Celebratio Mathematica

Mur­ray Rosen­blatt (1926–2019) was one of the top re­search­ers in prob­ab­il­ity and stat­ist­ics in the second half of the twen­ti­eth cen­tury. In 1964, he took a po­s­i­tion in the Math­em­at­ics De­part­ment at the Uni­versity of Cali­for­nia at San Diego (UC­SD), and he re­mained there for the rest of his ca­reer. He re­tired in 1994, and af­ter­words con­tin­ued to re­main act­ive in math­em­at­ics. I re­ceived my Ph.D. de­gree in Math­em­at­ics at UC­SD with Mur­ray in 1978. Oth­er art­icles in this volume of Cel­eb­ra­tio Math­em­at­ica will give a de­tailed ac­count of his ca­reer and the scope of his re­search. This write-up here will give a just few nar­row, spe­cif­ic per­son­al memor­ies.

Fairly early in his ca­reer, Mur­ray pub­lished sev­er­al dif­fer­ent pa­pers, on quite dif­fer­ent top­ics in prob­ab­il­ity and stat­ist­ics, that each in­spired an ex­tens­ive de­vel­op­ment of re­search by oth­er math­em­aticians. One of those pa­pers, one of his most fam­ous pa­pers over­all, was his art­icle “A cent­ral lim­it the­or­em and a strong mix­ing con­di­tion”, pub­lished in 1956 in the Pro­ceed­ings of the Na­tion­al Academy of Sci­ences [1]. In that pa­per, he gave a cent­ral lim­it the­or­em (CLT) for some se­quences of ran­dom vari­ables that sat­is­fy a cer­tain strong mix­ing con­di­tion, later known simply as the strong mix­ing con­di­tion, or as $\alpha$-mix­ing, the name that will be used here be­low. As Richard Olshen com­men­ted in the late 1970s, part of the mo­tiv­a­tion for that pa­per was the in­tent to ex­tend the meth­ods of stat­ist­ic­al in­fer­ence in time series ana­lys­is to ran­dom pro­cesses that do not sat­is­fy the usu­al “struc­tur­al” as­sump­tions such as in Markov pro­cesses, Gaus­si­an pro­cesses, and the clas­sic­al time series mod­els. Over the next sev­er­al years after that 1956 pa­per, nu­mer­ous oth­er sim­il­ar but stronger mix­ing con­di­tions were for­mu­lated by oth­er re­search­ers — in par­tic­u­lar, $\phi$-mix­ing, ab­so­lute reg­u­lar­ity ($\beta$-mix­ing), $\rho$-mix­ing, and $\psi$-mix­ing. Un­der those con­di­tions as well as un­der $\alpha$-mix­ing, and un­der vari­ations on those con­di­tions, there has since then been a vig­or­ous de­vel­op­ment, by many dif­fer­ent re­search­ers, of cent­ral lim­it the­ory and oth­er types of res­ults; that area of re­search re­mains act­ive to this day.

As a gradu­ate stu­dent at UC­SD in the 1970s, I be­came in­ter­ested in that area of prob­ab­il­ity the­ory. Strong mix­ing con­di­tions (plur­al), in­volving all five mix­ing con­di­tions men­tioned above and vari­ations on them, be­came the area of my Ph.D. thes­is as well as the area of al­most all of my re­search since then.

Al­though his re­search in­volved many dif­fer­ent areas of prob­ab­il­ity and stat­ist­ics, Mur­ray re­turned to strong mix­ing con­di­tions (plur­al) from time to time. He stud­ied in some de­tail such con­di­tions for Markov pro­cesses; in par­tic­u­lar, that sub­ject was the fo­cus of the fi­nal chapter of his book Markov pro­cesses: struc­ture and asymp­tot­ic be­ha­vi­or [3]. That chapter had sev­er­al in­ter­est­ing nug­gets, in­clud­ing what were ap­par­ently the first known ex­amples of strictly sta­tion­ary Markov se­quences (with real state space) that are $\rho$-mix­ing (and hence also $\alpha$-mix­ing) but for which (for all pos­it­ive in­tegers $n$) the $n$-step trans­ition dis­tri­bu­tions are al­most surely totally sin­gu­lar with re­spect to the (in­vari­ant) mar­gin­al dis­tri­bu­tion. In an­oth­er vein, much of Mur­ray’s book Sta­tion­ary se­quences and ran­dom fields [5] was de­voted to the study of es­tim­at­ors of spec­tral dens­ity, and oth­er re­lated es­tim­at­ors, for strictly sta­tion­ary ran­dom fields that sat­is­fy a ver­sion of $\alpha$-mix­ing.

