta <1$. Its sample mean can require a normalization different from $\sqrt{T}$. Long memory is dependence over long lags, rather than a changing marginal distribution.

In a second-order sense a weakly stationary process has short memory when its autocovariance is absolutely summable. Then its spectral density of a stationary process is continuous and its sample-mean variance has the usual finite long-run variance of a stationary process. A central limit theorem still needs additional conditions.

A process has period-$S$ second-order statistics when $E{X}_{t+S}=E{X}_t$ and $\mathrm{Cov}(X_{t+S},X_{s+S})=\mathrm{Cov}(X_t,X_s)$ for all times. Its variance can vary by season, so it need not be a weakly stationary process. A periodic covariance period need not be the fundamental period.

The difference $\Delta X_t=X_t-X_{t-1}$ can remove a unit-root stochastic trend. A seasonal difference operator instead subtracts an observation a full seasonal period earlier. Neither operation automatically removes a changing variance.

For an integer period $S$, ${\Delta }_S{X}_t=X_t-X_{t-S}$ uses the backshift operator as $1-B^S$. It annihilates a deterministic period-$S$ mean. If $X_t=m_t+{\epsilon }_t$ with $m_{t+S}=m_t$ and strong white noise, the result is the stationary moving average ${\epsilon }_t-{\epsilon }_{t-S}$.