An­oth­er early pa­per of Mur­ray’s that later in­spired much re­search by oth­er math­em­aticians was his art­icle “In­de­pend­ence and de­pend­ence”, in the Pro­ceed­ings of the Fourth Berke­ley Sym­posi­um on Math­em­at­ic­al Stat­ist­ics and Prob­ab­il­ity (1961) [2]. In that pa­per, he con­sidered among oth­er things a par­tic­u­lar sta­tion­ary Gaus­si­an se­quence $(Y_k,k\in \mathbb{Z})$ which has a stand­ard nor­mal mar­gin­al dis­tri­bu­tion and whose auto­co­v­ari­ances are pos­it­ive and de­cay to 0 at a cer­tain slow, non­sum­mable rate, and he ex­amined in sub­stan­tial de­tail the strictly sta­tion­ary ran­dom se­quence $(X_k,k\in \mathbb{Z})$ defined by the “in­stant­an­eous func­tion” $X_k=Y_k^2-1$ for $k\in \mathbb{Z}$. That par­tic­u­lar se­quence $(X_k)$ is centered with fi­nite second mo­ments; and for that se­quence, Mur­ray iden­ti­fied sev­er­al in­ter­est­ing prop­er­ties, in­clud­ing among oth­ers the fol­low­ing: (i) the vari­ance (second mo­ment) of the $n$-th par­tial sum blows up as $n\to \mathrm{\infty }$, and for each of those par­tial sums, there is a nice bound on the fourth mo­ment in re­la­tion to the second mo­ment; (ii) however, the par­tial sums, suit­ably nor­mal­ized, con­verge to a cer­tain nonde­gen­er­ate non­nor­mal law (which later be­came known as the Rosen­blatt dis­tri­bu­tion); and hence (iii) as an in­dir­ect con­sequence of the CLT in his 1956 pa­per dis­cussed above, that se­quence $(X_k)$ (and hence also the un­der­ly­ing Gaus­si­an se­quence $(Y_k)$) fails to sat­is­fy $\alpha$-mix­ing. That ex­ample il­lus­trated (in­dir­ectly) the power­ful role that the $\alpha$-mix­ing con­di­tion can play in cent­ral lim­it the­or­ems for de­pend­ent ran­dom vari­ables.

With re­gard to my own re­search, the 1961 pa­per of Mur­ray dis­cussed just above had a lim­ited dir­ect con­nec­tion, but in a cer­tain sense a quite sig­ni­fic­ant in­dir­ect con­nec­tion, as fol­lows: In the words of Richard Dav­is, that 1961 pa­per il­lus­trated the abid­ing in­terest that Mur­ray had in (counter-)ex­amples, which he found use­ful for il­lu­min­at­ing the lim­its of state­ments of cer­tain res­ults and, more gen­er­ally, dis­cov­er­ing in­sights in­to the “big­ger pic­ture of what is go­ing on”. Dur­ing my time as a gradu­ate stu­dent at UC­SD, some of the ques­tions that Mur­ray posed to me for pos­sible re­search were ques­tions as to wheth­er ex­amples could be con­struc­ted of strictly sta­tion­ary se­quences that sat­is­fied cer­tain strong mix­ing con­di­tions and be­haved in cer­tain ways — for ex­ample, in­volving “mix­ing rates”, or sat­is­fy­ing cer­tain “struc­tur­al” prop­er­ties to­geth­er with one strong mix­ing con­di­tion but not an­oth­er, or fail­ing to sat­is­fy a cent­ral lim­it the­or­em des­pite hav­ing cer­tain seem­ingly “nice” mo­ment and strong mix­ing prop­er­ties. My Ph.D. thes­is has the un­usu­al prop­erty that all of its five main res­ults as­ser­ted the ex­ist­ence of (counter-)ex­amples — of strictly sta­tion­ary se­quences with such com­bin­a­tions of prop­er­ties in­volving strong mix­ing con­di­tions. The con­struc­tion of fur­ther such (counter-)ex­amples in­volving strong mix­ing con­di­tions has since then been an on­go­ing ma­jor strand of my re­search.

The year 2003 brought a pe­cu­li­ar ex­per­i­ence in the cat­egory of “find­ing hid­den gems”, when I took a close look at Mur­ray’s pa­per “Uni­form er­godi­city and strong mix­ing”, pub­lished three dec­ades earli­er in 1972 in the Zeits­chrift für Wahr­schein­lich­keit­s­the­or­ie und ver­wandte Ge­bi­ete [4]. That pa­per con­tained some in­geni­ous ar­gu­ments that seemed to come from some spe­cial in­sight that Mur­ray had in­to the ef­fect­ive use of ba­sic op­er­at­or-the­or­et­ic tech­niques in the study of strong mix­ing con­di­tions for Markov pro­cesses. Of­fi­cially, the main res­ult proved in that pa­per in­volved, for strictly sta­tion­ary Markov se­quences, a con­nec­tion between the “uni­form er­godi­city” and $\alpha$-mix­ing con­di­tions. However, much oth­er in­form­a­tion was con­tained there as well. In par­tic­u­lar, among the vari­ous clev­er ar­gu­ments in that pa­per, the fol­low­ing fas­cin­at­ing state­ment was im­pli­citly proved without be­ing high­lighted with the la­bel “The­or­em” or “Co­rol­lary”: If a giv­en strictly sta­tion­ary Markov se­quence (with, say, real state space) is er­god­ic and aperi­od­ic, and its de­pend­ence coef­fi­cients $\alpha (1),\alpha (2),\dots$ as­so­ci­ated with the $\alpha$-mix­ing con­di­tion sat­is­fy for some $n\ge 1$ the in­equal­ity $\alpha (n)<1/4$ (the max­im­um pos­sible value for those par­tic­u­lar de­pend­ence coef­fi­cients), then that Markov se­quence is $\alpha$-mix­ing ($\alpha (n)\to 0$ as $n\to \mathrm{\infty }$). That res­ult, along with oth­er res­ults proved in that 1972 pa­per, are spelled out in a care­fully struc­tured, gen­er­ously de­tailed form in parts of Chapters 21 and 24 in volume 2 of my three-volume book series, In­tro­duc­tion to strong mix­ing con­di­tions [e1].

Out­side of math­em­at­ics, Mur­ray’s oth­er in­terests in­cluded art, mu­sic, and hik­ing. On a vis­it of mine to UC­SD a couple of dec­ades ago, he and I took sev­er­al hikes to­geth­er. On one of those hikes, in a canyon at the Tor­rey Pines State Nat­ur­al Re­serve Park in La Jolla, he poin­ted out some spe­cial things about the flora there.

For us gradu­ate stu­dents work­ing with Mur­ray, his wife Ady (Adylin, 1926–2009) was al­ways part of the pic­ture as well. Mur­ray and Ady gave us gen­er­ous sup­port and en­cour­age­ment dur­ing gradu­ate school and throughout the years af­ter­wards. They are sadly missed by all who knew them.

Richard Brad­ley (1950–) grew up in Ithaca, NY and Col­or­ado Springs, CO, and then went to, among oth­er places, MIT (B.S. in Math­em­at­ics, 1972), UC­SD (Ph.D. in Math­em­at­ics, 1978), and In­di­ana Uni­versity Bloom­ing­ton (Math­em­at­ics De­part­ment, 1980–present, re­tired 2015).